Hyperbolic Exterior Differential Systems and their Conservation Laws by ♭Robert Bryant, ^{♮}Phillip Griffiths, ^{♯}Lucas Hsu

♭ Department of Mathematics, Duke University, Durham, NC 27708 ♮ Institute for Advanced Study, Princeton, NJ 08540 ♯ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540

This research was supported in part by NSF Grant DMS 9205222 (Bryant), an NSERC Postdoctoral Fellowship (Hsu), and the Institute for Advanced Study (Griffiths and Hsu).

IAS Preprint: 1/25/94.

TABLE OF CONTENTS

Introduction

1. Hyperbolic Exterior Differential Systems 1.1 Basic definitions and some examples 1.2 The characteristic variety and the initial value problem 1.3 Prolongation of hyperbolic systems and their structure equations 1.4 Integration by the method of Darboux 1.5 Structure equations for hyperbolic systems of class s = 0 1.6 Hyperbolic Euler-Lagrange systems of class s = 0

2. Conservation Laws for Hyperbolic Systems of Class s = 0 2.1 General form of the conservation laws for hyperbolic systems of class s = 0 2.2 Determination of hyperbolic systems having the maximum number of classical conservation laws:

first steps 2.3 The two classes of hyperbolic systems having the maximum number of classical conservation laws 2.4 Moment conditions

3. Symplectic Hyperbolic Exterior Differential Systems 3.1 Definition and structure equations; Euler-Lagrange systems 3.2 The three classes of symplectic hyperbolic systems having an infinite number of classical conser-

vation laws 3.3 Euler-Lagrange representations of hyperbolic systems

References Introduction

This paper falls under the general subject of the geometric theory of differential equations. The theory, founded in the last century by Lie and Darboux and extensively developed by Goursat, Cartan, and others, seeks to understand differential equations through the study of their invariants under suitable groups of coordinate transformations (such as contact, point or gauge transformations). The goals of the theory are, first, to understand interesting special equations through explicit solutions or algorithms for solutions, or other properties such as conservation laws or estimates; and, second, to study the geometry of differential equations as a subject of interest in its own right (like Riemannian or CR geometry).

Much of the classical theory is centered on the attempt to describe the general solution of a given system of partial differential equations in some reasonably explicit way. The method of Darboux (a generalization of the method of characteristics which is so familiar in the study of hyperbolic systems) is perhaps the most successful of these techniques developed in the classical theory. However, the geometric methods developed during this period have significance even when no explicit general solution can be found. The application of these methods is in its infancy as regards modern issues in differential equations, such as inference of properties of solutions (especially global ones) rather than the explicit construction of solutions.

The variety of phenomena studied in the theory of partial differential equations is, of course, very great. To get to the deeper aspects of the subject, it is necessary to specialize to some extent. Thus, for example, while hyperbolic and elliptic systems share some very basic features, the sorts of interesting problems that one poses for these two classes of equations are very different. Naturally, this happens in geometry as well, with Euclidean and Lorentzian geometries bearing superficial resemblances, but having quite different deeper behaviors. It would be unrealistic to expect that the geometric method applied to the study of partial differential equations would not exhibit the same sort of divisions. For this reason, we have opted in this paper to specialize the class of equations to be studied.

In fact, we shall introduce the concept of a hyperbolic exterior differential system, a proper generalization of the classical case of second order hyperbolic PDE in the plane, and shall begin the study of the conservation laws and other geometric properties of such systems. The conservation laws provide an interesting intrinsic invariant of the hyperbolic system and are of use in understanding its solutions. We expect the class of hyperbolic exterior differential systems to provide a good case study within the general program.

Hyperbolic exterior differential systems are divided into the sets H_{s} of systems of class s where s may be any non-negative integer. The classes s = 1 and s = 3 include hyperbolic Monge-Ampere systems and second order hyperbolic equations for one unknown function z(x, y), respectively. The class s = 2 includes the case of a hyperbolic pair of first order equations for unknowns u(x, y) and v(x, y). The various classes are inter-related by the constructions of prolongation and (more subtly) integrable extensions.^{1}

A hyperbolic system of class s = 0 is given by a transverse pair of decomposable 2-forms Ω_{1}, Ω_{2} on a 4-manifold M—thus we have

\{ \begin{matrix}Ω_{1}∧Ω_{1} = 0 = Ω_{2}∧Ω_{2} \\ Ω_{1}∧Ω_{2} ≠ 0. \end{matrix}

This simple structure turns out to have a very rich geometry and appears in several guises in the course of this paper. Moreover, the application of the geometric method leads to some very special and interesting PDE systems, such as (1), (2), and (3) below.

In general, a hyperbolic exterior differential system (M, I) of class s is given by a differential ideal I on a manifold M of dimension s + 4 where I is generated algebraically by a rank s Pfaffian system I and a transverse pair of 2-forms that are decomposable modulo I. The k^{th} prolongation of (M, I) ∈ H_{s} is a hyperbolic system (M^{(k)}, I^{(k)}) ∈ H_{s+2k}. In order to understand the most interesting systems, which are those of classes s = 0, 1, 2 and 3, it seems to be advantageous to consider the whole set of hyperbolic exterior differential systems and its inter-relationships.

1A system of partial differential equations canonically gives rise to an exterior differential system. Differentiation of the PDE system then corresponds to prolongation of the exterior differential system. An integrable extension roughly corresponds to adjoining the primitive of a conservation law as a new variable—in the setting of exterior differential systems this may be done in a canonical way.

The most important objects associated to a hyperbolic system are the characteristic systems Ξ_{1}, Ξ_{2} and their prolongations Ξ_{1}^{(k)}, Ξ_{2}^{(k)}. Each of Ξ_{1}^{(k)} and Ξ_{2}^{(k)} is a Pfaffian system of rank s + 2 + k on the (s + 4 + 2k)-dimensional manifold M^{(k)} and their geometry—especially their integrable subsystems—is the dominant geometric feature of the hyperbolic system. Classically the characteristic systems arose in attempting to find explicit solutions of hyperbolic PDEs of Monge-Ampere type. Later they were used by Riemann to produce explicit integral formulas for certain special hyperbolic systems. The synthesizing concept of Darboux integrability was introduced by Darboux in 1870. A hyperbolic system is Darboux integrable at level k if there are rank 2 integrable systems

Δ_{1} ⊂ Ξ_{1}^{(k)}, Δ_{2} ⊂ Ξ_{2}^{(k)}

which are transverse to the ideal I^{(k)}. The solution to the Cauchy problem for such systems may be reduced to ODEs. Many interesting equations are Darboux integrable and first integrals may be found explicitly, leading to explicit forms for the general solution of the equation (well-known examples are the Liouville equation and the Weierstrass formulas for minimal surfaces).

An integral surface of a hyperbolic system (M, I) is an immersed surface S ⊂ M such that all the forms in I pull-back to be zero on S. The characteristic systems induce on any integral surface a pair of foliations by curves. The initial value problem for such a system is the problem of how to extend a given integral curve of I to an integral surface of I. We show that if the integral curve is ‘non-characteristic’ in the appropriate sense, then local solutions to the initial value problem always exist. More interestingly, the geometry of the characteristic curves allows us to intrinsically define a condition (which we call characteristic completeness) that is equivalent to the existence of global solutions to the initial value problem. The exterior differential system associated to many non-linear PDEs, such as the famous Fermi-Pasta-Ulam equation

z_{yy} − (k(z_{x}))^{2} z_{xx} = 0

where k : ℝ → ℝ is a diffeomorphism, are shown to admit unique global integral surfaces even though singularities necessarily develop for solutions to the PDE.

In this paper we are especially interested in the space C of conservation laws for a hyperbolic exterior differential system. From the general theory [BG_{1}] we know that (i) conservation laws have a normal form derived from the symbol of the exterior differential system, (ii) C is naturally given as the kernel of a linear differential operator (thus, we eliminate the trivial conservation laws), and (iii) C is filtered by subspaces C_{k} where Φ ∈ C_{k} means that Φ, assumed in normal form, is defined on M^{(k)}.^{2} One of the principal objectives of this paper is the study of conservation laws of hyperbolic systems of class s = 0.

Our main general results on conservation laws are:

(i) C_{2k} = C_{2k−1}, thus new conservation laws can be added only at odd levels;

(ii) Φ ∈ C_{2k+1} has a highest order part B = (B_{1}, B_{2}) with the properties that B uniquely determines Φ modulo C_{2k−1} and that for a pair Φ, Φ̄ ∈ C_{2k+1}

\{ \begin{matrix}d(B_{1}/B̄_{1}) ∈ Ξ_{1}^{(k)} \\ d(B_{2}/B̄_{2}) ∈ Ξ_{2}^{(k)}. \end{matrix}

2Traditionally, a conservation law is given by a 1-form φ involving the unknown function z together with its derivatives z_{x}, z_{y}, z_{xx}, … up to some finite order such that dφ = 0 whenever we substitute in a solution to the equation. For us conservation laws are given by closed 2-form Φ ∈ I; writing locally Φ = dφ gives the usual conservation law. In Section 2.1 we have included an introduction to the theory of conservation laws so that this paper can be read independently of [BG_{1}].

Thus, there is a direct relationship between conservation laws and integrable subsystems of the characteristic systems. This relationship has several consequences.

At one extreme, we can add (at least) two functions of two variables worth of new conservation laws when we pass from C_{2k−1} to C_{2k+1}, and this many are added if the hyperbolic system is Darboux integrable at this (or possibly a lower) level. The converse to this statement is quite plausible but we have not attempted to formulate and prove it in this paper. At the other extreme, if there are no integrable subsystems of the characteristic systems at level 2k + 1, then

dim C_{2k+1}/C_{2k−1} ≦ 2.

On the other hand, the s = 0 sine-Gordon system

(1)  \begin{aligned}u_{x} & = sin v \\ v_{y} & = sin u \end{aligned}

has rank one integrable subsystems of the characteristic systems and is shown to have the property that dim C_{0} = 1 while dim C_{1}/C_{0} = 3. Perhaps most interesting will be those hyperbolic systems which satisfy dim C = ∞ but dim C_{k} < ∞ for all k. The study of such systems will be the objective of a future paper in this series.

In this paper we shall work primarily with systems that are symmetric in that they exhibit symmetric behavior in the characteristic systems (the precise definition is given in Section 1.5). For these systems we shall determine those hyperbolic systems of class s = 0 for which the space C_{0} of level zero conservation laws—or as we shall say classical conservation laws—has infinite dimension.^{3} The word “determine” here has two meanings: First, we shall find the conditions imposed on the invariants of the system in order that dim C_{0} = ∞. In practice, this gives an algorithm for checking whether or not a given PDE system has an infinite number of classical conservation laws. Secondly, we shall derive a normal form for hyperbolic systems having dim C_{0} = ∞. Among the corollaries of our analysis is the result:

If dim C_{0} ≧ 7, then dim C_{0} = ∞. There is exactly one non-linear symmetric hyperbolic exterior differential system having an infinite number of classical conservation laws; namely, the exterior differential system associated to the s = 0 Liouville system

(2)  \begin{aligned}u_{y} & = e^{v} \\ v_{x} & = e^{u}. \end{aligned}

Among hyperbolic exterior differential systems of class s = 0 especially interesting are those of Euler-Lagrange type. Given a symplectic 4-manifold (M, Φ) and a 2-form Λ, the critical points of the functional

S → ∫_{S} Λ

defined on Φ-Lagrangian surfaces are solutions to an exterior differential system

Φ = Ψ = 0 where Ψ is a closed 2-form constructed from Λ. It turns out that this theory has a striking internal symmetry which exchanges the roles of Φ and Ψ. The s = 0 Goursat systems

3We should remark that unsymmetric systems for which dim C_{0} = ∞ have also been characterized. The results will be presented elsewhere.

(3)  \begin{aligned}u_{y} & = \frac{c\sqrt{uv}}{(x + y)} \\ v_{x} & = \frac{c\sqrt{uv}}{(x + y)} \end{aligned}  ,  c ∈ ℝ^{∗}

are ‘half’ of a certain one-parameter family of systems which may be uniquely characterized as being Euler-Lagrange in a two parameter family of geometrically distinct ways—the maximum possible for hyperbolic systems of class s = 0. A variant of Noether’s theorem gives an isomorphism between the symmetries and conservation laws modulo Φ, Ψ of Euler-Lagrange systems, and we show that in this case C_{0} is infinite dimensional if, and only if, the system is linearizable.^{4}

The study of Euler-Lagrange systems and other considerations suggests the importance of symplectic hyperbolic systems (M, Φ, I), this being a hyperbolic system (M, I) together with a symplectic form Φ such that Φ ∈ I. Automorphisms of such systems must preserve both I and Φ. We show that there are three classes of (symmetric) symplectic hyperbolic systems having an infinite number of classical conservation laws. These are:

Class A: consisting of systems that may be symplectically linearized;

Class B: consisting of symplectic hyperbolic systems that become Darboux integrable at level one (such as the system modeled on (2) above); and

Class C: consisting of symplectic hyperbolic systems modeled on the non-linear PDE system

\begin{aligned}u_{y} & = F (x, y)\sqrt{uv} \\ v_{x} & = F (x, y)\sqrt{uv}. \end{aligned}

Note that when F (x, y) = c/(x + y) we recover the s = 0 Goursat systems above.

Our technique of proof is to use the equivalence method of É. Cartan to introduce a canonical G-structure and class of pseudo-connections intrinsically associated to the hyperbolic system. This describes the intrinsic “geometry” associated to a PDE system up to contact equivalence, and associated to this geometry are the invariants or “curvatures” of the system. The condition that C_{0} have a certain structure—e.g., that there be infinitely many classical conservation laws—then imposes constraints on these invariants. Some of these constraints have direct geometric meaning—for example, if the characteristic systems do not each have an integrable subsystem then dim C_{0} ≦ 5. Others are expressed by algebraic conditions imposed on the various “curvatures”. As an application, using the general theory we may easily write down the explicit form of the conservation laws for explicitly given systems such as (1), (2), or (3). For example, for (2) the conservation laws are

(4)  φ = f(x)(du − e^{v}dy − e^{u}dx) + g(y)(dv − e^{u}dx − e^{v}dy)

where f and g are arbitrary functions, each of one variable.

A very interesting issue is the extent to which these intrinsic invariants relate to the more traditional estimates in PDE theory. The ultimate objective of each is, of course, to provide a means to understand the “solutions” of the system. As an illustration, we use the conservation laws (4) to infer that singularities of solutions to (2) are essentially of the form

4One of the unexpected discoveries of our study is the seeming ubiquity of linear hyperbolic systems. Of course, a highly non-linear PDE may be linearizable as an exterior differential system, as is the case for the system (3); moreover, an integrable extension of a non-linear PDE may linearize it, as is the case for (2). The fact that many interesting PDEs may be linearized in a variety of ways (hodographic transformation, Legendre transformations) is of course classical [CH]. The setting of exterior differential systems appears to synthesize and extend these classical methods.

u(ξ, t) = −c log ξ + (regular terms)
v(ξ, t) = c log ξ + (regular terms)

for some constant c > 0, in space-time coordinates (ξ, t) as ξ ↓ 0.

We also use the method of equivalence to answer a number of natural geometric questions concerning hyperbolic systems of class s = 0. Thus, in Section 1.5, we give necessary and sufficient conditions, expressed in terms of the torsion and curvature of the system, that it be linearizable. For example, this general result tells us that (3) is linearizable. This should be compared with the usual Goursat equation

z_{xy} = \frac{c\sqrt{z_{x}z_{y}}}{x + y}

which is obtained from the linearized form of (3) by a non-linear integrable extension; it is well-known that the usual Goursat equation is not linearizable within the class H_{1}. A linearizable system has a pair of intrinsic curvatures K and F and those for which K and F are constant, say K = c^{2} > 0 and F = Kβ are locally equivalent to the exterior differential system associated to the PDE system

(5)  \begin{aligned}u_{x} & = \frac{v}{cos^{1+β}(c(x + y))} \\ v_{y} & = \frac{u}{cos^{1−β}(c(x + y))}. \end{aligned}

Another natural question is: When is a hyperbolic exterior differential system Darboux integrable at level one? Both the systems (2) and (3) (with c = 2) satisfy these conditions and once the general result is known, the explicit integration may be carried out. (In fact, guided by the general theory, the explicit linearization and/or integration of examples such as the above may seem more straightforward than some of the calculations in the classical theory (cf. [Go]).) We show that there are, in fact, only two equivalence classes of hyperbolic systems of class s = 0 that are Darboux integrable at level one, namely that corresponding to (2) and linear equations with constant curvature.

We also want to remark that this may be related to the interesting recent paper of Anderson and Kamran [AK] which studies the conditions that a hyperbolic system of class s = 3 (e.g., a scalar second order hyperbolic equation for one function of two independent variables) be Darboux integrable at any level.

This paper is part of the general subject of the “geometry” associated to a differential equation. By geometry, we mean a G-structure together with an intrinsic class of pseudo-connections. The structure given by a 4-manifold together with a pair of transverse, non-integrable 2-plane fields is sufficiently simple that we are able to study it in an essentially self-contained manner using the general theory as a guide. It is, on the other hand, a very rich structure and the study in this paper is incomplete in two major aspects. The first is the analysis of systems that admit an infinite number of conservation laws of all levels. The second is the relation of the geometry of the unique global smooth integral surfaces “upstairs” with given non-characteristic initial data to the existence and uniqueness of global shock solutions to hyperbolic PDE systems of conservation laws with that same initial data. Both of these topics have been introduced and illustrated in the present work, but their satisfactory understanding is far from complete at this time.

1. Hyperbolic Exterior Differential Systems

1.1 Basic definition and examples

1.1.1 Some terminology. First, for convenience of the reader, we will recall some of the basic ideas of the theory of exterior differential systems. A much more extensive treatment of these concepts can be found in [BCG^{3}].

An exterior differential system is given by a manifold M and a differential ideal I ⊂ Ω^{∗}(M) in the algebra of smooth differential forms on M. We denote the exterior differential system by (M, I). Recall that by definition the differential ideal I is homogeneous in the sense that

I = ⊕_{q≥0}I^{q} ,

where I^{q} = I ∩ Ω^{q}(M) is the space of q-forms in I, and that I is closed under exterior differentiation d : I → I.

A symmetry of an exterior differential system (M, I) is a diffeomorphism f : M → M which satisfies f^{∗}I = I. An exterior differential system (M, I) is said to be equivalent to an exterior differential system (M̃, Ĩ) in case there is a diffeomorphism of M with M̃ which takes Ĩ to I.

Given a set θ_{1}, θ_{2}, … of forms of degrees q_{1}, q_{2}, …, we denote by {θ_{1}, θ_{2}, … } the algebraic ideal they generate in Ω^{∗}(M). The differential ideal generated by θ_{1}, θ_{2}, … is then denoted

I = {θ_{1}, θ_{2}, …; dθ_{1}, dθ_{2}, … }.

In practice, differential ideals are locally generated by a finite set of forms in this way. In case all the forms θ_{1}, θ_{2}, … have degree one, I is called a Pfaffian system.

An integral manifold of an exterior differential system (M, I) is an immersed submanifold f : N → M which satisfies f^{∗}θ = 0 for all θ ∈ I. When written out in local coordinates, this condition is a system of partial differential equations for the mapping f. Conversely, a sufficiently regular PDE system gives rise, in a canonical way, to an exterior differential system such that the solutions to the PDE system and the integral manifolds of the exterior differential system which satisfy a transversality condition are locally in one-to-one correspondence.

However, the notion of equivalence of exterior differential systems is different from that for partial differential equations. In particular, the symmetry group of an exterior differential system can be strictly larger than that of the PDE system from which it arises. This important point is explained and illustrated in Section 1.2 of [BG_{1}]; a consequence is that an exterior differential system has fewer invariants than the generating PDE system. Conforming to classical terminology, we shall call the symmetries of an exterior differential system contact transformations.

As always in the theory of exterior differential systems, we shall, without further mention, make suitable constant rank assumptions. For example, if θ_{1}, θ_{2}, … are 1-forms generating a Pfaffian system, then for each point x ∈ M the values θ_{1}(x), θ_{2}(x), … span a linear subspace I_{x} ⊂ T_{x}^{∗}M. We shall assume that dim I_{x} is constant (i.e., independent of x) and shall denote by I either the corresponding sub-bundle of T^{∗}M or the sub-module of Ω^{1}(M) generated by θ_{1}, θ_{2}, …; the context should make clear which of these interpretations to use. We shall generally use the notation

I = [θ_{1}, θ_{2}, …]

in either of the above two senses. (In general, the square bracket [θ_{1}, θ_{2}, …] will denote the linear span over the functions of a set θ_{1}, θ_{2}, … of differential forms.)

1.1.1

As another illustration of implicit constant rank assumptions, suppose we are given a Pfaffian system generated by I ⊂ Ω^{1}(M). Then the exterior derivative induces a C^{∞}(M)-linear mapping

δ : I → Ω^{2}(M)/{I},

where we recall our convention that {I} is the algebraic ideal generated by I (since the meaning is clear, we do not use the more correct but clumsy notation {I} ∩ Ω^{2}(M) in the denominator above). We shall assume without further mention that δ has pointwise constant rank. Then ker δ = I^{⟨1⟩} generates another Pfaffian system called the first derived system of I. Setting

I^{⟨2⟩} = (I^{⟨1⟩})^{⟨1⟩}

and so forth leads to the derived flag

I ⊃ I^{⟨1⟩} ⊃ I^{⟨2⟩} ⊃ ⋯

of I. We shall think of the I^{⟨k⟩} as either sub-bundles of T^{∗}M or sub-modules of Ω^{1}(M). This construction will play a central role below.

1.1.2 Hyperbolic systems. We will now define the main object of study of this paper, this being a hyperbolic exterior differential system—or briefly a hyperbolic system. This is a special type of exterior differential system which is meant to capture the essential features of the classical theory of hyperbolic (systems of) PDE in the plane. We shall give a more intrinsic definition below, but, informally stated, a hyperbolic system is an exterior differential system (M, I) where M is a manifold of dimension s+4 and I is a differential ideal with the property that every point of M lies in a neighborhood U on which there exists a coframing (i.e., a basis of 1-forms)

(θ; ω) = (θ^{1}, …, θ^{s}; ω^{1}, ω^{2}, ω^{3}, ω^{4})

so that, on U, the ideal I is generated algebraically in the form

I = {θ^{1}, …, θ^{s}; ω^{1} ∧ ω^{2}, ω^{3} ∧ ω^{4}}.

Such a coframing (θ; ω) will be referred to as an admissable local coframing for I. The condition that I be locally generated in the stated fashion implies that, for any admissable local coframing, there are relations of the form

dθ^{α} ≡ A^{α} ω^{1} ∧ ω^{2} + B^{α} ω^{3} ∧ ω^{4} mod {θ^{α}}

about which we will have more to say in Section 1.5.

It is not immediately apparent what a hyperbolic system in this sense has to do with hyperbolic PDE. However, as we will show by examples later in this section, many classical or well-known hyperbolic PDE systems for functions of two independent variables can be reformulated as hyperbolic systems in our sense.

From now on, I will denote a hyperbolic system on M. We will call s the class of the hyperbolic system under discussion. Note that s may be any non-negative integer. It is easy to see that our assumptions imply that I^{q} = Ω^{q}(M) for all q ≥ 3. In particular, it follows that the dimension of an integral manifold of I is at most 2. Integral surfaces of I will generally be referred to as solution surfaces of I.

We can give a more “intrinsic” description of what a hyperbolic system is, but first we want to make two remarks.

The first remark concerns the ambiguity in the choice of an admissable local coframing for I. If (θ; ω) and (θ̄; ω̄) are two admissable local coframings on the same domain in M, then, by inspection, we see that the span of the 1-forms θ^{α} and that of the 1-forms θ̄^{α} must be equal. Thus there must exist a globally defined

1.1.2 Pfaffian system I ⊂ T^{∗}M of rank s of which the θ^{α} in any local admissable coframing are local sections. Moreover, as is easy to see, there must exist non-zero functions λ and μ so that either

\begin{matrix}ω̄^{1} ∧ ω̄^{2} & ≡ λ ω^{1} ∧ ω^{2} \\ ω̄^{3} ∧ ω̄^{4} & ≡ μ ω^{3} ∧ ω^{4} \end{matrix}\} mod θ^{α}  or  \begin{matrix}ω̄^{1} ∧ ω̄^{2} & ≡ λ ω^{3} ∧ ω^{4} \\ ω̄^{3} ∧ ω̄^{4} & ≡ μ ω^{1} ∧ ω^{2} \end{matrix}\} mod θ^{α}.

Our second remark is about linear algebra. Let V be any vector space of dimension four. A nonzero 2-form Ω ∈ Λ^{2}V^{∗} is by definition decomposable if it can be written in the form Ω = ω^{1}∧ω^{2} for some ω^{1}, ω^{2} ∈ V^{∗}. For a non-zero Ω, decomposability is equivalent to the condition Ω∧Ω = 0. Because we are in dimension 4, a non-zero decomposable 2-form Ω = ω^{1}∧ω^{2} determines a 2-plane

Ω^{⊥} = {v ∈ V : ω^{1}(v) = ω^{2}(v) = 0}.

We will say that a pair of non-zero decomposable 2-forms (Ω_{1}, Ω_{2}) is transverse if the corresponding 2-planes are transverse: i.e., Ω_{1}^{⊥} ∩ Ω_{2}^{⊥} is a point. It is easy to see that this is equivalent to the condition Ω_{1}∧Ω_{2} ≠ 0. In this case, there always exists a basis (ω^{1}, ω^{2}, ω^{3}, ω^{4}) of V^{∗} so that Ω_{1} = ω^{1}∧ω^{2} and Ω_{2} = ω^{3}∧ω^{4}.

A pencil of 2-forms is, by definition, given by a line L ⊂ ℙ(Λ^{2}V^{∗}). If Φ_{1}, Φ_{2} ∈ Λ^{2}V^{∗} are independent 2-forms that generate the pencil, then elements of L may be represented as 2-forms Φ(μ) = μ_{1}Φ_{1}+μ_{2}Φ_{2} where μ = [μ_{1}, μ_{2}] are homogeneous coordinates in L ≅ ℙ^{1}. The decomposable elements of the pencil are given by solutions to the exterior equation

Φ(μ) ∧ Φ(μ) = 0.

Assuming that this equation is not identically satisfied for all μ, it is a homogeneous quadratic equation in μ_{1}, μ_{2}. We will say that the pencil L is hyperbolic in case this equation has distinct, real roots. In this case, it is not hard to see that there exists a transverse decomposable pair of generators for the pencil.

We now give the promised more intrinsic definition of a hyperbolic system.

Definition: A hyperbolic system of class s is given by an exterior differential system (M, I) where M is a manifold of dimension s + 4 and I = ⊕_{q>0}I^{q} is a differential ideal satisfying

(i) I^{1} = I is a Pfaffian system of rank s;

(ii) I^{2}/{I} is a hyperbolic pencil at each point.

More explicitly, for each x ∈ M, the subspace I_{x}^{⊥} ⊂ T_{x}M is a 4-dimensional vector space V with dual V^{∗} ≃ T_{x}^{∗}/I_{x}. The values

Φ(x) ∈ Λ^{2}(T_{x}^{∗}/I_{x}), Φ ∈ I^{2}/{I}

are well-defined. Condition (ii) then means that the 2-forms Φ(x) should then give a hyperbolic pencil.

Note that this condition has the effect of defining a ℙ^{1}-bundle L ⊂ ℙ(Λ^{2}(T^{∗}/I)), each fiber of which contains two special points, the decomposable elements of the pencil. These points fit together to form a smooth double cover of M. We are going to assume that this double cover is trivial, mainly for ease of exposition, though, in fact, this condition is satisfied in all of the interesting examples anyway.

In fact, by passing to a finite cover of M (with index at most 8), we can and shall assume that there exists a pair of non-vanishing, decomposable 2-forms Ω_{10} and Ω_{01} on M so that I is generated algebraically by the sections of I together with Ω_{10} and Ω_{01}. This necessarily implies that Ω_{10}∧Ω_{01} ≠ 0, since these 2-forms must restrict to each I_{x}^{⊥} to be a transverse pair. (The reason for the peculiar indexing of the Ω’s will be explained below.)

1.1.3

1.1.3 Some examples. We will now give some examples of hyperbolic systems, starting with those of class s = 0. Such a system I is defined on a 4-manifold M by a pair Ω_{10}, Ω_{01} of everywhere transverse decomposable 2-forms.

Example 1: Over a domain U ⊂ ℝ^{2} with coordinates (x, y) we consider a quasi-linear first order hyperbolic PDE system for two unknown functions u, v. Such a system is usually written in the form

(1)  A(x, y, u, v) ( \begin{matrix}u_{x} \\ v_{x} \end{matrix} ) + B(x, y, u, v) ( \begin{matrix}u_{y} \\ v_{y} \end{matrix} ) + C(x, y, u, v) = 0

where A and B are 2-by-2 matrices and C is a 2-by-1 column, with entries which are functions of (x, y, u, v) defined on some open set M ⊂ ℝ^{4}.

The usual condition that a solution (u, v) = (f(x, y), g(x, y)) be hyperbolic (or, more properly speaking, that (1) be hyperbolic at the solution (u, v) = (f(x, y), g(x, y))) is that the quadratic form

Q_{(f,g)}(ξ_{1}, ξ_{2}) = det( ξ_{1} A(x, y, f, g) + ξ_{2} B(x, y, f, g) ) = Q_{(f,g)}^{11} ξ_{1}^{2} + 2Q_{(f,g)}^{12} ξ_{1}ξ_{2} + Q_{(f,g)}^{22} ξ_{2}^{2}

have real, distinct linear factors, i.e., its discriminant Δ_{(f,g)} = Q_{(f,g)}^{11} Q_{(f,g)}^{22} − (Q_{(f,g)}^{12})^{2} should be negative.

To write (3) as an exterior differential system in the open set M ⊂ ℝ^{4}, we consider the pair of 2-forms Φ and Ψ defined by the matrix equation

( \begin{matrix}Φ^{1} \\ Φ^{2} \end{matrix} ) = A(x, y, u, v) ( \begin{matrix}du∧dy \\ dv∧dy \end{matrix} ) − B(x, y, u, v) ( \begin{matrix}du∧dx \\ dv∧dx \end{matrix} ) + C(x, y, u, v) dx ∧ dy .

The graph in M of a solution to (1) is easily seen to be an integral surface of the exterior differential system I generated by Φ^{1} and Φ^{2}. Conversely, integral surfaces of I on which dx∧dy is non-zero are locally graphs of solutions to (1). Thus, locally the solutions to the former are in one-to-one correspondence with those integral surfaces of the latter which satisfy a transversality condition.

Now,

(λ_{1} Φ^{1} + λ_{2} Φ^{2})^{2} = P (λ_{1}, λ_{2}) du ∧ dv ∧ dx ∧ dy.

where P = P^{11} λ_{1}^{2} + 2P^{12} λ_{1}λ_{2} + P^{22} λ_{2}^{2} is a quadratic form whose coefficients are functions of x, y, u, and v. Moreover, computation shows that its discriminant D = P^{11}P^{22} − (P^{12})^{2} has the property that, on the graph of a solution to (1), we have

D(x, y, f(x, y), g(x, y)) = Δ_{(f,g)}.

It follows that the pair of 2-forms Φ^{1}, Φ^{2} span a hyperbolic pencil (and thus define a hyperbolic exterior differential system of class s = 0) on the open subset of M which has the property that it contains the graphs of the hyperbolic solutions of (1).

Example 2: Consider the general hyperbolic exterior differential system I with s = 0 generated by a hyperbolic pencil of 2-forms {Φ^{1}, Φ^{2}}. In any local coordinate system (x, y, u, v), there are expressions

\begin{aligned}Φ^{1} & = A^{1} du ∧ dy + B^{1} dx ∧ du + C^{1} dv ∧ dy + D^{1} dx ∧ dv + E^{1} du ∧ dv + F^{1} dx ∧ dy \\ Φ^{2} & = A^{2} du ∧ dy + B^{2} dx ∧ du + C^{2} dv ∧ dy + D^{2} dx ∧ dv + E^{2} du ∧ dv + F^{2} dx ∧ dy \end{aligned}

where the coefficient functions A^{1}, etc., can be essentially arbitrary functions of x, y, u, and v (subject only to the conditions that the forms Φ^{1} and Φ^{2} generate a hyperbolic pencil). Then the integral surfaces of I to

1.1.4 which, say, x and y restrict to be independent functions are locally graphs of the form (x, y, u(x, y), v(x, y)) where u and v satisfy the pair of first order equations

\begin{aligned}0 & = A^{1} u_{x} + B^{1} u_{y} + C^{1} v_{x} + D^{1} v_{y} + E^{1} (u_{x}v_{y} − v_{x}u_{y}) + F^{1} \\ 0 & = A^{2} u_{x} + B^{2} u_{y} + C^{2} v_{x} + D^{2} v_{y} + E^{2} (u_{x}v_{y} − v_{x}u_{y}) + F^{2}. \end{aligned}

Note that if the functions E^{1} and E^{2} vanish identically in this coordinate system, this reduces to a quasi-linear first order hyperbolic PDE system as studied in Example 1. In fact, in Section 1.1.4 below we shall show that if the system I is real analytic in some local coordinate system, then each point of M lies in a neighborhood on which there exists a local coordinate system in which the generating 2-forms Φ^{1}, Φ^{2} have no du∧dv terms. Thus, at least in the real-analytic category (and, presumably in the smooth category as well, though we do not know this) the general hyperbolic system with s = 0 is locally equivalent to a hyperbolic pair of first-order quasi-linear PDE.

Example 3: Not all hyperbolic systems with s = 0 come directly from a first order system. Other natural constructions also yield these systems. Consider the second order Monge-Ampere equation

E (z_{xx}z_{yy} − z_{xy}^{2}) + A z_{xx} + 2B z_{xy} + C z_{yy} + D = 0

where A, B, C, D, and E are functions of x, y, p = z_{x}, and q = z_{y} only, i.e., they have no explicit dependence on z. In this case, the exterior differential system Ĩ generated in xypq-space M̃ by

\begin{matrix}Φ = & dp ∧ dx & + dq ∧ dy \\ Ψ = E dp ∧ dq + A dp ∧ dy + B ( & dq ∧ dy & + dx ∧ dp) + C dx ∧ dq + D dx ∧ dy \end{matrix}

has the following property: Integral surfaces on which dx∧dy is non-zero are locally graphs

(x, y) → (x, y, p(x, y), q(x, y)),

and Φ = 0 implies that there is (at least locally) a function z so that

p = z_{x}  and  q = z_{y} .

Then Ψ vanishes on this graph exactly when z(x, y) is a solution to the Monge-Ampere equation above. The condition that this solution be hyperbolic in the classical sense (see [CH]) is easily found to be equivalent to the condition that the graph lie in the open set where Φ and Ψ generate a hyperbolic pencil.

We shall study several explicit examples in later sections. However, this might be a good place to illustrate the notion of equivalence that we alluded to earlier. According to the general procedure that we have just outlined, the classical wave equation z_{yy} − z_{xx} = 0 corresponds to the ideal I_{1} on ℝ^{4} defined by

I_{1} = { dp ∧ dx + dq ∧ dy, dx ∧ dq − dp ∧ dy }

while the Monge-Ampere equation z_{xx} z_{yy} − z_{xy}^{2} = −1 corresponds to the ideal I_{2} on ℝ^{4} defined by

I_{2} = { dp ∧ dx + dq ∧ dy, dp ∧ dq + dx ∧ dy }.

Even though there is no change of variables in xyz-space which will convert one of these equations into the other, the ideals I_{i} are diffeomorphic: the diffeomorphism f : ℝ^{4} → ℝ^{4} defined by the formula

f(x, y, p, q) = (−p, y, x, q)

clearly has the property that f^{∗}(I_{1}) = I_{2} and f^{∗}(I_{2}) = I_{1}. Thus, the two systems are (globally) equivalent.

1.1.5

This example points out the importance of understanding when two systems are equivalent. Later sections of this paper will be directed at understanding how one can develop invariants which distinguish hyperbolic systems. The study of these invariants will then point out that several classically studied systems can be characterized by invariant properties. For example, in later sections our study will uncover, among others, the system

\begin{aligned}u_{x} − v_{x} & = sin(u + v) \\ u_{y} + v_{y} & = sin(u − v) \end{aligned}

which generates the famous Bäcklund transformation for the sine-Gordon equation z_{xy} = sin z, the system

\begin{aligned}u_{x} & = e^{v} \\ v_{y} & = e^{u} \end{aligned}

which is important in the study of the classical Liouville equation z_{xy} = e^{z}, and the system

\begin{aligned}u_{x} & = \frac{v}{x + y} \\ v_{y} & = \frac{u}{x + y} \end{aligned}

which turns out to be a ‘linearization’ of Goursat’s equation z_{xy} = 2\sqrt{z_{x} z_{y}}/(x + y).

The invariants to be discussed also influence the space of conservation laws of the systems in question, and other special systems will turn up in that context.

Example 4: Among the most interesting and important exterior differential systems are those that arise from critical points of functionals. In the present setting we consider a symplectic manifold (M, Φ) where dim M = 4 and Φ is a symplectic form on M. Recall that an immersed surface

f : S → M

is said to be Lagrangian if f^{∗}Φ = 0. Let Λ be another 2-form. Consider the functional on Lagrangian surfaces

(2)  L(f) = ∫_{S} f^{∗}(Λ) .

We will now construct an exterior differential system, called the Euler-Lagrange system E(Λ) associated to Λ, whose integral surfaces are the critical points of the functional (2).

To do this, we first write dΛ = Φ∧ψ for some (unique) 1-form ψ. (This form exists and is unique since Φ∧ : T^{∗}M → Λ^{3}T^{∗}M is an isomorphism.) We will say that (2) is non-degenerate in case the 2-form Ψ = dψ is a symplectic form on M, i.e., Ψ∧Ψ is nowhere vanishing.

We now define E(Λ) to be the ideal generated by Φ and Ψ. We are now going to show that a Lagrangian immersion f : S → M is critical for the functional (2) with respect to (compactly supported) variations of f through Φ-Lagrangian immersions if and only if it is an integral manifold of E(Λ).

To see this let f_{t} : S → M be a compactly supported variation through Φ-Lagrangian surfaces. Assuming for simplicity of notation that the f_{t} are imbeddings and setting S_{t} = f_{t}(S), a standard calculation gives

\frac{d}{dt} (∫_{S_{t}} Λ)_{t=0} = ∫_{S} v dΛ

1.1.6 where v is the variational vector field along S = S_{0}. Now, restricted to S we have

v (Φ ∧ ψ) = (v Φ) ∧ ψ
= dh ∧ ψ

for a suitable function h that depends linearly on v. The first equation follows from Φ|_{S} = 0, and the relation

v Φ|_{S} = dh

follows from the fact that the S_{t} are Φ-Lagrangian surfaces. By Stokes’ theorem

\frac{d}{dt} (∫_{S_{t}} Λ)_{t=0} = − ∫_{S} hΨ.

If this vanishes for all Φ-Lagrangian variations, then Ψ|_{S} = 0, as we wanted to show.

In case Φ and Ψ span a hyperbolic pencil, it follows that the Euler-Lagrange system E(Λ) is a hyperbolic exterior differential system of class s = 0.

We shall now give some examples of hyperbolic systems with s > 0.

Example 5: A general Monge-Ampere equation

E (z_{xx}z_{yy} − z_{xy}^{2}) + A z_{xx} + 2B z_{xy} + C z_{yy} + D = 0

as mentioned above, except where, now, the coefficients A, B, C, D, and E are functions of all five variables x, y, z, z_{x}, and z_{y}, may be written as an exterior differential system (M, I) on a 5-manifold. In fact, M is a suitable open set in the jet space J^{1}(ℝ^{2}, ℝ) with coordinates (x, y, z, p, q) and I is generated as a differential ideal by the contact 1-form

θ = dz − p dx − q dy

and the 2-form

Ψ = E dp ∧ dq + A dp ∧ dy + B(dq ∧ dy + dx ∧ dp) + C dx ∧ dq + D dx ∧ dy.

Algebraically I is generated by the 1-form θ and the 2-forms Θ = dθ and Ψ. As expected, the Monge-Ampere equation turns out to be hyperbolic in the usual sense if and only if Θ and Ψ generate a hyperbolic pencil modulo θ.

The relation between this example and the hyperbolic system (M̃, Ĩ) of class s = 0 constructed above is that, in the case the coefficients A, B, …, E do not depend on z, the system (M, I) is an integrable extension of (M̃, Ĩ) in the sense of Section 6 in [BG_{2}].

Explicitly, following the notation in Example 3, we note that φ = p dx + q dy satisfies dφ = Φ ∈ Ĩ. Thus, φ is closed modulo Ĩ. It follows that, on M^{5} = M̃ × ℝ with coordinate function z on the ℝ-factor, we may introduce a 1-form

θ = dz − p dx − q dy

on M. The exterior differential system generated by θ and Ψ (same notation in both examples) is then I as introduced above. We may think of the fibration M → M̃ as being given by the primitive of the 1-form giving a conservation law for Ĩ.

Example 6: Suppose that we consider the general first-order system for two functions of two unknowns:

\begin{aligned}F^{1}(x, y, u, v, u_{x}, u_{y}, v_{x}, v_{y}) & = 0 \\ F^{2}(x, y, u, v, u_{x}, u_{y}, v_{x}, v_{y}) & = 0. \end{aligned}

1.1.7 We shall naturally assume that these equations are non-degenerate in the sense that it is always possible to solve them locally for two of the partials. On ℝ^{8} with coordinates x, y, u, v, p, q, r, and s, consider the differential ideal generated by the two 1-forms

\begin{aligned}θ^{1} & = du − p & dx − q & dy \\ θ^{2} & = dv − r & dx − s & dy. \end{aligned}

Pull these forms back to the submanifold M^{6} ⊂ ℝ^{8} defined by the equations

F^{1}(x, y, u, v, p, q, r, s) = F^{2}(x, y, u, v, p, q, r, s) = 0.

Then they generate a rank 2 Pfaffian system I. Let I denote the ideal generated by {θ^{1}, θ^{2}, dθ^{1}, dθ^{2}}.

It is not difficult to show that a solution (u, v) = (f(x, y), g(x, y)) of the PDE system is hyperbolic in the usual PDE sense if and only if its 1-graph

(x, y) ↦ (x, y, f(x, y), g(x, y), f_{x}(x, y), f_{y}(x, y), g_{x}(x, y), g_{y}(x, y))

(which is clearly an integral manifold of I) lies in the open subset of M on which I is a hyperbolic exterior differential system of class s = 2.

Example 7: Now consider a single second-order hyperbolic equation

F (x, y, z, z_{x}, z_{y}, z_{xx}, z_{xy}, z_{yy}) = 0

which we assume to be non-degenerate, i.e., one can, at least locally, always solve this equation for one of the second order partials.

On ℝ^{8} with coordinates x, y, z, p (= z_{x}), q (= z_{y}), r (= z_{xx}), s (= z_{xy}), and t (= z_{yy}), we consider the Pfaffian system generated by the three 1-forms

\begin{aligned}θ^{0} & = dz − p & dx − q & dy \\ θ^{1} & = dp − r & dx − s & dy \\ θ^{2} & = dq − s & dx − t & dy. \end{aligned}

We pull these 1-forms back to the hypersurface M^{7} in ℝ^{8} defined by the equation

F (x, y, z, p, q, r, s, t) = 0

and let I denote the differential ideal generated by these forms. Now, dθ^{0} ≡ 0 mod θ^{0}, θ^{1}, θ^{2} as is easily verified, so it follows that I is generated algebraically by the forms {θ^{0}, θ^{1}, θ^{2}, dθ^{1}, dθ^{2}}. It is not difficult to show that the above equation is hyperbolic at a solution z = f(x, y) if and only if its 2-graph

(x, y) ↦ (x, y, f(x, y), f_{x}(x, y), f_{y}(x, y), f_{xx}(x, y), f_{xy}(x, y), f_{yy}(x, y))

(which is clearly an integral manifold of the system I) lies in the open subset of M on which I is a hyperbolic exterior differential system. Thus, (M, I) is hyperbolic of class s = 3.

Example 8: In the previous examples, we have shown how several classical hyperbolic PDEs can be recast as hyperbolic exterior differential systems. In fact, in geometry, this is frequently the reverse of the natural order. One generally encounters problems cast naturally in the form of a differential system and the “reduction” to an equation or system of equations can only be done after a somewhat arbitrary choice of coordinates. (One of the motives of studying exterior differential systems is to avoid having to make

1.1.8 these choices of coordinates.) In this example, we want to show how a hyperbolic system arises naturally in differential geometry. We will assume some familiarity with the language of moving frames.

Let (N^{3}, g) be an oriented Riemannian 3-manifold. Let π : F → N be its oriented orthonormal frame bundle. Thus, an element f ∈ F is a quadruple f = (x; e_{1}, e_{2}, e_{3}) where x = π(f) is a point of N and (e_{1}, e_{2}, e_{3}) is an oriented orthonormal basis of T_{x}N. There are canonical 1-forms ω_{i} defined on F by the rule

ω_{i}(v) = e_{i} · π′(v), for all v ∈ T_{f}F .

It is easy to show that π^{∗}(g) = ω_{1}^{2} + ω_{2}^{2} + ω_{3}^{2}. By the Fundamental Lemma of Riemannian geometry, there exist unique 1-forms ω_{ij} = −ω_{ji} (the connection forms) so that the following so-called structure equations hold:

dω_{i} = −ω_{ij} ∧ ω_{j} .

The 1-forms (ω_{1}, ω_{2}, ω_{3}, ω_{23}, ω_{31}, ω_{12}) then form a global coframing of F (which has dimension 6). One proves that the connection forms satisfy structure equations of the form

dω_{ij} = −ω_{ik} ∧ ω_{kj} + \frac{1}{2}R_{ijkl} ω_{k} ∧ ω_{l}

where R_{ijkl} = −R_{jikl} = −R_{ijlk}, represent the components of the Riemann curvature tensor.

Now let M^{5} ⊂ T N be the unit sphere bundle in the tangent bundle of N. There is a canonical map σ : F → M given by σ(x; e_{i}) = (x; e_{3}) and this map is a submersion. Looking at the definition of ω_{3} on F, it is clear that there exists a unique 1-form θ on M so that σ^{∗}(θ) = ω_{3}. Since

σ^{∗}(θ ∧ dθ^{2}) = ω_{3} ∧ (dω_{3})^{2} = −2 ω_{1} ∧ ω_{2} ∧ ω_{3} ∧ ω_{31} ∧ ω_{32} ≠ 0,

it follows that θ defines a contact structure on M. Moreover, because the fibers of σ are the leaves of the Pfaffian system generated by {ω_{1}, ω_{2}, ω_{3}, ω_{31}, ω_{32}}, the structure equations imply that there exist well-defined 2-forms Ω_{0}, Ω_{1}, Ω_{2} on M so that

\begin{aligned}σ^{∗}(Ω_{0}) & = ω_{1} ∧ ω_{2} \\ σ^{∗}(Ω_{1}) & = ω_{31} ∧ ω_{2} + ω_{1} ∧ ω_{32} \\ σ^{∗}(Ω_{2}) & = ω_{31} ∧ ω_{32}. \end{aligned}

In order to understand the geometric meaning of these forms, we consider the geometry of the integral surfaces of {θ, dθ}. For any immersed oriented surface f : S → N, there is a canonical lifting f̃: S → M defined by

f̃(s) = (f(s), ν(s))

where ν(s) ∈ T_{f(s)}N is the oriented normal to the oriented 2-plane f′(s)(T_{s}S) ⊂ T_{f(s)}N. It is easy to see that f̃^{∗}(θ) = 0, so that f̃: S → M is an integral surface of {θ, dθ}. Conversely, any integral surface g : S → M of θ which satisfies the condition that g^{∗}(Ω_{0}) ≠ 0 is easily seen to be of the form g = f̃ for some immersion f : S → N.

Moreover, a simple calculation shows that, first, f̃^{∗}(Ω_{0}) = dA is the area form induced on S by its immersion into N, second, f̃^{∗}(Ω_{1}) = 2H dA, where H is the mean curvature of the oriented immersion, and, third, f̃^{∗}(Ω_{2}) = K dA where K is the product of the principal curvatures of the immersion.^{5}

Now let λ_{0}, λ_{1}, and λ_{2} be constants (not all zero) and let

Ω = λ_{0} Ω_{0} + λ_{1} Ω_{1} + λ_{2} Ω_{2} .

1.1.9 Consider the ideal I on M generated by {θ, dθ, Ω}. It follows from the above discussion that the integral surfaces of I on which Ω_{0} is non-zero are exactly the canonical lifts of immersions into N which satisfy the relation

5Note that, unless the metric g is flat, this function K will not generally be the Gauss curvature of the induced metric f^{∗}(g) on S.

λ_{2} K + 2λ_{1} H + λ_{0} = 0.

This will be a hyperbolic system of class s = 1 exactly when the pair {dθ, Ω} generates a hyperbolic pencil modulo θ. A straightforward computation gives

(ξ_{1} Ω + ξ_{2} dθ)^{2} = ((λ_{1}^{2} − λ_{0}λ_{2}) ξ_{1}^{2} + ξ_{2}^{2}) (dθ)^{2}.

It follows that I will be hyperbolic if and only if λ_{1}^{2} − λ_{0}λ_{2} < 0.

Thus, for example, when (N, g) is flat Euclidean 3-space, setting (λ_{0}, λ_{1}, λ_{2}) = (1, 0, 1) gives the system whose integral manifolds are the surfaces in 3-space of constant Gauss curvature K = −1. Even in flat 3-space, however, there is no natural coordinate system for reducing this hyperbolic system to a partial differential equation. The introduction of the Tschebycheff coordinates usually associated with this problem (which are crucial in the proof of Hilbert’s theorem that there are no complete surfaces of constant negative curvature in Euclidean 3-space) depend, as we shall see, on understanding the conservation laws of this system in a coordinate-free manner. The relation of this system with the Bäcklund transformation for the sine-Gordon equation is well-known and will appear later in the paper.

In later sections, we will see further examples of hyperbolic systems with class s > 0 and will explore some of the relationships among systems of different classes. Although we are primarily interested in equations of class s = 0 or s = 1, it turns out that considering the more general case simplifies our study considerably.

1.1.4 A local normal form for hyperbolic systems of class s = 0. Let (M, I) be a hyperbolic system of class s = 0. We will show that in the real analytic case such a system is locally equivalent to the system associated to a quasi-linear first order hyperbolic PDE system exhibited in Example 1 in Section 1.1.3 above. We begin by proving the following

Proposition: Let (M, I) be a real analytic hyperbolic system of class s = 0. Then given any point p ∈ M there is a neighborhood U of p and a non-zero 2-form Ω defined on U satisfying

(i) Ω∧Ω = 0;

(ii) Ω∧Φ = 0 for all 2-forms Φ ∈ I;

(iii) Ω is integrable, i.e., dΩ ≡ 0 mod Ω.

Proof: We begin with a brief linear algebra discussion. Let V be a vector space of dimension 4 and ℙ = ℙ(Λ^{2}V^{∗}) the projective space associated to the vector space of 2-forms on V. Then dim ℙ = 5 and in ℙ the equation

Ω ∧ Ω = 0, Ω ∈ Λ^{2}V^{∗}

defines the Grassmannian 𝔾 of decomposable 2-planes. (More precisely, a point in ℙ is the line [Ω] spanned by a non-zero Ω ∈ Λ^{2}V^{∗}, and the validity of the above equation is independent of which point on the line we choose. We shall follow similar conventions for the remainder of this argument.) Let L ⊂ ℙ be a hyperbolic pencil with intersection

L ∩ 𝔾 = {[Ω_{1}], [Ω_{2}]}

where Ω_{1} and Ω_{2} are the two decomposable elements of the pencil, each of which is defined up to non-zero scalars. For a point [Ω] ∈ 𝔾 the equation

Ω ∧ Ω_{1} = 0

1.1.10 is equivalent to the condition that Ω and Ω_{1} have a common non-zero linear factor. Since the projectivized tangent space

ℙ(T_{[Ω_{1}]}𝔾) = {[Φ] ∈ ℙ : Φ ∧ Ω_{1} = 0},

we see that the above equation exactly defines

ℙ(T_{[Ω_{1}]}𝔾) ∩ 𝔾.

Thus the double intersection

S = ℙ(T_{[Ω_{1}]}𝔾) ∩ ℙ(T_{[Ω_{2}]}𝔾) ∩ 𝔾

is a smooth surface that defines the set of points [Ω] ∈ 𝔾 such that Ω has a common linear factor with each of Ω_{1} and Ω_{2}. For any such Ω we may choose a basis ω^{1}, ω^{2}, ω^{3}, ω^{4} for V^{∗} such that

(3)  \begin{aligned}Ω_{1} & = ω^{1} ∧ ω^{2} \\ Ω_{2} & = ω^{3} ∧ ω^{4} \\ Ω & = ω^{1} ∧ ω^{3}. \end{aligned}

For each point p ∈ M we may apply this construction to V = T_{p}M and obtain a surface S_{p} ⊂ ℙ(Λ^{2}T_{p}^{∗}M). In fact, there is an obvious 6-manifold Σ ⊂ ℙ(Λ^{2}T^{∗}M) given by ∪_{p∈M}S_{p}. To give local coordinates on Σ we choose a point (p_{0}, [Ω_{0}]) of Σ and a basis ω^{1}, ω^{2}, ω^{3}, ω^{4} in T_{p_{0}}^{∗}M for which (3) holds. We may extend the ω^{i} to a coframe near p_{0}, and then for (p, [Ω]) a point of Σ close to (p_{0}, [Ω_{0}]) we will have

(4)  Ω = (ω^{1} + u ω^{2}) ∧ (ω^{3} + v ω^{4})

for unique (u, v) ∈ ℝ^{2}. These u, v together with local coordinates near p_{0} on M will give a local coordinate system on Σ.

On Σ we have a tautological 2-form Ω, defined up to non-zero factors, and the exterior differential system that we want to consider is

(5)  dΩ ≡ 0 mod Ω

subject to the condition

ω^{1} ∧ ω^{2} ∧ ω^{3} ∧ ω^{4} ≠ 0.

An integral manifold of this system is then locally given by a section f : M → Σ such that f^{∗}Ω satisfies the conditions of the proposition. Locally we write

Ω = π_{1} ∧ π_{3}

where π_{1} = ω^{1} + u ω^{2} and π_{3} = ω^{3} + v ω^{4} are the linear factors appearing in (4). Equation (5) above then implies

\begin{aligned}dπ_{1} ∧ π_{1} ∧ π_{3} & = 0 \\ dπ_{3} ∧ π_{3} ∧ π_{1} & = 0. \end{aligned}

If we write

\begin{aligned}du & = u_{i} & ω^{i} \\ dv & = v_{i} & ω^{i} \end{aligned}

then solutions of the exterior differential system are given by solutions to the PDE system

\begin{aligned}u_{4} & = f & (u, v, u_{1}, u_{2}, u_{3}, v_{1}, v_{3}, v_{4}) \\ v_{2} & = g & (u, v, u_{1}, u_{2}, u_{3}, v_{1}, v_{3}, v_{4}) \end{aligned}

1.1.11 where f = g = 0 whenever u = v = 0. This is a determined PDE system for 2 unknown functions on an open set in ℝ^{4}, and by the Cauchy-Kowaleska Theorem it will have local solutions in the real analytic case. □

Remark: The above PDE system is in fact hyperbolic with characteristic variety given by the union of the two hyperplanes ω^{2} = 0 and ω^{4} = 0 in the tangent space at (p_{0}, [Ω_{0}]). Even though this characteristic variety is singular, it is possible that the methods of [Ya] might apply to give a C^{∞} result.

As an application of the proposition we have the

Corollary: Any real analytic hyperbolic system of class s = 0 is locally equivalent to the exterior differential system associated to a quasi-linear hyperbolic PDE system.

Proof: We may find local functions x, y such that Ω is a non-zero multiple of dx∧dy. If we then complete x, y to a local coordinate system x, y, u, v then each of the generators of the hyperbolic system will have no du∧dv term. The result then follows from the discussion below Example 2 in Section 1.1.3. □

An important special class of first order quasi-linear hyperbolic systems are the hyperbolic systems of conservation laws, which are PDE systems for functions u and v of x and y of the form

(6)  ∂_{x}(F(x, y, u, v)) + ∂_{y}(G(x, y, u, v)) = 0

where F and G are ℝ^{2}-valued functions. We may equivalently express (6) by the statement that the ℝ^{2}-valued 2-form

( \begin{matrix}Φ_{1} \\ Φ_{2} \end{matrix} ) = dF ∧ dy − dG ∧ dx = d(F dy − G dx)

vanishes on graphs (x, y) → (x, y, u(x, y), v(x, y)). We clearly have

dΦ_{1} = dΦ_{2} = 0

so that the exterior differential system associated to a hyperbolic system of conservation laws is generated by a pair of closed 2-forms. Using the above proposition we may prove a converse.

Proposition: A real analytic hyperbolic system (M, I) of class s = 0 is locally equivalent to the exterior differential system associated to a hyperbolic system of conservation laws if, and only if, I is generated by a pair of closed 2-forms.

Proof: Let Φ_{1} and Φ_{2} be closed generators of I. We want to show that on a neighborhood of each point of M there are functions x and y together with ℝ^{2}-valued functions F, G such that

( \begin{matrix}Φ_{1} \\ Φ_{2} \end{matrix} ) = d(F dy − G dx).

Let x, y, u, v be coordinates as in the above corollary and Φ ∈ I be any closed 2-form. Since Φ is locally exact we have

Φ = α ∧ dx + β ∧ dy + f_{1}dx ∧ dy = dγ

for some 1-forms α, β and γ, and a function f_{1}. Clearly, it suffices to show that γ may be chosen so that γ ≡ 0 mod dx, dy.

In the above expression for Φ we may assume that

\begin{aligned}α & = a du + b dv \\ β & = c du + e dv \end{aligned}

1.1.12 for suitable functions a, b, c and e. Looking at the coefficient of du∧dv∧dx in dΦ gives

a_{v} = b_{u}

so that

a = A_{u}, b = A_{v}

for a suitable function A. Then

Φ − d(A dx) = β ∧ dy + f_{2} dx ∧ dy

for some possibly new function f_{2}. Repeating the argument gives the existence of functions B and f such that

Φ − d(A dx + B dy) = f dx ∧ dy.

Taking the exterior derivatives of both sides gives

df ∧ dx ∧ dy = 0

and so f = f(x, y). Then by modifying A and B appropriately we obtain

Φ = d(A dx + B dy)

as desired. □

1.1.13 1.2 The characteristic variety and the initial value problem

The fundamental feature of hyperbolic (as opposed to, say, elliptic) PDE is the existence of the socalled “characteristics”. In the classical theory of (determined) hyperbolic PDE in two independent variables, one studies the initial value problem, which is usually posed as follows: Along a curve in the domain of the independent variables, one specifies initial data (the values of the dependent variables and their partial derivatives) which satisfy the so-called “strip conditions”. If this data is “non-characteristic” in an appropriate sense, then one hopes to prove that there is a unique solution to the PDE on an open neighborhood of the initial curve which agrees with the given initial data along that curve. (See [CH] for a discussion of these ideas.) Then the notion of a “characteristic” emerges as being a curve in the domain of a solution along which the initial value problem would not be well-posed.

In this section, we will explore this concept in the context of hyperbolic exterior differential systems.

1.2.1 Some linear algebra. Before turning to the details of hyperbolic systems, it is useful to briefly continue the linear algebra discussion on 4-dimensional vector spaces that we began in Section 1.1.2. We will keep the notation established there and let ℙ = ℙV denote the projective space of lines through the origin in V. An isomorphism V ≅ ℝ^{4} induces an identification ℙ ≅ ℙ^{3}, and all the statements below may be easily verified by choosing coordinates and making the appropriate calculations.

Let Ω_{1}, Ω_{2} ∈ Λ^{2}V^{∗} be decomposable 2-forms satisfying Ω_{1}∧Ω_{2} ≠ 0. The 2-planes Ω_{1}^{⊥}, Ω_{2}^{⊥} in V then give a pair of lines L_{1}, L_{2} in the projective space ℙ, and the condition Ω_{1}∧Ω_{2} ≠ 0 translates into the condition that L_{1} and L_{2} be skew lines. A 2-plane E ⊂ V gives a line L_{E} ⊂ ℙ and for each α = 1, 2 the conditions

(i) Ω_{α}|_{E} = 0
(ii) L_{α} meets L_{E}

are equivalent. Thus, the 2-planes for which (i) holds for α = 1, 2 are given by the lines L_{E} ⊂ ℙ which meet each of L_{1} and L_{2}.

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• • •  P  P_{1}  L_{E} = L_{P} P_{2} L_{2}  L_{1} Figure 1

Moreover, given any point P ∈ ℙ not on L_{1} or L_{2}, there is a unique line L_{P} passing through P and meeting each of L_{1} and L_{2} in points P_{1} and P_{2}. This gives a fibration

ℙ^{3} \ (L_{1} ∪ L_{2})

↓

L_{1} × L_{2}

1.2.1 with fibre ℙ^{1} \ {2 points} ≅ ℝ^{∗}. In particular, the set of lines meeting each of L_{1} and L_{2} is bijective to L_{1} × L_{2}. Back on V the statement is: The set of 2-planes E ⊂ V meeting each of the 2-planes Ω_{1}^{⊥}, Ω_{2}^{⊥} in a line is a ℙ^{1} × ℙ^{1}.

1.2.2 The Characteristic Variety. Now, recall (see [BCG^{3}]) that an integral element (of dimension q) of an exterior differential system (M, I) is a q-plane E ⊂ T_{x}M such that all the forms in I restrict to zero on E; i.e.,

θ(x)|_{E} = 0, θ ∈ I.

Intuitively, the q-dimensional elements are the candidates for tangent planes to q-dimensional integral manifolds.

Hyperbolic systems (which, as we have seen, satisfy I^{q} = Ω^{q}(M) for all q ≥ 3) have integral elements of dimensions q = 0, 1, 2, which we now want to describe. Let I = {θ^{1}, …, θ^{s}; Ω_{10}, Ω_{01}} be a hyperbolic exterior differential system on M^{s+4} and let x ∈ M. Then the subspace I_{x}^{⊥} ⊂ T_{x}M, defined as the set of v ∈ T_{x}M so that

θ_{x}^{1}(v) = ⋯ = θ_{x}^{s}(v) = 0,

is a vector space of dimension 4 and we denote by Ω_{1}, Ω_{2} the restrictions of Ω_{10}, Ω_{01} to I_{x}^{⊥}.

Integral elements of dimension one in T_{x}M are simply lines in T_{x}M which lie in I_{x}^{⊥} and are thus given by points P ∈ ℙI_{x}^{⊥}. By the linear algebra discussion above, integral elements of dimension two in T_{x}M are 2-planes in I_{x}^{⊥} which meet each of the 2-planes Ω_{1}^{⊥} and Ω_{2}^{⊥} and thus are given by lines L ⊂ ℙI_{x}^{⊥} meeting each of the skew lines L_{1} = ℙ(Ω_{1}^{⊥}) and L_{2} = ℙ(Ω_{2}^{⊥}). The condition that a point P ∈ ℙI_{x}^{⊥} lies on a unique line meeting each of L_{1} and L_{2} is that P not lie on either L_{1} or L_{2}.

We recall (again, see [BCG^{3}] for a more general discussion) that, by definition, the characteristic variety Ξ of the exterior differential system I consists of all integral elements of dimension one that do not lie in a unique integral element of dimension 2. It follows from the above discussion that the base point mapping Ξ → M (which sends each characteristic integral element to its base point) is a fibration with fibre ℙ^{1} ∪ℙ^{1}. Informally, we say that the characteristics of a hyperbolic exterior differential system define a ℙ^{1} ∪ℙ^{1} over each point of M.

This leads us to a very important definition for the subsequent theory. Recall our standing assumption from Section 1.1.2 that I can be generated on M by the sections of I and two globally defined, nonvanishing, decomposable 2-forms Ω_{10} and Ω_{01}. Clearly, we can choose a local adapted coframing (θ; ω) so that Ω_{10} = ω^{1}∧ω^{2} and Ω_{01} = ω^{3}∧ω^{4}.

Definition: The (0^{th}) characteristic system Ξ_{10} is the Pfaffian system generated by {θ^{1}, …, θ^{s}, ω^{1}, ω^{2}} while the (0^{th}) characteristic system Ξ_{01} is the Pfaffian system generated by {θ^{1}, …, θ^{s}, ω^{3}, ω^{4}}.

Note that each of the characteristic systems is a Pfaffian system of rank s+2. The importance of the characteristic systems is that a 1-dimensional integral element E_{1} of I is characteristic if and only if it lies in either the 2-plane Ξ_{10}^{⊥} or the 2-plane Ξ_{01}^{⊥}. In fact, we immediately see that the characteristic variety bundle Ξ → M decomposes into two disjoint ℙ^{1}-bundles as

Ξ = ℙ(Ξ_{10}^{⊥}) ∪ ℙ(Ξ_{01}^{⊥}).

Now, since, by definition, each of Ω_{01} and Ω_{10} vanish on any integral surface f : S → M of I, it follows that each of the characteristic systems Ξ_{10} and Ξ_{01} pulls back via such an f to be a Pfaffian system of rank 1. In particular, there are two everywhere transverse foliations F_{10} and F_{01} of S by curves so that the leaves of F_{10} map to integral curves of Ξ_{10} and the leaves of F_{01} map to integral curves of Ξ_{01}. As will be seen, in the classical cases, these foliations are merely the foliations of the domain of the solution by the so-called Monge characteristics.

1.2.2

Now a transverse pair of curve-foliations of a surface has no local geometry. However, the exterior differential system perspective shows that these are merely specializations to each solution surface of the “ambient” characteristic systems Ξ_{10} and Ξ_{01}. These Pfaffian systems do have local geometry and it is this geometry which strongly influences conservation laws. We shall explore it more deeply in the following sections.

Example 1: Consider the equation z_{xy} = f(x, y, z, z_{x}, z_{y}) which, as we showed in the last section, can be associated to the hyperbolic system I of class s = 1 on ℝ^{5}

I = {dz − p dx − q dy, (dp − f(x, y, z, p, q) dy) ∧ dx, (dq − f(x, y, z, p, q) dx) ∧ dy}.

Thus, the characteristic systems are given by

\begin{aligned}Ξ_{10} & = \{dz − p dx − q dy, (dp − f(x, y, z, p, q) dy), dx\} \\ Ξ_{01} & = \{dz − p dx − q dy, (dq − f(x, y, z, p, q) dx), dy\}. \end{aligned}

Of course, when we restrict to any solution of the above equation, Ξ_{10} (respectively, Ξ_{01}) pulls back to be the multiples of dx (respectively, dy). Thus the characteristic foliations in the xy-plane are simply the foliations by the x-lines and y-lines, just as we expected.

Example 2: Consider the Monge-Ampere equation z_{xx}z_{yy}−z_{xy}^{2} = −1 which gives rise to the hyperbolic system I of class s = 1 on ℝ^{5}

I = {dz − p dx − q dy, (dp − dy) ∧ (dq + dx), (dp + dy) ∧ (dq − dx)}

The characteristic systems are given by

\begin{aligned}Ξ_{10} & = \{dz − p dx − q dy, (dp − dy), (dq + dx) \} \\ Ξ_{01} & = \{dz − p dx − q dy, (dp + dy), (dq − dx) \}. \end{aligned}

Thus, for example, on the solution surface

(x, y, z, p, q) = (x, y, f(x)+xy, f′(x)+y, x)

(where f is an arbitrary smooth function of one variable), the foliation F_{10} is just the foliation induced by the slices x = const while the foliation F_{01} is the foliation induced by the slices 2y + f′(x) = const.

1.2.3 The initial value problem for hyperbolic systems. As mentioned earlier, the notion of an integral element of an exterior differential system was meant to capture the idea of an “infinitesimal” integral manifold. Since a characteristic integral element is, by definition, one for which the extension to a higher dimensional integral element is not unique, it is the infinitesimal version of the usual PDE notion of a characteristic.

In the case of a hyperbolic exterior differential system (M, I), the notion of “initial data satisfying a strip condition” corresponds exactly to an integral curve (i.e., one dimensional integral manifold) ϕ : C → M of the Pfaffian system I. We say that ϕ is non-characteristic if the tangent line ϕ′(t)(T_{t}C) ⊂ T_{ϕ(t)}M is not a characteristic integral element of I for any t ∈ C, i.e., if ϕ′(t)(T_{t}C) ∉ Ξ for all t ∈ C. As ought to be expected, this reduces in the classical cases to the notion of initial data being non-characteristic.

Example 2 (continued): A curve ϕ : (0, 1) → ℝ^{5} of the form

ϕ(t) = (x_{0}(t), y_{0}(t), z_{0}(t), p_{0}(t), q_{0}(t))

1.2.3 is an integral curve of I (i.e., satisfies the strip conditions) if and only if z′_{0}(t) = p_{0}(t) x′_{0}(t) + q_{0}(t) y′_{0}(t). It is non-characteristic if and only if it is not tangent to either of the distributions Ξ_{10}^{⊥} or Ξ_{01}^{⊥}, in other words, if and only if, for all 0 < t < 1, we have

\begin{matrix}(p′_{0}(t) − y′_{0}(t))^{2} + (q′_{0}(t) + x′_{0}(t))^{2} & ≠ 0, \\ (p′_{0}(t) + y′_{0}(t))^{2} + (q′_{0}(t) − x′_{0}(t))^{2} & ≠ 0. \end{matrix}

(Note that the condition x′_{0}(t)^{2} + y′_{0}(t)^{2} ≠ 0 is not required, which it would be if we were to think of ϕ as defining initial data satisfying a strip condition along a smooth curve in the domain of the independent variables.) Assuming that these inequalities are satisfied, we can construct an integral surface f : (0, 1) × (0, 1) → ℝ^{5} of I which satisfies f(t, t) = ϕ(t) for all 0 < t < 1 by setting

\begin{aligned}x(s, t) & = \frac{1}{2}(x_{0}(s) + x_{0}(t) − q_{0}(s) + q_{0}(t)) \\ y(s, t) & = \frac{1}{2}(y_{0}(s) + y_{0}(t) + p_{0}(s) − p_{0}(t)) \\ p(s, t) & = \frac{1}{2}(p_{0}(s) + p_{0}(t) + y_{0}(s) − y_{0}(t)) \\ q(s, t) & = \frac{1}{2}(q_{0}(s) + q_{0}(t) − x_{0}(s) + x_{0}(t)) \end{aligned}

and then defining, for some chosen t_{0} ∈ (0, 1),

z(s, t) = z_{0}(t_{0}) + ∫_{(t_{0},t_{0})}^{(s,t)} p dx + q dy

where the line integral is taken over any curve in (0, 1) × (0, 1) joining (t_{0}, t_{0}) to (s, t). The reader might check that the hypothesis that the curve ϕ be non-characteristic forces f to be an immersion (on the whole open square) and that the characteristic foliations are simply the foliations by the s- and t-slices in the square. We will comment further on the method we used to construct this solution in Section 1.4.

In fact, the general initial value problem for hyperbolic exterior differential systems has been studied quite extensively in the literature, see [Ka] and [Ya]. In particular, an elementary consequence of Theorem 5.1 in [Ka] is the following existence and uniqueness result:

Theorem: Let (M^{s+4}, I) be a smooth hyperbolic exterior differential system and let ϕ : (0, 1) → M be a non-characteristic smooth integral curve of I. Then there exists an open neighborhood U_{ϕ} ⊂ (0, 1) × (0, 1) of the diagonal and a smooth map f_{ϕ} : U_{ϕ} → M which is an integral surface of I, satisfies f_{ϕ}(t, t) = ϕ(t), and has the property that its characteristic foliation F_{10} is given by the slices t = const while its characteristic foliation F_{01} is given by the slices s = const. Moreover, if (Ũ_{ϕ}, f̃_{ϕ}) is any other pair with these properties, then f_{ϕ} and f̃_{ϕ} agree on any subsquare of the form (a, b) × (a, b) which lies in the intersection U_{ϕ} ∩ Ũ_{ϕ}.

Note that the condition of normalizing the characteristic foliations has the effect of removing the reparametrization ambiguity which normally affects integral manifolds of exterior differential systems. Also, note that while one would like to be able to say that the solution surface f is defined on the entire open unit square, this will not generally be the case. A very interesting problem is to try to characterize in terms of some sort of completeness, those systems (M, I) which have the property that every non-characteristic integral curve ϕ : (0, 1) → M has an extension as above with U_{ϕ} = (0, 1) × (0, 1). Example 2 above clearly does have this property, so it should be thought of as “characteristically complete”.

However, we caution the reader that the integral surface so constructed may not be representable as a graph of z as a function of x and y even if it is so representable in a neighborhood of the initial curve ϕ. Thus, from the PDE perspective, a given PDE may not have global existence for all initial data even when the corresponding hyperbolic exterior differential system is complete.

1.2.4

Example 2 (continued): Suppose that, for the Monge-Ampere equation above, we consider the initial curve given by the data

x_{0}(t) = cos t − sin t
y_{0}(t) = cos t + sin t
z_{0}(t) = 1
p_{0}(t) = cos t − sin t
y_{0}(t) = cos t + sin t,

with solution

x(s, t) = cos t − sin s
y(s, t) = cos s + sin t
z(s, t) = t − s + cos(t − s)
p(s, t) = cos s − sin t
q(s, t) = cos t + sin s.

Note that, as expected, this is an immersed surface in ℝ^{5}. However, it cannot be represented globally as a graph of z as a function of x and y. In fact, we have

dx ∧ dy = cos(t − s) dt ∧ ds.

Thus, for example, we cannot solve smoothly for s and t as functions of x and y in any region where cos(t−s) vanishes. In fact, restricting to the region where |t − s| < π/2, we can solve for z as a function of x and y, getting

z(x, y) = sin^{−1}(1 − \frac{1}{2}(x^{2} + y^{2})) + cos(sin^{−1}(1 − \frac{1}{2}(x^{2} + y^{2}))).

Note that this solution cannot be extended smoothly beyond the punctured disk 0 < x^{2} + y^{2} < 2.

Example 3: Let k be a smooth positive function on ℝ. Consider the non-linear equation

z_{yy} − (k(z_{x}))^{2} z_{xx} = 0,

known as the Fermi-Pasta-Ulam equation, or FPU equation, for short. The corresponding hyperbolic exterior differential system on ℝ^{5} with standard coordinates x, y, z, p, and q is

\begin{aligned}I & = \{ dz − p dx − q dy, dp ∧ dx + dq ∧ dy, dq ∧ dz + (k(p))^{2} dp ∧ dy\} \\ & = \{ dz − p dx − q dy, (dq + k(p) dp) ∧ (dx + k(p) dy), (dq − k(p) dp) ∧ (dx − k(p) dy) \}. \end{aligned}

Let K(p) be an antiderivative of k(p), and note that K is a strictly increasing function K : ℝ → ℝ which is a diffeomorphism of ℝ onto its image.

We are going to show that if the range of K is all of ℝ, then (ℝ^{5}, I) is complete in the above sense, in spite of the well-known fact (which we shall discuss later) that solutions to the FPU equation with compactly supported initial data develop singularities in finite time.

Now, the characteristic systems are

\begin{aligned}Ξ_{10} & = \{ dz − p dx − q dy, (dq + k(p) dp), (dx + k(p) dy) \} \\ Ξ_{01} & = \{ dz − p dx − q dy, (dq − k(p) dp), (dx − k(p) dy) \}. \end{aligned}

Suppose that ϕ : ℝ → ℝ^{5} is a non-characteristic integral curve of I. Thus,

ϕ(t) = (x_{0}(t), y_{0}(t), z_{0}(t), p_{0}(t), q_{0}(t))

where z′_{0}(t) = p_{0}(t) x′_{0}(t) + q_{0}(t) y′_{0}(t), but where, also, for all t, we have

(q′_{0}(t) ± k(p_{0}(t)) p′_{0}(t))^{2} + (x′_{0}(t) ± k(p_{0}(t)) y′_{0}(t))^{2} > 0.

1.2.5 We want to show how to construct f : ℝ^{2} → ℝ^{5} as in the above Theorem. First, if there is a solution, then there will be functions A, B, C and D on ℝ^{2} so that

\begin{aligned}dq+ & k(p) dp & = 2 A(s, t) & ds \\ dx+ & k(p) dy & = 2 B(s, t) & ds \end{aligned}  and  \begin{aligned}dq− & k(p) dp & = 2 C(s, t) & dt \\ dx− & k(p) dy & = 2 D(s, t) & dt. \end{aligned}

Since d(q + K(p)) is a multiple of ds, it follows that if there is to be a solution, then q + K(p) will have to be a function of s alone. Thus

q(s, t) + K(p(s, t)) = q(s, s) + K(p(s, s)) = q_{0}(s) + K(p_{0}(s)).

By similar reasoning using the other characteristic system, we see that

q(s, t) − K(p(s, t)) = q_{0}(t) − K(p_{0}(t)).

Solving these equations gives the only possibility for the functions p and q if there is to be a solution:

q(s, t) = \frac{1}{2}(q_{0}(s)+q_{0}(t)+K(p_{0}(s))−K(p_{0}(t))) and K(p(s, t)) = \frac{1}{2}(q_{0}(s)−q_{0}(t)+K(p_{0}(s))+K(p_{0}(t))).

Note that the assumption that the range of K be all of ℝ implies that this does, indeed define a (unique) function p : ℝ^{2} → ℝ. Now, we also have

\begin{aligned}dx & = B(s, t) ds + D(s, t) dt \\ k(p) dy & = B(s, t) ds − D(s, t) dt \end{aligned}

so it follows that B(s, t) = x_{s}(s, t) and D(s, t) = x_{t}(s, t). Substituting this into the second equation gives

dy = (k(p(s, t)))^{−1}(x_{s}(s, t) ds − x_{t}(s, t) dt).

Set λ(s, t) = log(k(p(s, t))), then differentiating this last equation and multiplying by e^{λ} gives the formula

2x_{st} ds ∧ dt + dλ ∧ (x_{s} ds − x_{t} dt) = 0.

and this expands to the linear equation (for x(s, t))

2x_{st} − λ_{t} x_{s} + λ_{s} x_{t} = 0.

Note also that we have the initial conditions x(t, t) = x_{0}(t) and x_{s}(t, t) − x_{t}(t, t) = k(p_{0}(t)) y′_{0}(t). Since λ is known, this equation is a linear hyperbolic equation for x in the st-plane, with initial conditions posed along the non-characteristic line s = t. By the usual existence and uniqueness theorems for linear equations, there exists a unique function x(s, t) on the whole st-plane satisfying this equation and the given initial conditions. Once x is known, the fact that it is a solution of the linear equation implies that the 1-form (k(p(s, t)))^{−1}(x_{s}(s, t) ds − x_{t}(s, t) dt) is closed. Hence, it is the exterior derivative of a function y. By choosing the additive constant correctly, we can make sure that, y(0, 0) = y_{0}(0), and then the equations above easily imply that y(t, t) = y_{0}(t) for all t ∈ ℝ. Finally, the function z(s, t) is constructed in the obvious way:

z(s, t) = z_{0}(0) + ∫_{(0,0)}^{(s,t)} p dx + q dy

(the path used for the line integral is immaterial since, by construction, p dx+q dy is a closed 1-form) and the strip conditions then imply that z(t, t) = z_{0}(t). It is now an elementary matter to check that the map f : ℝ^{2} → ℝ^{5}

f(s, t) = (x(s, t), y(s, t), z(s, t), p(s, t), q(s, t))

pulls back the forms in I to be zero. It is not, however, obvious that f is an immersion except along some open set containing the line s = t.

1.2.6 1.3 Prolongation of hyperbolic systems and their structure equations

In the theory of exterior differential systems, the operation of prolongation plays a central role. This operation is analagous to the process in classical PDE whereby one adjoins the derivatives of the unknown functions as new variables and then adjoins new partial differential equations to ensure that the new unknowns do, in fact, behave like the derivatives of the original unknowns. As usual, for a more complete explanation of the process of prolongation, we refer the reader to [BCG^{3}].

1.3.1 Prolongations. We denote by G_{2}(T M) the Grassmann bundle whose fibre over x ∈ M consists of all 2-planes E ⊂ T_{x}M. Sitting in G_{2}(T M) is the set G_{2}(I) of integral 2-planes of the hyperbolic system I. We then have the base point fibration

G_{2}(I) ⊂ G_{2}(T M)
↓ π
M

where, by our discussion above, each fibre of π is a ℙ^{1} × ℙ^{1}.

Recall that over G_{2}(T M) there is a tautological Pfaffian system J whose integral manifolds are the canonical lifts (Gauss maps) of immersed surfaces in M, as expressed by the diagram

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where f_{∗}(s) = T_{f(s)}(f(S)).

For a point (x, E) ∈ G_{2}(T M) we have by definition

J_{(x, E)} = π^{∗}(E^{⊥})

where E ⊂ T_{x}M is a 2-plane and E^{⊥} ⊂ T_{x}^{∗}M its codimension 2 annihilator. The restriction to G_{2}(I) ⊂ G_{2}(T M) of the canonical system J is, by definition, the (first) prolongation (M^{(1)}, I^{(1)}) of (M, I); thus

M^{(1)} = G_{2}(I)  and  I^{(1)} = J |_{M^{(1)}}.

We want to elucidate the local structure of the ideal I^{(1)}. To this end, we prove the following structure theorem.

Proposition: The prolongation of a hyperbolic system of class s is a hyperbolic system of class s+2.

Proof: Suppose that, on an open set U of M, we have chosen an admissable local coframing (θ; ω) for I as in Section 1.1.2. Thus, on U, we have

I = {θ^{1}, …, θ^{s}; ω^{1} ∧ ω^{2}, ω^{3} ∧ ω^{4}}.

If E_{0} ∈ G_{2}(I) is an integral element based in U, then, by changing basis in the ω^{i}, we can arrange that ω^{1}∧ω^{3} does not vanish on E_{0}. Let W ⊂ M^{(1)} denote the open subset of 2-dimensional integral elements of I with base point in U and on which the 2-form ω^{1}∧ω^{3} does not vanish. This is, of course, an open

1.3.1 neighborhood of E_{0} in M^{(1)}. For any E ∈ W, the space E^{⊥} ⊂ T_{π(E)}^{∗}M is spanned by s+2 1-forms. Since ω_{π(E)}^{1} and ω_{π(E)}^{3} are linearly independent on E, while all of the forms in I must vanish on E, it follows that E^{⊥} has a unique basis of the form

θ_{π(E)}^{1} , …, θ_{π(E)}^{s} , ω_{π(E)}^{2} − h_{20}(E) ω_{π(E)}^{1} , ω_{π(E)}^{4} − h_{02}(E) ω_{π(E)}^{3} .

The functions h_{20}, h_{02} thus defined on W define coordinates on the fibers of W (⊂ U^{(1)}) → U. Moreover, inspection, combined with the definition of I^{(1)}, shows that the 1-forms

θ_{10} = ω^{2} − h_{20} ω^{1}
θ_{01} = ω^{4} − h_{02} ω^{3}

lie in the Pfaffian system which generates I^{(1)}.^{6} Thus, on W ⊂ M^{(1)}, the first prolongation I^{(1)} is generated by the Pfaffian system

I^{(1)} = {θ^{1}, …, θ^{s}, θ_{10}, θ_{01}}.

By construction

ω^{1} ∧ ω^{2} = ω^{1} ∧ θ_{10}
ω^{3} ∧ ω^{4} = ω^{3} ∧ θ_{01}.

It follows that the algebraic ideal generated by the sections of I^{(1)} contains all of the forms in I. In particular, we have

dθ^{1} ≡ ⋯ ≡ dθ^{s} ≡ 0 mod {θ^{1}, …, θ^{s}, θ_{10}, θ_{01}}.

Moreover, since ω^{2} − h_{20} ω^{1} ≡ ω^{4} − h_{02} ω^{3} ≡ 0 mod {θ^{1}, …, θ^{s}, θ_{10}, θ_{01}} and since the 2-forms dω^{i} are well-defined on U, it follows that there are (unique) functions A^{i} so that

dω^{i} ≡ A^{i} ω^{1} ∧ ω^{3} mod {θ^{1}, …, θ^{s}, θ_{10}, θ_{01}}.

It follows that

\begin{matrix}dθ_{10} & ≡ −(dh_{20} + (A^{2} − h_{20}A^{1})ω^{3}) ∧ ω^{1} & ≡ −π_{20} ∧ ω^{1} \\ dθ_{01} & ≡ −(dh_{02} − (A^{4} − h_{02}A^{3})ω^{1}) ∧ ω^{3} & ≡ −π_{02} ∧ ω^{3} \end{matrix} \} mod \{θ^{1}, …, θ^{s}, θ_{10}, θ_{01}\}

where we have set π_{20} = dh_{20} + (A^{2} − h_{20}A^{1})ω^{3} and π_{02} = dh_{02} − (A^{4} − h_{02}A^{3})ω^{1}. It follows that on W the ideal I^{(1)} has the form

I^{(1)} = {θ^{1}, …, θ^{s}, θ_{10}, θ_{01}; π_{20} ∧ ω^{1}, π_{02} ∧ ω^{3} }.

Thus, I^{(1)} is a hyperbolic exterior differential system, as desired. □

We will let H_{s} denote the set of hyperbolic exterior differential systems of class s. Prolongation then gives a mapping

P : H_{s} → H_{s+2}.

This process can be iterated, leading to the successive higher prolongations, (M^{(k)}, I^{(k)}) ∈ H_{s+2k}.

We now want to remark on the relationship between the integral manifolds of I and those of I^{(1)}. Given any integral surface f : S → M of I, we will denote by f^{(1)} : S → M^{(1)} the canonical lift of f defined by the rule

f^{(1)}(s) = f′(s)(T_{s}S) = T_{f(s)}f(S).

1.3.2

6Here and elswhere, we adopt the common practice of writing simply ϕ instead of π^{∗}(ϕ) to denote the pullback via the submersion π : W → U whenever this abbreviation will not cause confusion. Since, in this case, W = U ×ℝ^{2} with h_{20} and h_{02} forming the coordinates on the ℝ^{2}-factor, confusion is almost impossible.

Since f is an immersion, so is f^{(1)}. Moreover, this latter map is clearly transverse to the fibers of the basepoint fibration π : M^{(1)} → M. By the tautological properties of the canonical system J on the Grassman bundle G_{2}(T M) it also follows that f^{(1)} : S → M^{(1)} is an integral surface of I^{(1)}.

Conversely, any integral surface g : S → M^{(1)} of I^{(1)} which is transverse to the fibers of π : M^{(1)} → M must be of the form g = f^{(1)} for some unique integral surface f : S → M of I; in fact, f = π∘g. Thus, the integral surfaces of I in M are in one-to-one correspondence with the integral surfaces of I^{(1)} in M^{(1)} which are transverse to the fibers of π. Obviously, this construction can be repeated, yielding integral surfaces f^{(k)} : S → M^{(k)} of I^{(k)} for all k ≥ 0.

1.3.2 Relations with the initial value problem. Regarding the initial value problem, it is important to note that a non-characteristic integral curve ϕ : C → M of I also has a canonical lifting to a non-characteristic integral curve ϕ^{(1)} : C → M^{(1)}. This lifting is defined by letting ϕ^{(1)}(t) be the unique 1-dimensional integral element of I which contains ϕ′(t)(T_{t}C). Again, every non-characteristic integral curve γ : C → M^{(1)} of I^{(1)} which is transverse to the fibers of π is of the form γ = ϕ^{(1)} where ϕ = π∘γ. As we shall see, this construction and its iterates ϕ^{(k)} : C → M^{(k)} will play an important role in Darboux’ method of integration.

1.3.3 Some examples. To illustrate the prolongation construction, consider the exterior differential system associated in the usual way to a second order hyperbolic equation^{7} as discussed in Example 7 in Section 1.1.3

F (x, y, z, p, q, r, s, t) = 0.

Here, we are following the classical notation: p = z_{x}, q = z_{y}, r = z_{xx}, s = z_{xy}, and t = z_{yy}. These are to be regarded as coordinates on the 2-jet space J^{2}(ℝ^{2}, ℝ) endowed with its canonical second order contact ideal J generated by the three 1-forms

θ^{0} = dz − p dx − q dy
θ^{1} = dp − r dx − s dy
θ^{2} = dq − s dx − t dy.

The pullback of this system to the 7-dimensional hypersurface M defined by the equation F = 0 will then be a hyperbolic exterior differential system (M, I) which lies in H_{3}.

In case the equation F = 0 happens to be a Monge-Ampere equation

(1)  E(rt − s^{2}) + Ar + 2Bs + Ct + D = 0

where A, B, C, D, E are functions of (x, y, z, p, q), we showed in the previous section how one may associate a hyperbolic exterior differential system (M_{0}, I_{0}) ∈ H_{1} to the equation. It turns out, as the reader may check, that the first prolongation of (M_{0}, I_{0}) ∈ H_{1} contains (M, I) ∈ H_{3} as a dense open set. More precisely, (M, I) is the open subset of (M_{0}^{(1)}, I_{0}^{(1)}) consisting of those integral elements on which dx∧dy ≠ 0.

Similarly, a first order hyperbolic system for two functions of two independent variables

(2)  \{ \begin{matrix}F (x, y, u, v, p, q, r, s) & = 0 \\ G(x, y, u, v, p, q, r, s) & = 0 \end{matrix}

where p = u_{x}, q = u_{y}, r = v_{x}, and s = v_{y} can always be expressed as a hyperbolic exterior differential system (M, I) ∈ H_{2}. However, in case (2) is a quasi-linear system, we saw in Section 1.1.3 that there is an associated hyperbolic exterior differential system (M_{0}, I_{0}) ∈ H_{0}. Again, it turns out that the open subset

1.3.3 of the first prolongation of this latter system consisting of the integral elements on which dx∧dy ≠ 0 is the exterior differential system (M, I) ∈ H_{2} canonically associated to (2).

7The general procedure for canonically associating an exterior differential system to a PDE system is explained in Chapter 1 of [BCG^{3}]. In the first example here, (x, y, z, p, q, r, s, t) are coordinates in the jet space J^{2}(ℝ^{2}, ℝ) and in the second example (x, y, u, v, p, q, r, s) are coordinates in J^{1}(ℝ^{2}, ℝ^{2}). In both cases the PDE defines a submanifold M of the jet space and the exterior differential system is generated as a differential system by the restriction to M of the contact system.

1.3.4 Partial Prolongations. Variations on the prolongation construction are possible and are ocassionally encountered in the theory. Suppose given a hyperbolic exterior differential system (M, I) with the property that the (ℙ^{1}∪ℙ^{1})-bundle Ξ → M can be written as a disjoint union

Ξ = M_{10}^{(1)} ∪ M_{01}^{(1)}

where each of M_{10}^{(1)} → M and M_{01}^{(1)} → M is a ℙ^{1}-bundle. (This can always be arranged by passing to a double cover of M if necessary.) Now, every 2-dimensional integral element E ⊂ T_{x}M of I can be written uniquely in the form E = L_{10} ⊕ L_{01} with L_{10} ∈ M_{10}^{(1)} and L_{01} ∈ M_{01}^{(1)}. It follows that there is a diagram of submersions

\begin{aligned}& & M^{(1)} \\ & ↙ & & ↘ \\ M_{10}^{(1)} & & & & M_{01}^{(1)} \\ & ↘ & & ↙ \\ & & M \end{aligned}  corresponding to  \begin{aligned}& & E \\ & ↙ & & ↘ \\ L_{10} & & & & L_{01} \\ & ↘ & & ↙ \\ & & x \end{aligned} .

It is not difficult to show that there exist canonically constructed hyperbolic exterior differential systems I_{10}^{(1)} and I_{01}^{(1)} on M_{10}^{(1)} and M_{01}^{(1)} respectively so that every integral surface f : S → M of I has canonical liftings f_{10}^{(1)} : S → M_{10}^{(1)} and f_{01}^{(1)} : S → M_{01}^{(1)} which are integral surfaces of I_{10}^{(1)} and I_{01}^{(1)} respectively. Thus, these partial prolongations are canonically defined and, for a system of class s, we have (M_{10}^{(1)}, I_{10}^{(1)}) and (M_{01}^{(1)}, I_{01}^{(1)}) in H_{s+1}.

Thus, hyperbolic exterior differential systems of classes s = 0, 1, 2, … form a very natural and interrelated set of exterior differential systems. Although we are primarily interested in the cases s = 0 and s = 1, it is convenient to set these in a general context as we have done.

1.3.5 The refined structure equations. We now want to derive more refined structure equations for the higher prolongations which will be fundamental in our later calculation of the conservation laws of (M^{s+4}, I). In fact, we will now show that the k^{th} prolongation satisfies a remarkable set of structure equations.

Proposition: Let (M, I) be a hyperbolic exterior differential system of class s. Then on the manifold M^{(k)} (of dimension s+4+2k) there is a coframing

θ^{1}, …, θ^{s} , θ_{10}, …, θ_{k0} , θ_{01}, …, θ_{0k} , π_{k+1,0}, ω_{10} , π_{0,k+1}, ω_{01}

such that I^{(k)} is generated as a differential ideal by θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k0}, θ_{01}, …, θ_{0k} which satisfies the structure equations

(3) (i) dθ_{k0} ≡ −π_{k+1,0} ∧ ω_{10} + T_{k0} ω_{01} ∧ θ_{01} mod {θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k0}} (ii) dθ_{0k} ≡ −π_{0,k+1} ∧ ω_{01} + T_{0k} ω_{10} ∧ θ_{10} mod {θ^{1}, …, θ^{s}, θ_{01}, …, θ_{0k}}.

We will discuss the meaning of these equations below. For now, we merely note that they immediately imply the earlier proposition from Section 1.3.1 to the effect that (M^{(k)}, I^{(k)}) is a hyperbolic system; i.e., dθ_{k0} and dθ_{0k} are decomposable modulo all of the θ’s. In fact, however, these structure equations go much further in that they give information on dθ_{k0} modulo only θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k0}; i.e., omitting θ_{01}, …, θ_{0k}. In later sections this fact (as well as others derivable from the above structure equations) will play an important role.

1.3.4 Proof : We are going to follow the notation of the earlier Proposition in Section 1.3.1. Recall that, by the definition of a hyperbolic system, we have, using Ω_{10} = ω_{10}∧θ_{10} and Ω_{01} = ω_{01}∧θ_{01}, that

(4)  dθ^{α} ≡ A_{10}^{α} ω_{10} ∧ θ_{10} + A_{01}^{α} ω_{01} ∧ θ_{01} mod {θ^{α}}.

For k = 1, equation (i) is

dθ_{10} ≡ −π_{20} ∧ ω_{10} + T_{10} ω_{01} ∧ θ_{01} mod {θ^{1}, …, θ^{s}, θ_{10}}.

Now Ω_{10} is defined on the original manifold M, and, since the 1-forms θ^{1}, …, θ^{s}, θ_{10}, ω_{10}, θ_{01}, ω_{01} are a basis for the semi-basic forms for the projection M^{(1)} → M, it follows that

dΩ_{10} ∈ Λ^{3}[θ^{1}, …, θ^{s}, θ_{10}, ω_{10}, θ_{01}, ω_{01}].

In fact, from the formula for dθ_{10} given above we have

dΩ_{10} ≡ −T_{10} ω_{10} ∧ ω_{01} ∧ θ_{01} mod {θ^{1}, …, θ^{s}, θ_{10}}.

Moreover, ω_{10} is actually defined on M = M^{(0)} (as opposed to θ_{10}, which is only defined on M^{(1)} even though it is semi-basic for the projection M^{(1)} → M^{(0)}). Thus we have

dω_{10} ∈ Λ^{2}[θ^{1}, …, θ^{s}, θ_{10}, ω_{10}, θ_{01}, ω_{01}].

Now using a similar argument for dω_{01} yields the formulae

(5)  \begin{matrix}dω_{10} ≡ & S_{10} ω_{01} ∧ θ_{01} & mod \{θ^{1}, …, θ^{s}, θ_{10}, ω_{10}\} \\ dω_{01} ≡ & S_{01} ω_{10} ∧ θ_{10} & mod \{θ^{1}, …, θ^{s}, θ_{01}, ω_{01}\}. \end{matrix}

From (5) and

dΩ_{10} = dω_{10} ∧ θ_{10} − ω_{10} ∧ dθ_{10}

we may infer that there exists a 1-form π_{20} on M^{(1)} such that

dθ_{10} ≡ −π_{20} ∧ ω_{10} mod Λ^{2}[θ^{1}, …, θ^{s}, θ_{10}, ω_{10}, θ_{01}, ω_{01}].

Thus

dθ_{10} ≡ −π_{20} ∧ ω_{10} + η ∧ ω_{10} + T_{10} ω_{01} ∧ θ_{01} mod {θ^{1}, …, θ^{s}, θ_{10}}

where η is a linear combination of θ_{01} and ω_{01}. Replacing π_{20} by π_{20} − η and relabeling we have

dθ_{10} ≡ −π_{20} ∧ ω_{10} + T_{10} ω_{01} ∧ θ_{01} mod {θ^{1}, …, θ^{s}, θ_{10}}.

This establishes (3) for the case k = 1.

Assuming by induction that (3) holds for some k ≧ 1, we shall establish it for k + 1.

On integral 2-planes E of I^{(k)} we have

π_{k+1,0} ∧ ω_{10}|_{E} = 0
ω_{10}|_{E} ≠ 0,

so that M^{(k+1)} is locally M^{(k)} × ℝ^{2} where ℝ^{2} has coordinates (h_{k+2,0}, h_{0,k+2}) and where

θ_{k+1,0} = π_{k+1,0} − h_{k+2,0} ω_{10}
θ_{0,k+1} = π_{0,k+1} − h_{0,k+2} ω_{01}.

1.3.5 In fact, the vanishing of θ_{k+1,0} and θ_{0,k+1} defines the 2-planes on which π_{k+1,0}∧ω_{10} and π_{0,k+1}∧ω_{01} restrict to zero but ω_{10}∧ω_{01} ≠ 0. From (1.i) we have on M^{(k+1)}

dθ_{k0} ≡ −θ_{k+1,0} ∧ ω_{10} + T_{k0} ω_{01} ∧ θ_{01} mod {θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k0}}.

We shall write this as

(6)  dθ_{k0} ≡ −θ_{k+1,0} ∧ ω_{10} + T_{k0} Ω_{01} mod {θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k0}}.

Taking the exterior derivative of this equation and substituting (1.i) for dθ_{j0} for j = 1, …, k−1 together with (6) gives

(7)  0 ≡ −dθ_{k+1,0} ∧ ω_{10} + χ ∧ ω_{01} ∧ θ_{01} mod {θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k+1,0}}.

We want to say a word about χ. It arises from the terms dT_{k0}∧Ω_{01} and T_{k0} dΩ_{01} and the coefficients of ω_{01}∧θ_{01} in dθ^{1}, …, dθ^{s}, dθ_{10}, …, dθ_{k0}. Since

dΩ_{01} ∈ Λ^{3}[θ_{10}, ω_{10}, θ_{01}, ω_{01}] mod {θ^{1}, …, θ^{s}}

we have

dΩ_{01} ≡ a ω_{10} ∧ ω_{01} ∧ θ_{01} mod {θ^{1}, …, θ^{s}, θ_{10}}.

From (7) we deduce that

χ ≡ 0 mod {θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k+1,0}, θ_{01}, ω_{10}, ω_{01}}

and hence we infer the existence of a form π_{k+2,0} on M^{(k+1)} such that

dθ_{k+1,0} ≡ −π_{k+2,0} ∧ ω_{10} + T_{k+1,0} ω_{01} ∧ θ_{01} mod {θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k+1,0}}.

This completes the induction step in the proof of (3). □

1.3.6 Interpretations—higher characteristic systems. We would like to discuss some of the meaning of the structure equations (3). First, we give an important definition.

Definition: The k^{th} characteristic systems of a hyperbolic system are the Pfaffian systems^{8}

Ξ_{10}^{(k)} = [θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k0}, π_{k+1,0}, ω_{10}]
Ξ_{01}^{(k)} = [θ^{1}, …, θ^{s}, θ_{01}, …, θ_{0k}, π_{0,k+1}, ω_{01}].

As explained above, each integral surface f : S → M of the original hyperbolic system has its k^{th}- prolongation f^{(k)} : S → M^{(k)} which is an integral surface of I^{(k)}, the hyperbolic exterior differential system generated as a differential system by θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k0}, θ_{01}, …, θ_{0k} on M^{(k)} (here k ≧ 1). It follows from (3) that

π_{k+1,0} ∧ ω_{10} = 0
Ξ_{10}^{(k)} = [θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k+1,0}, ω_{10}]
Ξ_{01}^{(k)} = [θ^{1}, …, θ^{s}, θ_{01}, …, θ_{0,k+1}, ω_{01}].

1.3.6 on f^{(k)}(S), so that Ξ_{10}^{(k)} induces on f^{(k)}(S) a Pfaffian system of rank one, and similarly for Ξ_{01}^{(k)}. Of course, these induce the same two characteristic foliations on solution surfaces to the hyperbolic exterior differential system as the 0^{th} characteristic systems. We will see in Section 1.4 that the geometry—meaning the derived flags, etc.—of the characteristic systems is an important feature of a hyperbolic exterior differential system.

8The Pfaffian system Ξ_{10}^{(k)} and Ξ_{01}^{(k)} are defined on M^{(k)}. When lifted to M^{(k+l)} for any l ≧ 1 they are given by

One interpretation of the structure equations (3) is that the characteristic systems Ξ_{10}^{(k)} and Ξ_{01}^{(k)} are coupled only at the first level. To explain this, we pass to the infinite prolongation M^{(∞)} with coframing^{9}

θ^{1}, …, θ^{s}; θ_{10}, θ_{20}, …; θ_{01}, θ_{02}, …; ω_{10}, ω_{01}

and write the structure equations (3) as

(8)  \{ \begin{matrix}(i) & dθ_{k0} ≡ −θ_{k+1,0} ∧ ω_{10} & + T_{k0} Ω_{01} & mod \{θ^{1}, …, θ^{s}, θ_{10}, …, θ_{k0}\} \\ (ii) & dθ_{0k} ≡ −θ_{0,k+1} ∧ ω_{01} & + T_{0k} Ω_{10} & mod \{θ^{1}, …, θ^{s}, θ_{01}, …, θ_{0k}\}. \end{matrix}

The characteristic systems are

Ξ_{10}^{(∞)} = [θ^{1}, …, θ^{s}, θ_{10}, θ_{20}, …, ω_{10}]
Ξ_{01}^{(∞)} = [θ^{1}, …, θ^{s}, θ_{01}, θ_{02}, …, ω_{01}],

and (5) and (8) say that the characteristic systems Ξ_{10}^{(∞)}, Ξ_{01}^{(∞)} are Frobenius up to the coupling terms S_{10} Ω_{01}, S_{01} Ω_{10} and T_{k0} Ω_{01}, T_{0k} Ω_{10}. It will turn out that the coefficients T_{k0}, T_{0k} are the fundamental relative invariants of hyperbolic systems of class s = 0.

In the case s = 0 the characteristic systems are disjoint and span all of the 1-forms on M^{(∞)}

Ξ_{10}^{(∞)} ∪ Ξ_{01}^{(∞)} = Ω^{1}(M^{(∞)})
Ξ_{10}^{(∞)} ∩ Ξ_{01}^{(∞)} = (0).

Moreover, as we shall see in Section 1.5 below, there are many interesting cases where S_{10} = S_{01} = 0 (cf. (5) above). In this case, it follows that: If all the T_{k0} and T_{0k} vanish, then each of Ξ_{10}^{(∞)} and Ξ_{01}^{(∞)} are Frobenius systems. We point this out here to emphasize the importance of these relative invariants.

1.3.7 Even further refinements. The structure equations discussed above are rather general and apply to hyperbolic systems of any class s. For the low values of s and with more information on the structure of the original ideal I, we can introduce further refinements.

Beginning in Section 1.5, we will concentrate mainly on the case s = 0 and will introduce considerable refinements via the method of equivalence. In the rest of this section, however, we want to comment on how they may also be refined when s = 1. Such a system is given on a 5-manifold locally as I = {θ, Ω_{10}, Ω_{01}}. Equation (4) simplifies to

dθ ≡ A_{10} Ω_{10} + A_{01} Ω_{01} mod {θ}.

If A_{10} = A_{01} = 0, then the 5-manifold M is foliated in codimension 1 by the leaves of the system θ = 0. Each integral manifold of I lies in one of these leaves and the system essentially reduces to a 1-parameter family of hyperbolic systems of class s = 0 (cf. Section 1.4 below).

If, say, A_{10} = 0 but A_{01} ≠ 0, then we may normalize so as to have the equation dθ ≡ Ω_{01} mod {θ}. In this case, the non-characteristic initial value problem for the system can be solved using only ODE techniques. Here is how this goes. Since θ∧dθ = θ∧Ω_{01} ≠ 0 but θ∧(dθ)^{2} = 0, it follows from the Pfaff-Darboux theorem

1.3.7 that every point of M lies in some open set U on which there exists a submersion f : U → ℝ^{3} so that θ restricted to U is a multiple of f^{∗}(dz −y dx). If ϕ : (0, 1) → U is a non-characteristic integral curve of I, then in particular it follows that f ∘ϕ : (0, 1) → ℝ^{3} is an immersion and that P = f^{−1}(f∘ϕ((0, 1))) ⊂ U is a smooth 3-manifold which is an integral manifold of θ and hence of Ω_{01}. Moreover, P clearly contains ϕ((0, 1)). The 2-form Ω_{10} restricts to P to have a Cauchy characteristic and the assumption that ϕ is non-characteristic implies that ϕ((0, 1)) is transverse to this characteristic line field. It follows that the union of these Cauchy characteristics passing through ϕ((0, 1)) is an integral surface of I passing through ϕ((0, 1)). (Of course, by uniqueness, there is only one such integral surface.)

9The infinite prolongation (M^{(∞)}, I^{(∞)}) of (M, I) is discussed in Section 2 of [BG_{1}].

Finally, if A_{10}A_{01} ≠ 0, we may normalize so as to have A_{10} = A_{01} = 1; i.e.,
(9)  dθ ≡ Ω_{10} + Ω_{01} mod {θ}.

It is easy to show that such a system is locally equivalent to the exterior differential system derived from a hyperbolic Monge-Ampere system as in Example 5 in Section 1.1.3 (cf. Appendix 2 to Section 2 in [BG_{2}]). The k^{th} prolongation then has a coframing

θ; θ_{10}, …, θ_{k0}, ω_{10}, π_{k+1,0}; θ_{01}, …, θ_{0k}, ω_{01}, π_{0,k+1}

satisfying the structure equations (3,i-ii) and it may also be shown that this coframing can be chosen so that, in addition to (9), we have

(10)  \begin{aligned}dθ_{k0} & ≡ −π_{k+1,0} ∧ ω_{10} mod \{θ, θ_{10}, …, θ_{k0}\} \\ dθ_{0k} & ≡ −π_{0,k+1} ∧ ω_{01} mod \{θ, θ_{01}, …, θ_{0k}\}. \end{aligned}

Thus, all of the invariants T_{k0} and T_{0k} vanish identically and the coupling between the characteristic systems only occurs at the 0^{th} level through equation (9).

1.3.8 1.4 Integration by the method of Darboux

1.4.1 Riemann invariants. Let (M, I) be a hyperbolic system of class s. We have discussed how to associate to such a system its prolongations (M^{(k)}, I^{(k)}), which are hyperbolic systems of class s + 2k, and its characteristic systems

Ξ_{10}^{(k)}, Ξ_{01}^{(k)} ⊂ Ω^{1}(M^{(k)}),

which are Pfaffian systems of rank s+2+k. The solution surfaces of (M, I) and the solution surfaces of (M^{(k)}, I^{(k)}) are in one-to-one correspondence (where the latter must satisfy a transversality condition), and the characteristic systems induce on each solution surface a pair of foliations by characteristic curves.

As we mentioned in Section 1.2, the characteristic foliations on each solution surface carry no local geometry. However, the characteristic systems Ξ_{10}^{(k)} and Ξ_{01}^{(k)} do, indeed, carry local geometry. In fact the integrability properties of the characteristic systems in the ambient manifolds M^{(k)} play a crucial role in the classical integration methods. In particular, the study of integrable subsystems of the characteristic systems turns out to be very fruitful, and leads directly to the method of Darboux ([Go]), which we describe below.

First, a brief historical perspective. In the early days—beginning over two centuries ago—a primary interest was finding explicit solutions of PDEs. Somewhat later the issue was to prove existence either by finding explicit solutions or by giving on algorithm, based on integration, for finding solutions (cf. the Poisson integral formula, etc.). Soon thereafter, this form of existence proof was extended to the nineteenth century concept of “integration of the equation”, which meant to reduce finding the solution to (at most) solving a sequence of ordinary differential equations. In this connection the following notion arose.

Definition : A (generalized) Riemann invariant of I is a codimension one foliation F on M^{(k)} such that, for any function f constant on the leaves of F (we say that f belongs to F), its differential df lies in either Ξ_{10}^{(k)} or Ξ_{01}^{(k)}.

The importance of Riemann invariants stems from the following fact. Let f belong to a Riemann invariant F of I, with, say df ∈ Ξ_{10}^{(k)}, and let ϕ: S → M be an integral surface of I. Then the pull back function f∘ϕ^{(k)}: S → ℝ clearly has the property that it is constant on each of the curves in the characteristic foliation F_{10}. Thus, f functions as a sort of “conservation law” for characteristic curves on solutions. The knowledge of such functions (when they exist) frequently yields important information about the behavior of solutions to the original system.

We should perhaps say a word about our choice of the fundamental object in the notion of a Riemann invariant. One usually thinks of a Riemann invariant as a function of the variables and their derivatives which is constant on the leaves of one of the characteristic foliations on every solution of the equation. However, if f is a Riemann invariant in this more classical sense, then any function of f, say g ∘ f where g : ℝ → ℝ is arbitrary, is also a Riemann invariant, with essentially the same level sets as f. However, it turns out that it is the level sets of f rather than the function f itself which is important in applications. On the open set where df is non-vanishing, knowing these level sets is equivalent to knowing a codimension one foliation. For this reason, we have taken the foliation as the fundamental object, preferring to identify Riemann invariants which determine the same level sets.

The above definition generalizes the classical situations in which the notion of a Riemann invariant is often discussed: a first order hyperbolic system for two unknown functions u(x, y) and v(x, y) and a second order hyperbolic equation for one unknown z(x, y). These give hyperbolic systems of classes s = 2 and s = 3 respectively, and the classical Riemann invariants in these cases belong to the special case k = 0. Their use in producing integral formulas for solutions to certain hyperbolic equations is standard and well known [CH].

Before proceeding, we should like to discuss the possibility that a function f belonging to a Riemann invariant F might be constant on all solution surfaces. Recall that a hyperbolic system I is given by a rank s Pfaffian system I together with a pair of 2-forms Ω_{10} and Ω_{01} that are well defined and decomposable

1.4.1 modulo {I}. The Pfaffian system I induces a rank s Pfaffian system, still denoted by I, on each M^{(k)}. This abuse of notation is justified for our present purposes, since it is easy to see that integrable subsystems of I on M^{(0)} and I on M^{(k)} are in one-to-one correspondence.^{10}

Recalling our notation I^{⟨i⟩} for the i^{th} derived system of the Pfaffian system I, let I^{⟨∞⟩} = ∩_{i}I^{⟨i⟩} be the

largest integrable subsystem of I. Since I^{⟨∞⟩} is an integrable subsystem of I, locally M is a product

M = N × U

where U ⊂ ℝ^{n} (n = rank I^{⟨∞⟩}) has coordinates t = (t^{1}, …, t^{n}). It is easy to see that I induces on each N × {t} a hyperbolic system I_{t} such that the solution surfaces of I are just the solution surfaces of some I_{t} for a fixed t. Thus, if I^{⟨∞⟩} ≠ 0 we essentially have the situation of a family of hyperbolic systems depending on a parameter.

For this reason we shall make the standing assumption that the infinite derived system I^{⟨∞⟩} is trivial. Since it is well known that the infinite derived system I^{⟨∞⟩} contains any integrable subsystem of I, it follows from our assumption that I contains no integrable subsystems. Under this assumption we have the result.

Proposition : The Pfaffian systems I^{(k)} contain no integrable subsystems for k ≥ 0.

Proof : This follows immediately since the structure equations in the last section clearly imply that, for all k ≥ 0,

I^{(k)}^{⟨1⟩} = I^{(k−1)}.

Note that I^{(0)} = I and now we apply induction to conclude that I^{(k)}^{⟨∞⟩} = 0. □

This result implies that no non-constant Riemann invariant can be constant on all integral surfaces of I. To see this, note that, if f is a Riemann invariant of I defined on M^{(k)} and if df_{x} ≠ 0, then df is not a section of I^{(k)} on a neighborhood of x, and hence that there must exist a nearby point y ∈ M^{(k)} so that df_{y} ∉ I_{y}^{(k)}. In particular, df_{y} does not vanish on (I_{y}^{(k)})^{⊥} and hence there exists a 2-dimensional integral element E ⊂ (I_{y}^{(k)})^{⊥} of I^{(k)} on which df_{y} does not vanish. Now f will not be constant on any integral surface S of I which passes through y and satisfies T_{y}S = E. (Such an integral surface always exists by the existence theorem in Section 1.2.3.)

The classical theory of Riemann invariants, as well as the even more classical but less well-known theories of Ampere, Monge, Laplace, Darboux, Goursat, etc (cf. [Go]), thus focuses attention on integrable subsystems of the characteristic systems. We shall see that studying the conservation laws does the same thing.

1.4.2 The method of Darboux. We now introduce the main concept from the classical theory.

Definition : A hyperbolic system (M, I) is integrable in the sense of Darboux at level k if there are rank 2 integrable subsystems

\{ \begin{matrix}Δ_{10} ⊂ Ξ_{10}^{(k)} \\ Δ_{01} ⊂ Ξ_{01}^{(k)} \end{matrix}

θ(x, y, dx) = ∑_{α} f_{α}(x, y)θ^{α}(x, dx), x ∈ U, y ∈ V

dθ = 0.

1.4.2 which satisfy Δ_{10} ∩ I^{(k)} = Δ_{01} ∩ I^{(k)} = (0).

10The point is this: In a domain U ⊂ ℝ^{n} with coordinates x = (x^{1}, …, x^{n}), let θ^{α} = θ^{α}(x, dx) be a set of everywhere linearly independent one forms. Suppose that we have another connected domain V ⊂ ℝ^{n} with coordinates y = (y^{1}, …, y^{n})

and a linear combination satisfying Then the f_{α}(x, y) are constant in y.

We will now explain the method of Darboux for solving the initial value problem for hyperbolic systems which satisfy this hypothesis.

Theorem: Suppose that (M, I) is integrable in the sense of Darboux at level k and that f : (0, 1) → M is a non-characteristic integral curve of I. Then there exists an open set U ⊂ ℝ^{2} which contains the diagonal interval Δ = {(t, t) : 0 < t < 1} and an integral surface F : U → M of I so that F (t, t) = f(t) for all 0 < t < 1. Moreover, F can be chosen so that the characteristic foliations induced on U are the coordinate slices and this makes F unique on some neighborhood of Δ. Finally, F can be constructed from f by a procedure involving ordinary differential equations alone.

Proof: Since f is non-characteristic, the tangent space T_{f(t)}f((0, 1)) lies in a unique 2-dimensional element of I which we shall denote by f^{(1)}(t) ∈ M^{(1)}. Now, f^{(1)} : (0, 1) → M^{(1)} satisfies our hypothesis with respect to (M^{(1)}, I^{(1)}), i.e., it is a non-characteristic integral immersion of I^{(1)}. By repeating this process, we eventually arrive at f^{(k)} : (0, 1) → M^{(k)}, which is a non-characteristic integral immersion of I^{(k)}. Clearly, it now suffices to prove the theorem for the case k = 0, and then apply it to the initial data f^{(k)}.

Thus, let us suppose that Δ_{10} ⊂ Ξ_{10} and Δ_{01} ⊂ Ξ_{01} are rank 2 integrable subsystems which satisfy Δ_{10} ∩ I = Δ_{10} ∩ I = (0). Fix a t_{0} ∈ (0, 1) and let x, y, u, and v be functions on a neighborhood V of f(t_{0}) which have the property that, on V, the system Δ_{10} is spanned by dx and dy while Δ_{01} is spanned by du and dv. It now follows from the structure equations and the very definitions of Ξ_{10} and Ξ_{01} that I in V is generated by the sections of I and the two 2-forms dx∧dy and du∧dv. Note that, also, by construction, for each p ∈ V the space I_{p}^{⊥} is transverse to the fibers of the the submersion (x, y, u, v) : V → ℝ^{4}.

Now, there is a a δ > 0 so that 0 < t_{0}−δ < t_{0}+δ < 1 and so that f((t_{0}−δ, t_{0}+δ)) ⊂ V. By the assumption that f is non-characteristic, f is transverse to the fibers of both of the mappings (x, y) : V → ℝ^{2} and (u, v) : V → ℝ^{2}. It follows that, by shrinking δ, we may assume that (x, y) ∘ f and (u, v) ∘ f are smooth embeddings of the interval (t_{0}−δ, t_{0}+δ) into ℝ^{2}. Set

Γ_{10} = (x, y)∘f((t_{0}−δ, t_{0}+δ))  and  Γ_{01} = (u, v)∘f((t_{0}−δ, t_{0}+δ))

These are two smooth curves in the xy- and uv-planes respectively. Set

M_{Γ} = (x, y, u, v)^{−1}(Γ_{10} × Γ_{01}) ⊂ V ⊂ M.

Now M_{Γ} is a smooth manifold of dimension s + 2 which contains f((t_{0}−δ, t_{0}+δ)). Moreover, by constuction, since the image of M_{Γ} under the (x, y) map and the (u, v) map is a curve, it follows that dx∧dy and du∧dv both vanish identically on M_{Γ}. In particular, it follows that the pull back of the system I to M_{Γ} is generated algebraically by the sections of I. In other words, I pulls back to be a rank s integrable system I_{Γ} on M_{Γ}.

It follows that there is a unique 2-dimensional leaf L_{Γ} of I_{Γ} which contains the curve f((t_{0}−δ, t_{0}+δ)). (Note that the construction of the leaves of an integrable Pfaffian system can be accomplished by solving ODE alone, PDE methods are not required.) This surface L_{Γ} is, by construction, an integral surface of I which contains f((t_{0}−δ, t_{0}+δ)). It is also transverse to the fibers of the submersion M_{Γ} → Γ_{10} × Γ_{01}. As a result, by shrinking δ once more, we may assume that L_{Γ} maps diffeomorphically onto the open square

(x, y)∘f((t_{0}−δ, t_{0}+δ)) × (u, v)∘f((t_{0}−δ, t_{0}+δ)).

We now define F on the open square (t_{0}−δ, t_{0}+δ) × (t_{0}−δ, t_{0}+δ) by letting F (s, t) be the point of L_{Γ} which maps to

(x(f(s)), y(f(s)), u(f(t)), v(f(t))).

Local uniqueness of this extension is easily established and then an elementary patching argument shows that F can be defined on an entire neighborhood of the “diagonal” interval. □

1.4.3

It is important to remark that finding the integrable subsystems of the characteristic systems is a routine matter. In fact, one merely computes the derived systems Ξ_{10}^{(k)⟨j⟩} and Ξ_{01}^{(k)⟨j⟩}. Since we clearly have, for some j and j′,

Ξ_{10}^{(k)} = Ξ_{10}^{(k)⟨0⟩} ⊇ Ξ_{10}^{(k)⟨1⟩} ⊇ Ξ_{10}^{(k)⟨2⟩} ⋯ ⊇ Ξ_{10}^{(k)⟨j⟩} = Ξ_{10}^{(k)⟨j+1⟩} = Ξ_{10}^{(k)⟨∞⟩}
Ξ_{01}^{(k)} = Ξ_{01}^{(k)⟨0⟩} ⊇ Ξ_{01}^{(k)⟨1⟩} ⊇ Ξ_{01}^{(k)⟨2⟩} ⋯ ⊇ Ξ_{01}^{(k)⟨j′⟩} = Ξ_{01}^{(k)⟨j′+1⟩} = Ξ_{01}^{(k)⟨∞⟩}

and since any integrable subsystem of a Pfaffian system lies inside its last derived system, the test for integrability by the method of Darboux can be carried out effectively, using only differentiations and algebraic manipulations.

1.4.3 An example—the f-Gordon equation. Our main interest in Darboux integrability in this paper is its effect on the computation of conservation laws (to be defined in Section 2.1 below). However, classically, the importance of Darboux’ method was that it often lead to explicit formulas for solutions to important PDEs. We will illustrate this by an example drawn from the classical literature.

First, however, we would like to remark that the integration of the Monge-Ampere equation z_{xx} z_{yy} − z_{xy}^{2} = −1 accomplished in Example 2 of Section 1.2 was done by the method of Darboux, for this equation happens to be integrable by the method of Darboux at level 0, as can be seen immediately by the formulae

\begin{aligned}Ξ_{10} & = \{dz − p dx − q dy, d(p − y), d(q + x) \} \\ Ξ_{01} & = \{dz − p dx − q dy, d(p + y), d(q − x) \}. \end{aligned}

This is, of course, not surprising because, as we saw in Section 1.1, this system is (globally) equivalent to the hyperbolic system generated by the classical wave equation z_{xy} = 0.

We now turn to a more interesting example. Consider the so-called f-Gordon equation z_{xy} = f(z). As this is an equation of Monge-Ampere type, we may construct a corresponding hyperbolic system of the form (ℝ^{5}, I) ∈ H_{1} where, with the usual notation,

I = {dz − p dx − q dy, (dp − f(z) dy) ∧ dx, (dq − f(z) dx) ∧ dy}.

The characteristic systems are

\begin{aligned}Ξ_{10} & = \{dz − p dx − q dy, dp − f(z) dy, dx \} \\ Ξ_{01} & = \{dz − p dx − q dy, dq − f(z) dx, dy \}. \end{aligned}

We may now compute the derived flag for these characteristic systems. Assuming that f′(z) ≠ 0, which we shall for the rest of this section, the only integrable subsystems of Ξ_{10} and Ξ_{01} are of rank one and are generated by dx and dy, respectively.

We will now determine the conditions on f that the f-Gordon equation be integrable by the method of Darboux at level one. As explained in Section 1.3, the first prolongation (M^{(1)}, I^{(1)}) of the exterior differential system (M, I) associated to the f-Gordon equation is obtained locally by introducing new coordinates r (= h_{20}) and t (= h_{02}) and setting

\begin{aligned}θ_{10} & = dp − f(z) dy & − r dx \\ θ_{01} & = dq − f(z) dx & − t dy. \end{aligned}

From

\begin{matrix}dθ_{10} & ≡ −(dr & − f′(z)p dy) ∧ dx \\ dθ_{01} & ≡ −(dt & − f′(z)q dx) ∧ dy \end{matrix}\} mod θ

1.4.4 we infer that

\begin{matrix}Ξ_{10}^{(1)} = & [dx, dz − q dy, dp − f(z) dy, dr & − f′(z)p dy] \\ Ξ_{01}^{(1)} = & [dy, dq − p dx, dq − f(z) dx, dt & − f′(z)q dx]. \end{matrix}

First, we compute that Ξ_{10}^{(1)⟨1⟩} = [dx, dp − f(z) dy, dr − f′(z)p dy]. Now it is not difficult to compute that

\begin{matrix}d(dp − f(z) & dy) ≡ −f′(z) & dz ∧ dy \\ d(dr − f′(z)p & dy) ≡ −f′′(z)p & dz ∧ dy \end{matrix}\} mod Ξ_{10}^{(1)⟨1⟩}

while of course d(dx) = 0. Since we have assumed that f′ is non-vanishing, we see that

Ξ_{10}^{(1)⟨2⟩} = [ dx, (dr − f′(z)p dy) − (f′′(z)/f′(z))p (dp − f(z) dy)]
= [ dx, dr − (f′′(z)/f′(z))p dp − (f′(z)^{2} − f′′(z)f(z))p/f′(z) dy].

We can now compute that in order for the third derived system to have rank two, it is necessary and sufficient that f′(z)^{2} − f′′(z)f(z) = 0. The general solution of this relation is

f(z) = A e^{Bz}

for some constants A and B. Since we are assuming that f′(z) is non-zero, neither A nor B can vanish. By scaling z and x and y, we may then easily reduce to the case

f(z) = ez.

In this case

\begin{aligned}Ξ_{10}^{(1)⟨2⟩} = Ξ_{10}^{(1)⟨∞⟩} & = [dx, dr & − p dp] & = [dx, & d(r & − \frac{1}{2}p^{2})] \\ Ξ_{01}^{(1)⟨2⟩} = Ξ_{01}^{(1)⟨∞⟩} & = [dy, dt & − q dq] & = [dy, & d(t & − \frac{1}{2}q^{2})]. \end{aligned}

This proves the classical result that Liouville’s equation z_{xy} = e^{z} is integrable by the method of Darboux.

In fact, it is a theorem of Lie [Li] that, up to local equivalence, this equation and the wave equation z_{xy} = 0 are the only f-Gordon equations integrable by the method of Darboux at any level.

The explicit expressions above show that on solution surfaces we have

\begin{aligned}d & (r − \frac{1}{2}p^{2}) ∧ dx & = 0 \\ d & (t − \frac{1}{2}q^{2}) ∧ dy & = 0. \end{aligned}

These relations may be used to show that the general solution of the Liouville equation on any rectangle of the form

R = { (x, y) ∈ ℝ^{2} | a < x < b, c < y < d }

can be written in the form

(1)  ez = \frac{2X′(x)Y′(y)}{(X(x) + Y (y))^{2}}

where X : (a, b) → ℝ and Y : (c, d) → ℝ are arbitrary smooth maps subject to the open conditions that X′(x)Y′(y) > 0 and X(x)+Y (y) > 0 for all (x, y) ∈ R. The calculation is a little complicated, perhaps because the more fundamental equation is the s = 0 Liouville system

\begin{aligned}u_{y} & = e^{v} \\ v_{x} & = e^{u}. \end{aligned}

1.4.5 As we shall show in the Section 1.5, this latter system is more easily integrated by the method of Darboux, giving (1) above as an immediate consequence.

Finally, we would like to mention an example, (cf. Chapter 3 of Darboux [Da]), of an equation integrable by the method of Darboux at level k but not at level k − 1. This is the linear equation

z_{xy} = −\frac{k(k + 1)}{(x − y)^{2}} z

defined on the half-plane x > y. For k not an integer, it turns out that this equation cannot be integrated by the method of Darboux, but for k ≥ 0 an integer, it turns out that applying the method of Darboux yields the general solution in the form

z(x, y) = (x − y)^{k+1} \frac{∂^{2k}}{∂x^{k} ∂y^{k}} (\frac{X(x) − Y (y)}{x − y})

where X and Y are arbitrary functions of one variable.

1.4.4 Semi-integrability by Darboux’ method. In closing this section, we want to remark that there is actually a generalization of the method of Darboux which only depends on there being a transverse rank 2 integrable subsystem in one of the characteristic systems in order to solve the initial value problem via ODE methods. Since this will not play an important role in this paper, we only give a sketch of the method.

Let us say that a hyperbolic system (M, I) is semi-integrable in the sense of Darboux if there exists an integrable rank 2 subsystem Δ ⊂ Ξ_{10} which satisfies Ξ_{10} = Δ ⊕ I. In this case, it easily follows that any point of M lies in an open set U on which I can be generated in the form

I = {θ^{1}, …, θ^{s}; dx ∧ dy, ω^{3} ∧ ω^{4}}.

Suppose, now that we are given a non-characteristic integral curve ϕ : (0, 1) → U of I. The hypothesis that ϕ be non-characteristic implies that the map (x, y) ∘ ϕ : (0, 1) → ℝ^{2} is an immersion. By shrinking domains appropriately, we can assume that this map is an embedding, which we shall. Set Γ = (x, y) ∘ ϕ((0, 1)) ⊂ ℝ^{2}, and let

M_{Γ} = (x, y)^{−1}(Γ).

Then M_{Γ} is a smooth hypersurface in M which contains ϕ((0, 1)). Let I_{Γ} denote the induced exterior differential system on M_{Γ}. If we let an overbar denote the pull back of forms from M to M_{Γ}, then we see that

I_{Γ} = {θ̄^{1}, …, θ̄^{s}; ω̄^{3} ∧ ω̄^{4}}.

It follows that there exists a non-zero vector field X on M_{Γ} (unique up to scalar multiples) which satisfies

θ̄^{1}(X) = ⋯ = θ̄^{s}(X) = ω̄^{3}(X) = ω̄^{4}(X) = 0.

(The reader familiar with the theory of exterior differential systems will recognize X as spanning the Cauchy characteristic distribution on I_{Γ}.) Note that, by our assumption that ϕ be non-characteristic, the 1-forms ϕ^{∗}(ω^{3}) and ϕ^{∗}(ω^{4}) do not vanish simultaneously at any point of (0, 1). Thus, it follows that for every t, the vectors X(ϕ(t)) and ϕ′(t)(∂/∂t) are linearly independent. It then follows that there is an open neighborhood R of (0, 1) × {0} in (0, 1) × (−1, 1) so that the mapping f : R → M defined by

f(t, s) = exp_{sX}(ϕ(t)), (t, s) ∈ R

is well-defined and an immersion. By its very construction, f is an integral surface of I which solves the initial value problem for ϕ.

Clearly, this method can be generalized to the case where, for some k ≥ 0, the ideal (M^{(k)}, I^{(k)}) is semi-integrable by the method of Darboux. Thus, in all of these cases, the initial value problem can be solved by ODE methods.

1.4.6 1.5 Structure equations for hyperbolic systems of class s = 0

From the general theory of conservation laws for exterior differential systems (cf. [BG_{1}]), it is known that the symbol of an exterior differential system I determines an algebraic normal form for its conservation laws. The space C of conservation laws of I is then isomorphic to the space of closed forms in this algebraic normal form. In Chapter 2 of this paper, we shall need to do some rather explicit computations for hyperbolic systems of class s = 0 using this normal form. In this section, as preparation for those computations, we are going to use the extra assumption s = 0 to refine the general structure equations derived in Section 1.3 above.

1.5.1 Symmetry and non-degeneracy. Recall that, according to our definitions, a hyperbolic system I of class s = 0 on a 4-manifold M is locally generated by a pair of decomposable 2-forms Ω_{10} and Ω_{01} on M which are defined up to non-zero multiples and with the property that Ω_{10}∧Ω_{01} ≠ 0. As usual, we shall let Ξ_{10} and Ξ_{01} denote the characteristic systems.

We first want to say a word about the “generality” of hyperbolic systems with s = 0. Working locally, we can imagine a hyperbolic system with s = 0 as a pair of 2-plane distributions on a neighborhood of the origin in ℝ^{4}. Since the bundle G_{2}(T ℝ^{4}) of 2-planes in the tangent spaces at points of ℝ^{4} is a smooth bundle of fiber dimension 4, it follows that, locally, a choice of a hyperbolic system with s = 0 depends on the choice of 4 + 4 = 8 functions of four variables, these functions being subject only to some open conditions which ensure that the two distributions are transverse. On the other hand, we want to identify two such hyperbolic systems if they differ only by some diffeomorphism of ℝ^{4}. Since a local diffeomorphism of ℝ^{4} depends on a choice of 4 functions of four variables, it seems reasonable to guess that the “moduli space” of equivalence classes of local hyperbolic systems with s = 0 modulo diffeomorphisms “depends” on 8 − 4 = 4 arbitrary functions of four variables. In particular, we should expect there to be differential invariants attached to a hyperbolic system with s = 0 just as the Riemannian curvature tensor is attached to a Riemannian metric. In the next subsection, we will develop a mechanism for computing these invariants, analogous to the construction of the Levi-Civita connection and its curvature in Riemannian geometry.

The simplest hyperbolic exterior differential system of class s = 0 occurs when both Ξ_{10} and Ξ_{01} are integrable, and we will henceforth refer to this case as the trivial case. In the trivial case, we may choose local coordinates (x, y, u, v) such that the system I is generated by

Ω_{10} = du ∧ dx  and  Ω_{01} = dv ∧ dy.

This is the exterior differential system arising from the (trivial) PDE system u_{y} = v_{x} = 0, whose solutions are u = u(x) and v = v(y). This is essentially the s = 0 version of the classical wave equation in characteristic coordinates.

Before going on to study the non-trivial cases, we want to first remark on a general hypothesis that we will be assuming in order to simplify the exposition. Strictly speaking, we are considering a structure which has slightly more information than a hyperbolic system with s = 0. In fact, we are also imposing a choice of which of the two characteristic systems we want to call Ξ_{10} and which we want to call Ξ_{01}. Thus, we might think of the structure we are studying primarily as a pair Ξ = (Ξ_{10}, Ξ_{01}) of transverse rank 2 Pfaffian systems on M^{4}. In some sense, the pair ^{∗}Ξ = (Ξ_{01}, Ξ_{10}) should be thought of as the “opposite” structure, the operation Ξ ↦ ^{∗}Ξ defining an involution on the space of structures that we are studying.

From the structure equations below, we will extract certain so-called “relative invariants” of the pair Ξ. In modern terminology, a relative invariant is a “natural” section σ_{Ξ} of a “natural” line bundle L_{Ξ} associated to Ξ where “natural” means that, whenever one has a diffeomorphism f : M_{1} → M_{2} which induces an isomorphism f^{∗}(Ξ_{2}) = Ξ_{1} of hyperbolic systems with s = 0, there is also canonically determined an isomorphism f^{∗}L_{Ξ_{2}} = L_{Ξ_{1}} which satisfies f^{∗}σ_{Ξ_{2}} = σ_{Ξ_{1}}. (We will give examples of such invariants in the next subsection.)

1.5.1

In the present case, the involution Ξ ↦ ^{∗}Ξ clearly exchanges each relative invariant with an ‘opposite’ relative invariant ^{∗}σ. Thus, the relative invariants are naturally grouped into pairs (or, ocassionally, singlets when a relative invariant happens to be its own opposite).

Definition: A hyperbolic exterior differential system of class s = 0 is symmetric in case the relative invariants in each pair (σ, ^{∗}σ) are either both zero or both non-zero.

For example, for a symmetric system, the integrability properties of the two characteristic systems will be the same (in particular, the derived flags will have the same ranks). For simplicity, we shall concentrate on symmetric systems in this paper. Thus, henceforth, we shall assume that all of our systems are symmetric in this sense.

In the non-trivial case, our general assumption that the system be symmetric implies that each of Ξ_{10} and Ξ_{01} will be non-integrable.^{11} We shall say that a hyperbolic system with s = 0 is non-degenerate if this condition is satisfied. For the rest of this section, we shall assume that all of our hyperbolic systems are non-degenerate.

1.5.2 A G-structure for the non-degenerate case and its invariants. Let us now assume that we have a non-degenerate hyperbolic system (M, I) of class s = 0. The first derived system of each characteristic system then has rank 1. This suggests that we consider local coframings (η^{1}, η^{2}, η^{3}, η^{4}) on M satisfying the conditions

(1)  Ξ_{10} = { η^{1}, η^{2} }  Ξ_{01} = { η^{3}, η^{4} }

and

(2)  dη^{1} ≡ 0 mod η^{1}, η^{2}  dη^{3} ≡ 0 mod η^{3}, η^{4}.

Thus, η^{1} and η^{3} span the first derived systems of the characteristic systems Ξ_{10} and Ξ_{01}, respectively. If (η̄^{1}, η̄^{2}, η̄^{3}, η̄^{4}) is another coframing on the same domain with properties (1) and (2), then the formula for transition between the two coframings takes the form

( \begin{matrix} & η̄^{1} \\ & η̄^{2} \\ & η̄^{3} \\ & η̄^{4} \end{matrix} ) = ( \begin{matrix}a_{1}^{1} & 0 & 0 & 0 \\ a_{1}^{2} & a_{2}^{2} & 0 & 0 \\ 0 & 0 & a_{3}^{3} & 0 \\ 0 & 0 & a_{3}^{4} & a_{4}^{4} \end{matrix} ) ( \begin{matrix}η^{1} \\ η^{2} \\ η^{3} \\ η^{4} \end{matrix} )

where the a_{j}^{i} are arbitrary subject to the obvious condition that a_{i}^{i} ≠ 0 (or else the transition matrix would not be invertible).

Now, for any coframing which satisfies (1) and (2), there must exist functions A and C so that

(3)  dη^{2} ≡ A η^{3} ∧ η^{4} mod η^{1}, η^{2}  dη^{4} ≡ C η^{1} ∧ η^{2} mod η^{3}, η^{4}.

By our assumptions, Ξ_{10} and Ξ_{01} are non-integrable. It follows from this and (2) that neither A nor C can vanish. Using this, we can construct a coframing, say (η̄^{1}, η̄^{2}, η̄^{3}, η̄^{4}) = (C η^{1}, η^{2}, A η^{3}, η^{4}) which satisfies properties (1) and (2) and also the equations Ā = C̄ = 1. We shall say that a coframing is 1-adapted to I if it satisfies the conditions (1), (2), and (3) with A = C = 1.

1.5.2

11In any case, if one of the systems is integrable, then the system is semi-integrable in the sense of Darboux and, as we pointed out in Section 1.4, the initial value problem be solved by ODE methods anyway.

If η = (η^{1}, η^{2}, η^{3}, η^{4}) is a 1-adapted coframing on a domain U ⊂ M, then any other coframing on U, say η̄ = (η̄^{1}, η̄^{2}, η̄^{3}, η̄^{4}) is seen to be 1-adapted to I if and only if there exist functions a_{1}^{2}, a_{3}^{4}, a_{2}^{2} ≠ 0, and a_{4}^{4} ≠ 0 on U so that

( \begin{matrix} & η̄^{1} \\ & η̄^{2} \\ & η̄^{3} \\ & η̄^{4} \end{matrix} ) = ( \begin{matrix}a_{4}^{4}/a_{2}^{2} & 0 & 0 & 0 \\ a_{1}^{2} & a_{2}^{2} & 0 & 0 \\ 0 & 0 & a_{2}^{2}/a_{4}^{4} & 0 \\ 0 & 0 & a_{3}^{4} & a_{4}^{4} \end{matrix} ) ( \begin{matrix}η^{1} \\ η^{2} \\ η^{3} \\ η^{4} \end{matrix} ) .

Such “transition matrices” take values in a certain 4-dimensional lower triangular subgroup of GL(4, ℝ) which we shall henceforth denote G. Thus, the local coframings which are 1-adapted to I are the local sections of a principal G-bundle B → M which is a subbundle of the bundle of all coframes of M. In other words, B is a G-structure on M in the usual sense. We will refer to B as the G-structure associated to (or determined by) the non-degenerate hyperbolic exterior differential system I. Moreover, one can clearly recover I from a knowledge of B.

Now we shall apply the equivalence method^{12} to the G-structure B in order to understand its invariants. Accordingly, we write the structure equations on B in the form

(4)  d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}ϕ_{44}−ϕ_{22} & 0 & 0 & 0 \\ ϕ_{21} & ϕ_{22} & 0 & 0 \\ 0 & 0 & ϕ_{22}−ϕ_{44} & 0 \\ 0 & 0 & ϕ_{43} & ϕ_{44} \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}T^{1} \\ T^{2} \\ T^{3} \\ T^{4} \end{matrix} )

where, in the terminology of the equivalence method, the ϕ_{ij} are the pseudo-connection forms and the T^{i} are the torsion terms (which are semi-basic^{13}). These forms are not uniquely determined by these equations, and, following the usual method of equivalence, we now want to understand how modifications of the pseudoconnection forms can be employed to simplify the torsion terms.

Now, by the defining properties of the G-structure B, we have

dω^{1} ≡ 0  mod ω^{1}, ω^{2}
dω^{2} ≡ ω^{3} ∧ ω^{4} mod ω^{1}, ω^{2}
dω^{3} ≡ 0  mod ω^{3}, ω^{4}
dω^{4} ≡ ω^{1} ∧ ω^{2} mod ω^{3}, ω^{4}

so

T^{1} ≡ 0  mod ω^{1}, ω^{2}
T^{2} ≡ ω^{3} ∧ ω^{4} mod ω^{1}, ω^{2}
T^{3} ≡ 0  mod ω^{3}, ω^{4}
T^{4} ≡ ω^{1} ∧ ω^{2} mod ω^{3}, ω^{4}.

It follows that there exist 1-forms χ_{1}, χ_{2}, χ_{3}, and χ_{4} which are linear combinations of the ω^{i} so that

\begin{aligned}T^{2} & = ω^{3} ∧ ω^{4} + χ_{1} ∧ ω^{1} + χ_{2} ∧ ω^{2}, \\ T^{4} & = ω^{1} ∧ ω^{2} + χ_{3} ∧ ω^{3} + χ_{4} ∧ ω^{4}. \end{aligned}

The equations for dω^{2} and dω^{4} can therefore be written in the form

\begin{aligned}dω^{2} & = −(ϕ_{21} − χ_{1}) ∧ ω^{1} − (ϕ_{22} − χ_{2}) ∧ ω^{2} + ω^{3} ∧ ω^{4} \\ dω^{4} & = −(ϕ_{43} − χ_{3}) ∧ ω^{3} − (ϕ_{44} − χ_{4}) ∧ ω^{4} + ω^{1} ∧ ω^{2}. \end{aligned}

It follows that we may assume that the ϕ_{ij} have been chosen so that

T^{2} = ω^{3} ∧ ω^{4}  and  T^{4} = ω^{1} ∧ ω^{2},

1.5.3 so we assume this from now on. This condition still does not determine the ϕ_{ij} since making the replacements

12The general equivalence method is explained in Appendix 1 to Section 2 of [BG_{2}]. Fortunately, however, the full complexity of the method will not be needed in this simple case.

13I.e., these terms have the form T^{i} = T_{jk}^{i} ω^{j} ∧ω^{k} for some functions T_{jk}^{i} = −T_{kj}^{i} on B.

( \begin{matrix}ϕ_{21} \\ ϕ_{22} \\ ϕ_{43} \\ ϕ_{44} \end{matrix} ) ↦ ( \begin{matrix}ϕ_{21} + a_{1} ω^{1} + a_{2} ω^{2} \\ ϕ_{22} + a_{2} ω^{1} + a_{3} ω^{2} \\ ϕ_{43} + c_{3} ω^{3} + c_{4} ω^{4} \\ ϕ_{44} + c_{4} ω^{3} + c_{1} ω^{4} \end{matrix} )

in the above equations will clearly not affect T^{2} or T^{4}. However, the above congruences on T^{1} and T^{3} imply that

\begin{aligned}T^{1} & ≡ T_{13}^{1} ω^{1} ∧ ω^{3} + T_{14}^{1} ω^{1} ∧ ω^{4} mod ω^{2} \\ T^{3} & ≡ T_{31}^{3} ω^{3} ∧ ω^{1} + T_{32}^{3} ω^{3} ∧ ω^{2} mod ω^{4} \end{aligned}

and the above replacements can be chosen so that T_{13}^{1} = T_{14}^{1} = T_{31}^{3} = T_{32}^{3} = 0. Note that the only replacements of the above form which preserve these latter conditions are ones with a_{2} = a_{3} = c_{4} = c_{1} = 0.

The upshot of this discussion is that, for the G-structure we have associated to a non-degenerate hyperbolic system with s = 0, there is a choice of pseudo-connection so that the structure equations take the form

(5)  ( \begin{matrix}dω^{1} \\ dω^{2} \\ dω^{3} \\ dω^{4} \end{matrix} ) = − ( \begin{matrix}ϕ_{44}−ϕ_{22} & 0 & 0 & 0 \\ ϕ_{21} & ϕ_{22} & 0 & 0 \\ 0 & 0 & ϕ_{22}−ϕ_{44} & 0 \\ 0 & 0 & ϕ_{43} & ϕ_{44} \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}ω^{2}∧(p_{1} ω^{1}+p_{3} ω^{3}+p_{4} ω^{4}) \\ ω^{3}∧ω^{4} \\ ω^{4}∧(q_{3} ω^{3}+q_{1} ω^{1}+q_{2} ω^{2}) \\ ω^{1}∧ω^{2} \end{matrix} ) .

Moreover, with the structure equations in this form, the 1-forms ϕ_{22} and ϕ_{44} are unique, the form ϕ_{21} is determined up to the addition of a multiple of ω^{1}, and the form ϕ_{43} is determined up to the addition of a multiple of ω^{3}. As we shall see in the next subsection, no further reduction of this G-structure can be made without making some non-vanishing assumptions on the invariants.

To complete the discussion of the structure equations, it will be necessary to compute their “Bianchi identities” by differentiating the equations in (5).^{14} We will not give the details of the calculations (which are straightforward, if tedious), but shall describe the results. First of all, differentiation of the equations (5) and reduction of the results modulo various combinations of the ω^{i} shows that there are relations of the form

(6)  \begin{matrix}dp_{1} & ≡ p_{1} ϕ_{22} & − q_{2} ϕ_{43} \\ dp_{3} & ≡ p_{3} (3ϕ_{22}−2ϕ_{44}) & + p_{4} ϕ_{43} \\ dp_{4} & ≡ p_{4} (2ϕ_{22}) \\ dq_{3} & ≡ q_{3} ϕ_{44} & − p_{4} ϕ_{21} \\ dq_{1} & ≡ q_{1} (3ϕ_{44}−2ϕ_{22}) & + q_{2} ϕ_{21} \\ dq_{2} & ≡ q_{2} (2ϕ_{44}) \end{matrix} \} mod ω^{1}, ω^{2}, ω^{3}, ω^{4}.

We shall use the notation ∇p_{4} to mean dp_{4} − 2p_{4} ϕ_{22}, i.e., the semi-basic part of the exterior derivative of p_{4}, and similarly for the other quantities.

If we now introduce “curvature” 2-forms Φ_{22}, Φ_{44}, Φ_{21}, and Φ_{43} by the equations

(7)  \begin{aligned}dϕ_{22} & = −ϕ_{21} ∧ (p_{3} ω^{3} + p_{4} ω^{4}) + q_{3} ω^{1} ∧ ω^{2} + \frac{1}{2}p_{3}q_{1} ω^{2} ∧ ω^{4} + Φ_{22} \\ dϕ_{44} & = −ϕ_{43} ∧ (q_{1} ω^{1} + q_{2} ω^{2}) + p_{1} ω^{3} ∧ ω^{4} + \frac{1}{2}q_{1}p_{3} ω^{4} ∧ ω^{2} + Φ_{44} \\ dϕ_{21} & = −ϕ_{21} ∧ (ϕ_{44} − 2ϕ_{22} − p_{1} ω^{2}) + Φ_{21} \\ dϕ_{43} & = −ϕ_{43} ∧ (ϕ_{22} − 2ϕ_{44} − q_{3} ω^{4}) + Φ_{43}, \end{aligned}

1.5.4 the exterior derivatives of the equations (5) become

14The reader may want to skip the remainder of this subsection on first reading, instead going on directly to Section 1.5.3.

These “Bianchi” calculations are somewhat technical and will be more meaningful once the reader can see that they are needed.

(8)

0 = (Φ_{22} − Φ_{44}) ∧ ω^{1} − (∇p_{1} ∧ ω^{1} + ∇p_{3} ∧ ω^{3} + ∇p_{4} ∧ ω^{4} − q_{3}p_{3} ω^{3} ∧ ω^{4}) ∧ ω^{2}
0 = −Φ_{21} ∧ ω^{1} − Φ_{22} ∧ ω^{2} − ω^{1} ∧ ω^{2} ∧ ω^{3}
0 = (Φ_{44} − Φ_{22}) ∧ ω^{3} − (∇q_{3} ∧ ω^{3} + ∇q_{1} ∧ ω^{1} + ∇q_{2} ∧ ω^{2} − p_{1}q_{1} ω^{1} ∧ ω^{2}) ∧ ω^{4}
0 = −Φ_{43} ∧ ω^{3} − Φ_{44} ∧ ω^{4} − ω^{3} ∧ ω^{4} ∧ ω^{1}.

(Note that because ϕ_{21} and ϕ_{43} are not canonical, the expression ∇p_{3} is actually only well-defined modulo ω^{3}. However, since this term only occurs wedged with ω^{3}, the resulting term is well defined. A similar comment applies to the other ambiguities caused by the ambiguity in the pseudo-connection.) The identities (8) now give relations among the coefficients of the derivatives of the primary invariants (i.e., the torsion coefficients) and the curvature coefficients. It is not useful to write these out here; the form (8) will suffice for our purposes.

A little exterior algebra shows that the relations (8) imply that Φ_{22} and Φ_{44} are semi-basic 2-forms, i.e., they are quadratic expressions in the ω^{i}. In fact,

(9)  Φ_{22} = −κ_{1} ∧ ω^{1} − κ_{2} ∧ ω^{2}  and  Φ_{44} = −κ_{3} ∧ ω^{3} − κ_{4} ∧ ω^{4}

where

κ_{i} = k_{ij}ω^{j}

for functions k_{ij}, suitably skew-symmetrized so as to be well-defined. Using this plus some more exterior algebra, it follows that there are 1-forms ϕ_{211} and ϕ_{433} so that

(10)  \begin{aligned}Φ_{21} & = −(κ_{1} − ω^{3}) ∧ ω^{2} − ϕ_{211} ∧ ω^{1} \\ Φ_{43} & = −(κ_{3} − ω^{1}) ∧ ω^{4} − ϕ_{433} ∧ ω^{3}. \end{aligned}

1.5.3 Relative invariants. In order to interpret the coefficients in the torsion terms, it is important to understand how the quantities p_{1}, p_{3}, p_{4} and q_{1}, q_{2}, q_{3} vary on the fibres of B → M. This information can be read off from the relations (6).

It follows from (6) that p_{4} and q_{2} are what is known in the classical literature as relative invariants—i.e., they are well-defined as sections of suitable line bundles over M. For example, the expression σ = p_{4} (ω^{3}∧ω^{4})^{2} is a well-defined section of the square of the determinant bundle of the characteristic system Ξ_{01}. Its opposite (in the sense of the involution discussed in Section 1.5.1) is ^{∗}σ = q_{2} (ω^{1}∧ω^{2})^{2}, a section of the ‘opposite’ bundle, i.e., the square of the determinant bundle of the characteristic system Ξ_{10}. Moreover, calculation using (6) shows that, for example, τ = (p_{4}q_{1}+q_{2}q_{3}) (ω^{1}∧ω^{2})^{3} is a well-defined section of the cube of the determinant bundle of Ξ_{10} with ‘opposite’ invariant ^{∗}τ = (q_{2}p_{3}+p_{4}p_{1}) (ω^{3}∧ω^{4})^{3}.

Thus, even though p_{4} and q_{2} are not well-defined on M, their zero loci make sense on M. Note also that if p_{4} (resp. q_{2}) vanishes identically, then p_{3} and q_{3} (resp. q_{1} and p_{1}) become relative invariants, a fact to which we shall return later.

1.5.4 Structure reduction in the generic case. According to (6), on the open set M^{∗} ⊂ M which is the complement of the zero loci of the relative invariants p_{4} and q_{2}, there exists a G_{1}-substructure B_{1} ⊂ B defined by the equations

(p_{4})^{2} = 1,  (q_{2})^{2} = 1,  p_{3} = 0,  q_{1} = 0

where G_{1} is the group consisting of the diagonal matrices in G whose diagonal entries are each ±1. Thus, up to a finite group ambiguity (of order 4), this G_{1}-structure defines a canonical coframing on M^{∗}. The invariants of this coframing are then invariants of the original hyperbolic system. Note that any symmetry of the

1.5.5 hyperbolic system I must preserve the open set M^{∗} and, on this open set, must preserve the G_{1}-structure B_{1}. Since preserving this latter structure is essentially equivalent to preserving a coframing on M^{∗}, it follows that the group of symmetries of I on M^{∗} is of dimension at most 4.

Conversely, it is not hard to show that hyperbolic systems satisfying these conditions with a 4-parameter symmetry group do exist: Let H be a Lie group of dimension 4 and choose any basis (η^{1}, …, η^{4}) of its leftinvariant 1-forms. Form the ideal I = {η^{1}∧η^{2}, η^{3}∧η^{4}}. This will yield a hyperbolic system which clearly does have (at least) a 4-parameter symmetry group. If the group H is sufficiently “generic” among 4-dimensional Lie groups and the basis (η^{i}) is chosen sufficiently generically, it can be shown that the resulting system will have its relative invariants p_{4} and q_{2} be non-zero, so that its group of symmetries is exactly of dimension 4.

What the corresponding PDE look like is a very interesting question. Also particularly interesting is the problem of knowing whether the ideals (H, I) are complete for the initial value problem in the sense of Section 1.2.3.

1.5.5 Normal forms. We now want to interpret the vanishing of the torsion coefficients in (5) in terms of integrability of various bundles intrinsically associated to the original hyperbolic system I and use this to derive (local) normal forms in various special cases. We will then use the structure equations to develop a test for ‘linearizability’ and to classify the systems which are, in some sense, the most “homogeneous” among non-degenerate hyperbolic systems with s = 0.

We begin our first interpretation by noting that the rank 2 Pfaffian system Θ = {ω^{1}, ω^{3}} spanned by the first derived systems of Ξ_{10} and Ξ_{01} is well-defined. Indeed, from the structure equations, the 2-form

Ω = ω^{1} ∧ ω^{3}

itself is well-defined, since the scalings of ω^{1} and ω^{3} cancel. Of course, Ω is a relative invariant, being a section of the determinant bundle of Θ and having the property that ^{∗}Ω = −Ω. The integrability of Ω has the following interpretation:

Proposition: For any non-degenerate hyperbolic system with s = 0, the system Θ = {ω^{1}, ω^{3}} is Frobenius if and only if p_{4} = q_{2} = 0. Moreover, I has the property that p_{4} and q_{2} vanish identically if and only if every point of M has a neighborhood U on which there exists a coordinate system (x, y, u, v) : U → ℝ^{4} and functions A, B, C, and D on U satisfying AB ≠ 1, C_{v} ≠ 0, D_{u} ≠ 0 as well as A_{v} = B_{u} = 0 so that, on U,

I = { (du − C dy) ∧ (dx − A dy), (dv − D dx) ∧ (dy − B dx) }.

Proof: The fact that the system Θ is differentially closed if and only if p_{4} and q_{2} vanish is immediate from the structure equations (5). It remains to verify that the promised coordinate system exists and has the properties claimed for it. First, since Θ is Frobenius, it follows that every point of M has a neighborhood U on which there exist functions x and y so that Θ = {dx, dy}. Thus, for a 1-adpated coframing on this neighborhood, we must have η^{1} and η^{3} be linear combinations of dx and dy. By making a linear change of variables in x and y, we can assume that η^{1}∧dy and η^{3}∧dx are non-zero. It follows that there are functions A and B so that η^{1} is a multiple of dx − A dy and η^{3} is a multiple of dy − B dx. Since η^{1}∧η^{3} is non-zero, it follows that AB ≠ 1.

Next, the systems {η^{1}, η^{2}, η^{3}} and {η^{1}, η^{3}, η^{4}} are clearly Frobenius on U, so there must be functions u and v so that

{η^{1}, η^{2}, η^{3}} = {dx, dy, du}  and  {η^{1}, η^{3}, η^{4}} = {dx, dy, dv}.

It follows that there are functions C and D so that

Ξ_{10} = {η^{1}, η^{2}} = {dx − A dy, du − C dy}  and  Ξ_{01} = {η^{3}, η^{4}} = {dy − B dx, dv − D dx}.

1.5.6 Now, by (2), it follows that A_{v} = B_{u} = 0, and the non-degeneracy assumption implies that neither C_{v} nor D_{u} can vanish.

Finally, any hyperbolic system which locally can be put in the form we have just derived is clearly a non-degenerate hyperbolic system (with s = 0) for which Θ is Frobenius. □

As we remarked before, in the case where p_{4} and q_{2} vanish identically, so that Θ is integrable, then p_{3} and q_{1} become relative invariants. If we have information about their vanishing or non-vanishing, we can refine the normal form given above:

Proposition: Suppose that (M^{4}, I) is a non-degenerate hyperbolic system satisfying the condition that p_{4} and q_{2} vanish identically. Any point in the open set in M where p_{3} and q_{1} are non-zero lies in a neighborhood on which there exists a coordinate system (x, y, u, v) : U → ℝ^{4} and two functions C and D with C_{v} and D_{u} non-zero so that, on U,

I = { (du − C dy) ∧ (dx − u dy), (dv − D dx) ∧ (dy − v dx) }.

On the other hand, p_{3} and q_{1} vanish identically on a neighborhood of a point in M if and only if that point lies in a neighborhood U with coordinates and functions C and D as above so that, on U

I = { (du − C dy) ∧ dx, (dv − D dx) ∧ dy }.

Proof: Construct a local coordinate system of the type guaranteed by the first proposition. Because p_{4} = q_{2} = 0, the structure equations imply ω^{1}∧dω^{1} = p_{3} ω^{1}∧ω^{2}∧ω^{3} and ω^{3}∧dω^{3} = q_{1} ω^{3}∧ω^{4}∧ω^{1}.

Suppose first, that p_{3} and q_{1} are non-zero at a point in M (and hence on a neighborhood of this point), then using the fact that ω^{1} = λ (dx−A dy) for some non-zero function λ, we compute that ω^{1}∧dω^{1} = λ^{2} dx∧dy∧dA = p_{3} ω^{1}∧ω^{2}∧ω^{3}. It follows that {ω^{1}, ω^{2}, ω^{3}} = {dx, dy, dA} and hence that we may take u = A. Applying the same argument to ω^{3}, and using the assumption that q_{1} ≠ 0, we see that we may take v = B. This is the first statement we wanted to prove.

On the other hand, if p_{3} and q_{1} vanish identically on a neighborhood of the point in question, then these structure equations imply that {ω^{1}} and {ω^{3}} are each integrable separately. It then follows that we can choose our initial functions x and y so that ω^{1} is a multiple of dx and ω^{3} is a multiple of dy. The rest of the construction proceeds as before. □

The coordinate systems constructed in the course of the proofs of the above two propositions are not canonical. However, an examination of the proofs shows that the ambiguity in the choice of coordinates is, first, the choice of x and y subject to the condition that Ω = λ dx∧dy, which involves a choice of 2 arbitrary functions of two variables, and then a choice of u and v subject to conditions which determine each of these two functions up to a choice of an arbitrary function of three variables. Thus, the “coordinate ambiguity” (sometimes known as the “gauge group”) in the above normal forms depends only on functions of three variables. Since the normal forms involve 2 arbitrary functions of four variables (namely, C and D), it is reasonable to say that the “moduli space” of hyperbolic systems with p_{4} = q_{2} = 0 “depends” on 2 arbitrary functions of four variables. This is not entirely unexpected, of course, because the conditions p_{4} = q_{2} = 0 represent two conditions on the 4 arbitary functions of four variables on which the “moduli space” of general hyperbolic systems with s = 0 “depends”. What is, perhaps, surprising is that imposing the further conditions p_{3} = q_{1} = 0 does not lower this “generality”. The space of such structures still “depends” on 2 arbitrary functions of four variables.

We would also like to note that the normal form of the first proposition is quite useful for doing calculations. For example, it is easy to calculate that, in an open set U with local coordinates as in that

1.5.7 proposition, the coframing

\begin{aligned}η^{1} & = D_{u} & (dx − A dy) \\ η^{2} & = & (du − C dy) \\ η^{3} & = C_{v} & (dy − B dx) \\ η^{4} & = & (dv − D dx) \end{aligned}

is 1-adpated to I. Thus, for example, we have the following simple formula:

Ω = (1 − AB)D_{u}C_{v} dx ∧ dy.

In particular, Ω will be closed, a condition equivalent to the conditions p_{4} = q_{2} = p_{1} = q_{3} = 0, if and only if

((1 − AB)D_{u}C_{v})_{u} = ((1 − AB)D_{u}C_{v})_{v} = 0.

In other words (1 − AB)D_{u}C_{v} must be a (non-zero) function of x and y only. Note that this condition still leaves 1 arbitrary function of four variables free (either D or C can still be chosen arbitrarily subject only to the condition that D_{u}C_{v} ≠ 0), another suprising “function count”, given that the closure of Ω is four conditions on the invariants. Even imposing the further condition that p_{3} = q_{1} = 0, so that we can take A = B = 0 in the normal form, still leaves C and D subject only to the single condition that D_{u}C_{v} should be a function of x and y only.

1.5.6 Linear systems. As another example of the use of the invariants of B to understand normal forms, we want to give a characterization of linear systems of PDE in terms of these invariants. Before stating the characterization, it is useful to first get an idea of what we might want to prove by computing the invariants for the general linear first order hyperbolic system for two functions of two variables.

Consider the general such PDE system

A ( \begin{matrix}u_{x} \\ v_{x} \end{matrix} ) + B ( \begin{matrix}u_{y} \\ v_{y} \end{matrix} ) + C ( \begin{matrix}u \\ v \end{matrix} ) = ( \begin{matrix}0 \\ 0 \end{matrix} ) ,

where A, B and C are 2-by-2 matrices with entries which are functions of x and y. The assumption that this system is hyperbolic is equivalent to the condition that A and B be everywhere linearly independent matrices which are simultaneously diagonalizable, i.e., there should exist invertible matrices P and Q (with entries which are functions of x and y) so that

PAQ = ( \begin{matrix}a_{1} & 0 \\ 0 & a_{2} \end{matrix} )  and  PBQ = ( \begin{matrix}b_{1} & 0 \\ 0 & b_{2} \end{matrix} )

(of course, a_{1}b_{2} − a_{2}b_{1} ≠ 0). This implies that we can make a change of dependent variables u and v, writing

( \begin{matrix}u \\ v \end{matrix} ) = Q ( \begin{matrix}z \\ w \end{matrix} )

and then the above equations reduce to the form

( \begin{matrix}a_{1} z_{x} + b_{1} z_{y} \\ a_{2} w_{x} + b_{2}w_{y} \end{matrix} ) + P^{−1}CQ ( \begin{matrix}z \\ w \end{matrix} ) = 0.

Now, every point of the xy-plane has a neighborhood on which there exist coordinates s and t so that

a_{1} \frac{∂}{∂x} + b_{1} \frac{∂}{∂y} = a \frac{∂}{∂t}  and  a_{2} \frac{∂}{∂x} + b_{2} \frac{∂}{∂y} = b \frac{∂}{∂s}

1.5.8 for some non-zero functions a and b.^{15} Making this change of coordinates, the above equation takes the form

( \begin{matrix}z_{t} \\ w_{s} \end{matrix} ) = ( \begin{matrix}c_{11} & c_{12} \\ c_{21} & c_{22} \end{matrix} ) ( \begin{matrix}z \\ w \end{matrix} )

where the c_{ij} are functions of s and t. Now setting c_{11} = −f_{s} and c_{22} = −g_{t} and then defining u = e^{f}z and v = e^{g}w, the system reduces to the form

u_{t} = P v  and  v_{s} = Qu

for some functions P and Q of s and t.

The corresponding exterior differential system on M = ℝ^{4} with coordinates s, t, u, and v is generated by

I = {(du − P v dt) ∧ ds, (dv − Qu ds) ∧ dt}.

It is easy to see that this is a non-degenerate system if and only if P (s, t) and Q(s, t) are non-zero. By reversing s and t if necessary, we may even assume that P and Q are positive, which we shall do from now on.

In order to understand the form of the invariants for the general first order linear hyperbolic system for two functions of two variables, it is therefore enough to understand the form of the invariants for this latter system. Note that the second proposition above implies that such a system must have all of its primary invariants equal to zero: p_{4} = q_{2} = p_{3} = q_{1} = p_{1} = q_{3} = 0. Let B → M be the G-structure associated to the above I. Consider the 1-adapted coframing η with components

\begin{aligned}η^{1} & = & Q & ds \\ η^{2} & = & ( & du − P v dt) \\ η^{3} & = & P & dt \\ η^{4} & = & ( & dv − Qu ds) \end{aligned}

as a (global) section of B. Of course, we have η^{∗}(ω^{i}) = η^{i} by the tautological properties of the forms ω^{i}. Let us use φ_{ij} to denote η^{∗}(ϕ_{ij}). The first and third structure equations give

\begin{aligned}(φ_{22} − φ_{44}) ∧ Q ds & = (φ_{22} − φ_{44}) ∧ η^{1} = dη^{1} = Q_{t} dt ∧ ds \\ (φ_{44} − φ_{22}) ∧ P dt & = (φ_{44} − φ_{22}) ∧ η^{3} = dη^{3} = P_{s} ds ∧ dt \end{aligned}

while the second and fourth coupled with the above formulae give

\begin{aligned}−φ_{22} ∧ η^{2} & = dη^{2} + φ_{21} ∧ η^{1} − η^{3} ∧ η^{4} & ≡ 0 mod η^{1} \\ −φ_{44} ∧ η^{4} & = dη^{4} + φ_{43} ∧ η^{3} − η^{1} ∧ η^{2} & ≡ 0 mod η^{3} \end{aligned}

which, in particular, imply that φ_{22} = f_{221}η^{1} + f_{222}η^{2} and φ_{44} = f_{443}η^{3} + f_{333}η^{4} for some functions f_{iij}. Substituting these formulae into the preceding formulae and then solving yields

φ_{22} = −(P_{s}/P) ds  and  φ_{44} = −(Q_{t}/Q) dt.

From this, we get the formulae

η^{∗}(Φ_{22}) = (log P)_{st} ds ∧ dt  and  η^{∗}(Φ_{44}) = −(log Q)_{st} ds ∧ dt.

1.5.9 In particular, it follows that in (9), we must have k_{ij} = 0 for all ij-pairs except possibly 13 and 31. It is this last observation which provides the key to characterizing linear systems.

15Just choose s with ds ≠ 0 to be constant on the flow lines of the first vector field and t with dt ≠ 0 to be constant on the flow lines of the second vector field.

Proposition: A non-degenerate hyperbolic system (M^{4}, I) satisfies p_{4} = q_{2} = p_{3} = q_{1} = p_{1} = q_{3} = 0 and Φ_{22}+Φ_{44} = F ω^{1}∧ω^{3} for some function F if and only if I is locally the hyperbolic system associated to a linear first order hyperbolic system for two functions of two variables. In particular, if I satisfies these hypotheses, then every point of M lies in a coordinate chart (x, y, u, v) : U → ℝ^{4} in which I has generators of the form

I = {(du − e^{2λ}v dy) ∧ dx, (dv − e^{2μ}u dx) ∧ dy}

for some functions λ and μ of x and y alone.

Proof: We have already shown that a hyperbolic system which arises from a linear first order hyperbolic PDE system for two functions of two variables satisfies these invariant conditions. It remains to prove the converse. Consider the structure equations (5-10), substituting the identities p_{4} = q_{2} = p_{3} = q_{1} = p_{1} = q_{3} = 0. The equations (8) reduce to

0 = (Φ_{22} − Φ_{44}) ∧ ω^{1}
0 = −Φ_{21} ∧ ω^{1} − Φ_{22} ∧ ω^{2} − ω^{1} ∧ ω^{2} ∧ ω^{3}
0 = (Φ_{44} − Φ_{22}) ∧ ω^{3}
0 = −Φ_{43} ∧ ω^{3} − Φ_{44} ∧ ω^{4} − ω^{3} ∧ ω^{4} ∧ ω^{1}.

The first and third of these equations together imply that Φ_{22}−Φ_{44} is a multiple of both ω^{1} and ω^{3} and hence of ω^{1}∧ω^{3}. On the other hand, by hypothesis, Φ_{22}+Φ_{44} = F ω^{1}∧ω^{3}, thus implying

\begin{aligned}dϕ_{22} & = k_{13} ω^{1} ∧ ω^{3} \\ dϕ_{44} & = k_{31} ω^{3} ∧ ω^{1}. \end{aligned}

We are now going to show that every point of M lies in a local coordinate system as in the statement of the Proposition. To produce this coordinate system, we proceed as follows: First, note that because the primary invariants are zero, it follows that ω^{1} and ω^{3} are separately integrable. Now fix a point of B and choose functions x and y on a neighborhood V of the point so that ω^{1} is a multiple of dx and ω^{3} is a multiple of dy. By suitably restricting the domain V, we can assume that the map (x, y) : V → ℝ^{2} is a submersion onto a rectangle of the form R = (x_{0}, x_{1}) × (y_{0}, y_{1}) in the xy-plane and that the fibers of this map are connected and contractible.

By construction, the 2-forms dϕ_{22} and dϕ_{44} are closed multiples of dx∧dy and hence are of the form dϕ_{22} = L(x, y) dx∧dy and dϕ_{44} = M(x, y) dx∧dy for some functions L and M on R.^{16} Now, there clearly exist functions λ and μ on R so that L = 2λ_{xy} and M = −2μ_{xy}. It follows then that

d(ϕ_{22} + λ_{x} dx − λ_{y} dy) = d(ϕ_{44} − μ_{x} dx + μ_{y} dy) = 0.

Thus, there exist functions s and t on V so that

\begin{aligned}ϕ_{22} & = ds & − λ_{x} dx & + λ_{y} dy \\ ϕ_{44} & = dt & + μ_{x} dx & − μ_{y} dy. \end{aligned}

Now, using the structure equations and the fact that ω^{1}∧dx = 0, we compute

d(e^{t−s−μ−λ}ω^{1}) = e^{t−s−μ−λ}(−ϕ_{44} + ϕ_{22} + dt − ds − dμ − dλ) ∧ ω^{1} = −2e^{t−s−μ−λ}(μ_{x} + λ_{x}) dx ∧ ω^{1} = 0.

1.5.10 Thus, the non-vanishing 1-form e^{t−s−μ−λ}ω^{1} is closed and a multiple of dx. Thus, e^{t−s−μ−λ}ω^{1} = X′(x) dx = dX ≠ 0 for some function X on R. Replacing x by X, we can assume that X = x, so we do this. Using a similar argument applied to ω^{3} and replacing y by a function Y if necessary we can arrange

16For convenience, for the rest of the proof, we will simplify our notation by simply writing F instead of F (x, y) or (x, y)^{∗}(F) as notation for the pullback via (x, y) of a function F on R.

\begin{aligned}ω^{1} & = e^{s−t+μ+λ} dx \\ ω^{3} & = e^{t−s+μ+λ} dy. \end{aligned}

In particular, we now have Ω = e^{2(λ+μ)}dx∧dy. (Note that this change of variables in x and y will likely change the boundaries of the rectangle R, but this is not important.) Now we can compute that

d(e^{f+λ}ω^{1} ∧ ω^{3} ∧ ω^{2}) = (ds − ϕ_{22}) ∧ ω^{1} ∧ ω^{3} ∧ ω^{2} = 0.

It follows that there is a function u, unique up to addition of an arbitrary function of x and y so that

e^{s+λ}ω^{1} ∧ ω^{3} ∧ ω^{2} = Ω ∧ du.

Thus, there exists a function P on V so that

ω^{2} ≡ e^{−s−λ}(du − P dy) mod dx.

Similarly, there exist functions v and Q so that

ω^{4} ≡ e^{−t−μ}(dv − Q dx) mod dy.

Now, using the structure equations, we compute that

0 = ω^{1} ∧ (dω^{2} + ϕ_{22} ∧ ω^{2} − ω^{3} ∧ ω^{4}) = e^{−t+μ} dx ∧ dy ∧ d(P − e^{2λ}v).

It follows that P = e^{2λ}v + P_{0} where P_{0} is a function of x and y alone. Similarly, we see that Q = e^{2μ}u + Q_{0} where Q_{0} is a function of x and y alone. Now, by the existence theory for linear hyperbolic PDE, there exist functions u_{0} and v_{0} on R which satisfy the equations

(u_{0})_{y} = e^{2λ}v_{0} + P_{0}  and  (v_{0})_{x} = e^{2λ}u_{0} + Q_{0}.

Replacing u and v by u + u_{0} and v + v_{0}, we get a new coordinate system where P_{0} = Q_{0} = 0. Now the functions x, y, u, and v are constant on the fibers of the submersion V → M and furnish the desired coordinate system on an open neighborhood of the base point in M of the point in B that we initially fixed. But, by construction,

I = { ω^{1} ∧ ω^{2}, ω^{3} ∧ ω^{4} } = { (du − e^{2λ}v dy) ∧ dx, (dv − e^{2μ}u dx) ∧ dy },

as we wanted to show. □

Example. Let us illustrate this result by applying it to the FPU equation

z_{yy} − (k(z_{x}))^{2} z_{xx} = 0

introduced as Example 3 in Section 1.2.3. Recall that k is assumed to be a smooth positive function on ℝ. Since there is no explicit z-dependence in this equation, we can associate to it the s = 0 exterior differential system I on xypq-space defined by

I = { (dq + k(p) dp) ∧ (dx + k(p) dy), (dq − k(p) dp) ∧ (dx − k(p) dy) }.

1.5.11 (Since x and y appear linearly in the generating forms, it is not surprising that this system should turn out to be linear.) The condition that this system be non-degenerate is easily seen to be that k′(p) ≠ 0, so we assume this from now on. It is then easy to compute that the coframing

\begin{aligned}η^{1} & = k′(p)/(4k(p)^{2}) & (k(p) dp + dq) \\ η^{2} & = & (k(p) dy + dx) \\ η^{3} & = k′(p)/(4k(p)^{2}) & (k(p) dp − dq) \\ η^{4} & = & (k(p) dy − dx) \end{aligned}

is 1-adapted. Since η^{1} and η^{3} are clearly integrable and since Ω = η^{1}∧η^{3} is clearly closed, we must have p_{4} = q_{2} = p_{3} = q_{1} = p_{1} = q_{3} = 0. Further computation then reveals that

\begin{aligned}dϕ_{22} & = & K η^{1} ∧ η^{3} \\ dϕ_{44} & = & K η^{3} ∧ η^{1} \end{aligned}

where

K = \frac{4(k′′′(p)k′(p)k(p)^{2} − 3k′′(p)k′(p)^{2}k(p) − k′′(p)^{2}k(p)^{2} + 4k′(p)^{4})}{k′(p)^{4}}.

In particular, Φ_{22} + Φ_{44} ≡ 0, so all the conditions for linearity are fulfilled.

Before ending this subsection, we would like to comment on the geometric meaning of the invariants in the linear case. As the structure equations derived in the course of the proof make clear, the quantities

Ω = ω^{1} ∧ ω^{3} = e^{2λ+2μ} dx ∧ dy = −^{∗}Ω
g = ω^{1} ∘ ω^{3} = e^{2λ+2μ} dx ∘ dy = ^{∗}g
\frac{1}{2}(Φ_{22} − Φ_{44}) = K ω^{1} ∧ ω^{3} = (λ + μ)_{xy} dx ∧ dy
\frac{1}{2}(Φ_{22} + Φ_{44}) = F ω^{1} ∧ ω^{3} = (λ − μ)_{xy} dx ∧ dy

are invariants of the system

I = {(du − e^{2λ}v dy) ∧ dx, (dv − e^{2μ}u dx) ∧ dy}.

Note, in particular, that K is the curvature of the pseudo-Riemannian metric g.

The cases where K and F are constant are particularly interesting since these are precisely the cases where the (pseudo-)group of local automorphisms of the structure acts transitively on B. Depending on the signum of K, there are three models for complete pseudo-Riemannian metrics of constant curvature:

g_{K} = \{ \begin{matrix}\frac{dx ∘ dy}{cos^{2}(c(x + y))} & where |c(x + y)| < π/2 and K = c^{2} > 0, \\ dx ∘ dy & if K = 0, \\ \frac{dx ∘ dy}{cos^{2}(c(x − y))} & where |c(x − y)| < π/2 and K = −c^{2} < 0. \end{matrix}

For each allowable value of K, there is a 1-parameter family of inequivalent homogeneous linear hyperbolic systems corresponding to the value of F. For example, when K = 0, we get the systems

I_{0,F} = { (du − e^{F xy}v dy) ∧ dx, (dv − e^{−F xy}u dx) ∧ dy},

while when K = c^{2} > 0 and F = Kβ, we get the systems

I_{K,β} = \{ (du − \frac{v}{cos^{1+β}(c(x + y))} dy) ∧ dx, (dv − \frac{u}{cos^{1−β}(c(x + y))} dx) ∧ dy\} .

1.5.12 The case of K < 0 is similar. (Note, by the way, that the geometric difference between the K > 0 and K < 0 cases is that, in the former, a “time-like” curve (i.e., one on which the metric restricts to be negative) can cross all of the null curves while a “space-like” one cannot. In the latter, the reverse is true.)

The corresponding linear PDE, in various coordinate systems and in the s = 1 version as well, were studied extensively by Euler and Poisson.^{17}

As our final remark about these linear systems with K and F constant, we note these systems are precisely the ones for which the group of symmetries acts transitively on B. In particular, in these cases, there can be no canonical subbundle of B which is preserved under all symmetries. Hence, there cannot be any canonical structure reduction in these cases. This shows that, without making some assumptions about the torsion terms in (5) or the curvature terms in (9) or (10), there will be no canonical reduction of the G-structure.

1.5.7 The first prolongation. We are now going to apply the structure equations derived so far to study the first prolongation of (M, I). Our goal in this subsection is to prove that any non-degenerate hyperbolic system with s = 0 which is integrable by the method of Darboux at level 1 is locally equivalent to one of two possible hyperbolic systems.

First, we want to describe a natural submersion B → M^{(1)} which will be used to express the characteristic systems on M^{(1)} in terms of the structure equations on B. First, we give an invariant description: Let E ⊂ T B be the codimension 2 distribution defined by the equations ω^{2} = ω^{4} = 0. By its very construction, for each b ∈ B, the image subspace

π^{(1)}(b) = π_{∗}(E_{b}) ⊂ T_{π(b)}M

is a 2-dimensional integral element of I and hence is an element of M^{(1)}. Thus, we have defined a mapping π^{(1)} : B → M^{(1)}. As we shall see, this is a submersion onto the open set of integral elements of I on which the 2-form Ω = ω^{1}∧ω^{3} is non-zero and the π^{(1)}-pullback of the Pfaffian system I^{(1)} is just the Pfaffian system generated by the 1-forms ω^{2} and ω^{4}.

Explicitly, one can see this as follows: If η = (η^{i}) is a 1-adapted coframing on an open set U ⊂ M, then on B_{U} = π^{−1}(U) ⊂ B, there exist unique functions s_{2}, s_{4}, r_{2} ≠ 0, and r_{4} ≠ 0, so that

( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = ( \begin{matrix}r_{2}/r_{4} & 0 & 0 & 0 \\ −s_{2}/r_{2} & 1/r_{2} & 0 & 0 \\ 0 & 0 & r_{4}/r_{2} & 0 \\ 0 & 0 & −s_{4}/r_{4} & 1/r_{4} \end{matrix} ) ( \begin{matrix}η^{1} \\ η^{2} \\ η^{3} \\ η^{4} \end{matrix} ) .

(In fact, the map (π, r_{2}, r_{4}, s_{2}, s_{4}) : B_{U} → U×ℝ^{∗}×ℝ^{∗}×ℝ×ℝ is a diffeomorphism.) Substituting these formulae into the structure equations (4) and expanding shows that there are congruences

\begin{matrix}ϕ_{22} & ≡ & (1/r_{2}) & dr_{2} \\ ϕ_{44} & ≡ & (1/r_{4}) & dr_{4} \\ ϕ_{21} & ≡ & (r_{4}/r_{2}^{2}) & ds_{2} \\ ϕ_{43} & ≡ & (r_{2}/r_{4}^{2}) & ds_{4} \end{matrix} \} mod η^{1}, η^{2}, η^{3}, η^{4}.

z_{xy} = \frac{β′}{(x − y)} z_{x} − \frac{β}{(x − y)} z_{y} .

1.5.13 If (e_{1}, e_{2}, e_{3}, e_{4}) is the frame field on U dual to the coframing η, then one computes that, for x = π(b),

17For example, see Chap. 3 of Livre 4 of [Da], where Darboux studies the equation for z as a function of x and y in the half-plane x − y > 0. As this is a second order equation with no explicit z dependence, the methods of Section 1.1 show how to associate to it a hyperbolic system of class s = 0. This system will clearly be linear and one easily sees that, as β and β′ vary, this gives a two parameter family of linearizable systems with K and F constant.

π^{(1)}(b) = span { e_{1}(x) + s_{2}(b) e_{2}(x), e_{3}(x) + s_{4}(b) e_{4}(x) },

which is clearly an integral element of I. Moreover, it follows that

(π^{(1)})^{∗}(I^{(1)}) = {η^{2} − s_{2} η^{1}, η^{4} − s_{4} η^{3}} = {ω^{2}, ω^{4}},

as we wanted to show.

Now, from (5), and the definitions given in Section 1.3.6, the structure equations

\begin{aligned}dω^{2} & ≡ −ϕ_{21} ∧ ω^{1} + ω^{3} ∧ ω^{4} mod ω^{2} \\ dω^{4} & ≡ −ϕ_{43} ∧ ω^{3} + ω^{1} ∧ ω^{2} mod ω^{4} \end{aligned}

yield these formulae for the characteristic systems:

\begin{aligned}Ξ_{10}^{(1)} & = \{ω^{2}, ω^{1}, ϕ_{21}\}, \\ Ξ_{01}^{(1)} & = \{ω^{4}, ω^{3}, ϕ_{43}\}. \end{aligned}

By the structure equations (5), (7), and (10), we see that

\begin{aligned}dω^{2} − ω^{3} ∧ ω^{4} ≡ dω^{1} ≡ dϕ_{21} & ≡ 0 mod ω^{2}, ω^{1}, ϕ_{21} \\ dω^{4} − ω^{1} ∧ ω^{2} ≡ dω^{3} ≡ dϕ_{43} & ≡ 0 mod ω^{4}, ω^{3}, ϕ_{43} \end{aligned}

so it follows that

\begin{aligned}Ξ_{10}^{(1)⟨1⟩} & = \{ω^{1}, ϕ_{21}\}, \\ Ξ_{01}^{(1)⟨1⟩} & = \{ω^{3}, ϕ_{43}\}. \end{aligned}

Now, if Darboux’ method is to succeed at level 1 (i.e., if both the level 1 characteristic systems are to contain completely integrable subsystems of rank 2), then both of these latter systems must be completely integrable. However, by the structure equations (5), (7), and (10) we see that

\begin{matrix}dω^{1} ≡ & −(p_{3} ω^{3} + p_{4} ω^{4}) & ∧ ω^{2} \\ dϕ_{21} ≡ & −(κ_{1} − ω^{3}) & ∧ ω^{2} \end{matrix} \} mod ω^{1}, ϕ_{21}

and

\begin{matrix}dω^{3} ≡ & −(q_{1} ω^{1} + q_{2} ω^{2}) & ∧ ω^{4} \\ dϕ_{43} ≡ & −(κ_{3} − ω^{1}) & ∧ ω^{4} \end{matrix} \} mod ω^{3}, ϕ_{43}.

Thus, it follows that the systems Ξ_{10}^{(1)⟨1⟩} and Ξ_{01}^{(1)⟨1⟩} are completely integrable if and only if

p_{3} = p_{4} = q_{1} = q_{2} = 0  and  (k_{13} − 1) = (k_{31} − 1) = k_{14} = k_{32} = 0.

We are now going to show that, up to diffeomorphism, there are essentially only two systems satisfying these conditions. To do this, we assume that the above equations hold and set, according to (6),

\begin{aligned}d & p_{1} & = & p_{1} & ϕ_{22} + ∇ & p_{1} & = & p_{1} & ϕ_{22} + & p_{11} & ω^{1} + & p_{12} & ω^{2} + & p_{13} & ω^{3} + & p_{14} & ω^{4} \\ d & q_{3} & = & q_{3} & ϕ_{44} + ∇ & q_{3} & = & q_{3} & ϕ_{44} + & q_{33} & ω^{3} + & q_{34} & ω^{4} + & q_{31} & ω^{1} + & q_{32} & ω^{2}. \end{aligned}

1.5.14 Now, on account of the above vanishing assumptions, equations (7) and (8) simplify dramatically. In fact,

(8) becomes

\begin{aligned}0 & = (Φ_{22} − Φ_{44}) ∧ ω^{1} − ∇p_{1} ∧ ω^{1} ∧ ω^{2} \\ 0 & = −Φ_{22} ∧ ω^{2} − ω^{1} ∧ ω^{2} ∧ ω^{3} \\ 0 & = (Φ_{44} − Φ_{22}) ∧ ω^{3} − ∇q_{3} ∧ ω^{3} ∧ ω^{4} \\ 0 & = −Φ_{44} ∧ ω^{4} − ω^{3} ∧ ω^{4} ∧ ω^{1}. \end{aligned}

These equations can now be solved for Φ_{22} and Φ_{44}, yielding the identity p_{14} = q_{32} and the formulae

\begin{aligned}Φ_{22} & = ω^{1} ∧ ω^{3} + p_{13} & ω^{2} ∧ ω^{3} + (b + f) ω^{2} ∧ ω^{4} \\ Φ_{44} & = ω^{3} ∧ ω^{1} + q_{31} & ω^{4} ∧ ω^{1} + (b − f) ω^{4} ∧ ω^{2} \end{aligned}

where we have written 2b for the common value of q_{32} and p_{14} and f is a yet to be determined function. Now, a straightforward differentiation using the structure equations so far yields

0 = d(d(ϕ_{22} − ϕ_{44})) ≡ −(p_{13} ϕ_{21} + q_{31} ϕ_{43}) ∧ ω^{1} ∧ ω^{3} mod ω^{2}, ω^{4},

so it follows that we must have p_{13} = q_{31} = 0. We can now compute

\begin{matrix}0 = d(dϕ_{22}) & ≡ −(f + b) ϕ_{21} ∧ ω^{1} ∧ ω^{4} \\ 0 = d(dϕ_{44}) & ≡ −(f − b) ϕ_{21} ∧ ω^{1} ∧ ω^{4} \end{matrix} \} mod ω^{2}, ω^{3}.

Thus, b = f = 0. We now compute

\begin{aligned}0 = d(dq_{3}) & ≡ q_{34} & ω^{1} ∧ ω^{2} mod ω^{3}, ω^{4} \\ 0 = d(dp_{1}) & ≡ p_{12} & ω^{3} ∧ ω^{4} mod ω^{1}, ω^{2} \end{aligned}

which implies q_{34} = 0 and p_{12} = 0. Finally, we compute

0 = d(dϕ_{22}) = (q_{33} − p_{1}) ω^{1} ∧ ω^{2} ∧ ω^{3}
0 = d(dϕ_{44}) = (p_{11} − q_{3}) ω^{1} ∧ ω^{3} ∧ ω^{4}

which yields q_{33} = p_{1} and p_{11} = q_{3}.

At this point, we have structure equations

(11)  d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}ϕ_{44} − ϕ_{22} & 0 & 0 & 0 \\ ϕ_{21} & ϕ_{22} & 0 & 0 \\ 0 & 0 & ϕ_{22} − ϕ_{44} & 0 \\ 0 & 0 & ϕ_{43} & ϕ_{44} \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}−p_{1} ω^{1}∧ω^{2} \\ ω^{3}∧ω^{4} \\ −q_{3} ω^{3}∧ω^{4} \\ ω^{1}∧ω^{2} \end{matrix} )

with

(12)  \begin{aligned}dϕ_{22} & = ω^{1} ∧ ω^{3} + q_{3} ω^{1} ∧ ω^{2} \\ dϕ_{44} & = ω^{3} ∧ ω^{1} + p_{1} ω^{3} ∧ ω^{4} \end{aligned}

and

(13)  \begin{aligned}dp_{1} & = p_{1} ϕ_{22} + & q_{3} & ω^{1} \\ dq_{3} & = q_{3} ϕ_{44} + & p_{1} & ω^{3}. \end{aligned}

From these equations, we see that the 2-forms Ψ_{10} = q_{3} ω^{1}∧ω^{2} and Ψ_{01} = p_{1} ω^{3}∧ω^{4} are well-defined on M. From the equations (13), we see that if there is a point x_{0} ∈ M where both of these 2-forms vanish, then

1.5.15 both q_{3} and p_{1} vanish on the fiber above x_{0} and hence on all the connected components of B which intersect this fiber. In particular, Ψ_{10} and Ψ_{01} must vanish identically on the entire connected component of M which contains x_{0}.

Now, the equations (11–13) contain, in particular, the equations

\begin{aligned}dω^{1} & = −(ϕ_{44} − ϕ_{22}) ∧ ω^{1} − & p_{1} & ω^{1} ∧ ω^{2} \\ dω^{3} & = −(ϕ_{22} − ϕ_{44}) ∧ ω^{3} − & q_{3} & ω^{3} ∧ ω^{4} \end{aligned}

and

d(ϕ_{44} − ϕ_{22}) = −2 ω^{1} ∧ ω^{3} − q_{3} ω^{1} ∧ ω^{2} + p_{1} ω^{3} ∧ ω^{4}.

The significance of these equations is that they imply that there is a canonical ℝ^{∗}-bundle F → M with connection form θ = ϕ_{44}−ϕ_{22} with the following property: To every integral manifold S ⊂ M of the exterior differential system with independence condition (I, ω^{1}∧ω^{3}), the bundle F restricts to be the coframe bundle of the “characteristic Lorentzian metric” g = ω^{1}∘ω^{3} and its connection form restricts to be the Levi-Civita connection form of this metric. Moreover, this characteristic metric has constant curvature and is locally isometric to the complete metric of this curvature, namely

g_{0} = \frac{dx ∘ dy}{cos^{2}(x + y)}.

Let us first dispose of the connected components of M where Ψ_{10} and Ψ_{01} vanish identically. Over such a component, the above structure equations simplify considerably. Since all of the primary invariants (even q_{3} and p_{1}) are now zero, and since, by inspection, we have Φ_{22} +Φ_{44} = 0, it follows by our linearization result that the system I represents a linearizable system of PDE. Moreover, F = 0 and K = 1. Thus, the system is locally equivalent to the constant curvature linear example

I_{1,0} = \{ (du − \frac{v}{cos(x + y)} dy) ∧ dx, (dv − \frac{u}{cos(x + y)} dx) ∧ dy\} ,

defined in the domain in xyuv-space given by the inequality |x + y| < π/2.

Applying the method of Darboux and going through the calculations then shows that every solution of the system

u_{y}(x, y) = sec(x + y) v(x, y) v_{x}(x, y) = sec(x + y) u(x, y)

can be written in the form

u(x, y) = g(y) sec(x + y)+f(x) tan(x + y)+f′(x) v(x, y) = f(x) sec(x + y)+g(y) tan(x + y)+g′(y)

for some functions of one variable f and g and that these are unique up to a replacement of the form

(f(x), g(y)) ↦ (f(x) + a sin(x) + b cos(x), g(y) + a cos(y) + b sin(y))

where a and b are constants. We leave to the reader the task of showing how to determine the functions f and g from non-characteristic initial data.

Example (continued) Before going on to consider the non-linear possibility, let us consider the FPU equation that we earlier saw was linear. We computed that

K = \frac{4(k′′′(p)k′(p)k(p)^{2} − 3k′′(p)k′(p)^{2}k(p) − k′′(p)^{2}k(p)^{2} + 4k′(p)^{4})}{k′(p)^{4}}

1.5.16 and now know that K = 1 is the necessary and sufficient condition that this equation be integrable by the method of Darboux at level 1. Now, the equation K ≡ 1 is a third order differential equation for k(p). Inspection shows that if k(p) is a positive solution to this equation, then so is c_{2} k(c_{1}p + c_{0}) for any three constants c_{0}, c_{1} ≠ 0, and c_{2} > 0. The transformations k(p) ↦ c_{2} k(c_{1}p + c_{0}) with c_{1} ≠ 0 and c_{2} > 0 form a 3-parameter group Γ which acts effectively on the space of positive solutions of this third order equation. Up to equivalence under this group action, only two of these solutions have a positive dimensional stabilizer, namely

k(p) = p^{−2}  and  k(p) = p^{− \frac{2}{3}} ,

both only defined for p ≠ 0. It is not hard to show that, aside from these two so-called “singular solutions”, up to the action of Γ there are only two other solutions. The first is the unique function k : ℝ → (0, 1) defined implicitly by the equation

p = \frac{2}{\sqrt{k(p)}} + log (\frac{1 − \sqrt{k(p)}}{1 + \sqrt{k(p)}})

and the second is the unique function k : (−∞, 0) → (1, ∞) defined implicitly by the equation

p = \frac{2}{\sqrt{k(p)}} + log (\frac{\sqrt{k(p)} − 1}{\sqrt{k(p)} + 1}) .

Thus, this gives the complete list of FPU equations which are integrable by the method of Darboux at the first level.

As an explicit example of the use of the method of Darboux in this case, let us consider the case k(p) = p^{−2}. Consider the differential system I defined in the region p > 0 in xypq-space by the 2-forms

I = { (dq + p^{−2} dp) ∧ (dx + p^{−2} dy), (dq − p^{−2} dp) ∧ (dx − p^{−2} dy) }

which is the differential system associated to the equations

p_{y} −  q_{x} = 0
q_{y} + (p^{−3}/3)_{x} = 0

which model the dynamics of a polytropic perfect gas [FX]. The initial conditions are specified along y = 0 in the form

p(x, 0) = p_{0}(x) > 0
q(x, 0) = q_{0}(x).

Applying the method of Darboux then leads to the following recipe for solving this initial value problem. Define a new function s by setting

s(x) = ∫_{0}^{x} \frac{1}{2}p_{0}(ξ) dξ.

Note that, because p_{0} > 0, the function s is strictly increasing. If we now reparametrize the initial curve (x, y, p, q) = (x, 0, p_{0}(x), q_{0}(x)) in terms of s, we can write

(x, y, p, q) = (2α(s), 0, 1/α′(s), β′(s))

1.5.17 for some functions α and β defined on the range of s. Then the integral surface of I containing this curve is given by

x(s, t) = α(s) + α(t) + β(t) − β(s) + \frac{1}{2}(s − t)(β′(s) + β′(t) + α′(t) − α′(s))
y(s, t) = 2(s − t)/(α′(s) + α′(t) + β′(t) − β′(s))
p(s, t) = 2/(α′(s) + α′(t) + β′(t) − β′(s))
q(s, t) = \frac{1}{2}(β′(s) + β′(t) + α′(t) − α′(s))

where the four functions are defined in the region in the st-plane defined by the inequality

(α′(s) + α′(t) + β′(t) − β′(s)) > 0.

Note that this certainly includes the line s = t which corresponds to the original initial curve.

Now, it is well-known that, under suitable hypotheses on the initial data, solutions of an FPU equation will develop “shocks”. In our terminology, this corresponds to the failure of the solution surface constructed above to be representable as a graph, i.e., in the form p = P (x, y) and q = Q(x, y) for some functions P and Q defined on the whole xy-plane. Of course, this will generally happen when the map (s, t) ↦ (x(s, t), y(s, t)) fails to be a diffeomorphism from the st-plane to the xy-plane. In order to understand this behavior, it is helpful to simplify the above formulae by introducing new functions a and b by a = β + α and b = β − α so that the above formulae become

x(s, t) = a(t) − b(s) + \frac{1}{2}(s − t)(a′(t) + b′(s))
y(s, t) = 2(s − t)/(a′(t) − b′(s))
p(s, t) = 2/(a′(t) − b′(s))
q(s, t) = \frac{1}{2}(a′(t) + b′(s)).

The condition for characteristic completeness is then that we have a′(t) > b′(s) for all s and t in the range of definition. In other words, the graph of a′ must lie strictly above the graph of b′. One easily computes that

dx ∧ dy = −2\frac{(a′(t) − b′(s) − (t − s)a′′(t))(a′(t) − b′(s) − (t − s)b′′(s))}{(a′(t) − b′(s))^{2}} ds ∧ dt.

It follows that, in order that dx∧dy not vanish (so that the mapping (s, t) ↦ (x(s, t), y(s, t)) is at least a local diffeomorphism), we must have a′(t) > b′(s) + (t − s)b′′(s) for all s and t in the range of definition, i.e., the graph of a′ must lie above all the tangent lines to the graph of b′, as well as a′(t) + (s − t)a′′(t) > b′(s) for all s and t in the range of definition, i.e., the graph of b′ must lie below all the tangent lines to the graph of a′. These conditions clearly cannot be met for any non-constant functions a and b which satisfy the conditions a′(t) ≡ a_{0} and b′(s) ≡ b_{0} for all s and t satisfying |s|, |t| > M, in other words, for initial data which are compactly supported perturbations of constant initial data. Thus, this analysis recovers the well-known fact (see, for instance, [La]) that solutions of this FPU equation with initial condition which are “small” perturbations of constant initial data must develop shocks.

Now let us turn to the analysis of the components of M on which Ψ_{10} and Ψ_{01} do not vanish simultaneously. Thus, restricting our attention to one such component and calling it M for convenience, we see that q_{3} and p_{1} do not vanish simultaneously at any point of B. Now, it follows from (13) that at any point of B where q_{3} vanishes, its differential dq_{3} is a non-zero multiple of ω^{3}. In particular, the locus q_{3} = 0 is a smooth hypersurface in B which contains all of the fibers of B → M which it intersects. Moreover, since dq_{3} = p_{1} ω^{3} along this locus, it follows that this hypersurface is a union of leaves of the foliation of B defined

1.5.18 by ω^{3} = 0. In particular the image of this hypersurface in M, which is the locus where Ψ_{10} vanishes, is a countable union of closed leaves of the characteristic foliation defined by ω^{3} = 0. A similar picture prevails for the locus where Ψ_{01} vanishes. It is a countable union of closed leaves of the characteristic foliation defined by ω^{1} = 0.

Suppose, first of all, that we are at a point z of M where neither Ψ_{10} nor Ψ_{01} vanish. Then, by the structure equations, we can choose a coframing η = (η^{1}, η^{2}, η^{3}, η^{4}) on a neighborhood of z which is a section of B and also satisfies p_{1} = q_{3} = 1. This coframing then satisfies the following equations (which are clearly specializations of (11–13)).

d ( \begin{matrix}η^{1} \\ η^{2} \\ η^{3} \\ η^{4} \end{matrix} ) = − ( \begin{matrix}φ_{44} − φ_{22} & 0 & 0 & 0 \\ φ_{21} & φ_{22} & 0 & 0 \\ 0 & 0 & φ_{22} − φ_{44} & 0 \\ 0 & 0 & φ_{43} & φ_{44} \end{matrix} ) ∧ ( \begin{matrix}η^{1} \\ η^{2} \\ η^{3} \\ η^{4} \end{matrix} ) + ( \begin{matrix}−η^{1}∧η^{2} \\ η^{3}∧η^{4} \\ −η^{3}∧η^{4} \\ η^{1}∧η^{2} \end{matrix} )

with

\begin{aligned}dφ_{22} & = η^{1} ∧ η^{3} + η^{1} ∧ η^{2} \\ dφ_{44} & = η^{3} ∧ η^{1} + η^{3} ∧ η^{4} \end{aligned}

and

\begin{aligned}0 & = φ_{22} + η^{1} \\ 0 & = φ_{44} + η^{3} \end{aligned}

and the ideal I is generated by {η^{1}∧η^{2}, η^{3}∧η^{4}}.

Since η^{1} and η^{3} are integrable but dη^{1} ≠ 0 and dη^{3} ≠ 0, we may introduce coordinates (x, y, p, q) with p and q positive, such that

η^{1} = \frac{dx}{p}  and  η^{3} = \frac{dy}{q} .

From the structure equations, we have

Ω_{10} = η^{1} ∧ η^{2} = −dη^{1} − η^{1} ∧ η^{3} = (\frac{dp}{p} + \frac{dy}{q}) ∧ \frac{dx}{p}

and similarly

Ω_{01} = η^{3} ∧ η^{4} = −dη^{3} − η^{3} ∧ η^{1} = (\frac{dq}{q} + \frac{dx}{p}) ∧ \frac{dy}{q} .

Setting p = e^{−u} and q = e^{−v} gives

Ω_{10} = −(du − e^{v} dy) ∧ e^{u} dx
Ω_{01} = −(dv − e^{u} dx) ∧ e^{v} dy

and therefore the system models the s = 0 Liouville system

\begin{aligned}u_{y} & = e^{v} \\ v_{x} & = e^{u}. \end{aligned}

Now, although the coordinate chart on a neighborhood of z that we have constructed is not canonical, the 1-forms η^{1} and η^{3} are actually well-defined. Any two local coordinate charts (x, y, p, q) and (x̄, ȳ, p̄, q̄) with p, q, p̄, and q̄ all positive and satisfying

η^{1} = \frac{dx}{p} = \frac{dx̄}{p̄}  and  η^{3} = \frac{dy}{q} = \frac{dȳ}{q̄}

1.5.19 must be related on some (possibly smaller) neighborhood of z by relations of the form

(x, y, p, q) = (X(x̄), Y (ȳ), X′(x̄)p̄, Y′(ȳ)q̄)

where X and Y are functions of one variable with positive derivative. Thus, on the open set in M which is the complement of the loci Ψ_{10} = 0 and Ψ_{01} = 0, the hyperbolic structure is actually locally homogeneous and induces a special atlas of charts which carry the system I into the s = 0 Liouville system.

Now let us consider the case where z ∈ M lies in the locus where Ψ_{10} vanishes. Let y be any function on a neighborhood of z satisfying y(z) = 0 but (dy)_{z} ≠ 0 and with the property that dy∧ω^{3} = 0. Then we may restrict to smaller neighborhood V of z with the property that y = 0 defines the zero locus of Ψ_{10} in V. Regarding y as a function on B_{V} by pull-back, we see that q_{3} and y have the same zero locus in B_{V} and have non-vanishing differentials there. It follows that the ratio r = q_{3}/y is a smooth non-vanishing function on B_{V} and calculation shows that

dr ≡ r ϕ_{44} mod ω^{3}.

It follows that we can choose a section (η^{1}, η^{2}, η^{3}, η^{4}) of B_{V} → V which lies in the locus p_{1} = r = 1. In other words, this section will have p_{1} = 1 and q_{3} = y. Thus, we will have structure equations

d ( \begin{matrix}η^{1} \\ η^{2} \\ η^{3} \\ η^{4} \end{matrix} ) = − ( \begin{matrix}φ_{44} − φ_{22} & 0 & 0 & 0 \\ φ_{21} & φ_{22} & 0 & 0 \\ 0 & 0 & φ_{22} − φ_{44} & 0 \\ 0 & 0 & φ_{43} & φ_{44} \end{matrix} ) ∧ ( \begin{matrix}η^{1} \\ η^{2} \\ η^{3} \\ η^{4} \end{matrix} ) + ( \begin{matrix}−η^{1}∧η^{2} \\ η^{3}∧η^{4} \\ −y η^{3}∧η^{4} \\ η^{1}∧η^{2} \end{matrix} )

with

\begin{aligned}dφ_{22} & = η^{1} ∧ η^{3} + y η^{1} ∧ η^{2} \\ dφ_{44} & = η^{3} ∧ η^{1} + η^{3} ∧ η^{4} \end{aligned}

and

\begin{aligned}0 & = φ_{22} + y η^{1} \\ dy & = y φ_{44} + η^{3} \end{aligned}

and the ideal I is generated by {η^{1}∧η^{2}, η^{3}∧η^{4}}. Now, since η^{1} is integrable, but dη^{1} ≠ 0, it is easy to see that there exist functions x and p > 0 on a possibly smaller neighborhood of z so that η^{1} = p dx, and by construction, there exists a function Q on this neighborhood so that η^{3} = Q dy. However, looking at the equation dy = y φ_{44} + η^{3} on the locus y = 0 shows that we must have Q = 1 on this locus. Thus, we may write Q = 1 + y q, where q is some smooth function on a neighborhood of z. Substituting this into the equations just above shows that

φ_{22} = −y p dx  and  φ_{44} = −q dx.

Substituting these relations into the formulae for dφ_{22} and dφ_{44} and then solving for η^{1}∧η^{2} and η^{3}∧η^{4} yields the expressions

η^{1} ∧ η^{2} = dx ∧ (dp − pq dy)  and  η^{3} ∧ η^{4} = dy ∧ (dq − p(1 + yq) dx)

It follows that the four functions x, y, p, and q form a coordinate system on a neighborhood of z. Moreover, the system in xypq-space defined by

J = {dx ∧ (dp − pq dy), dy ∧ (dq − p(1 + yq) dx)}

is easily seen to be a non-degenerate hyperbolic system away from the hypersurfaces defined by p = 0 and 1 + yq = 0.

1.5.20

There is a similar normal form for this non-linear system in a neighborhood of a point on the locus where Ψ_{01} vanishes. Details will be left to the reader.

In conclusion, we have shown that there are essentially only two types of non-degenerate hyperbolic systems with s = 0 which are integrable by the method of Darboux at level one. The first type is linear and the second is non-linear, being locally equivalent to the s = 0 Liouville system away from a hypersurface. We shall refer to it henceforth as the extended s = 0 Liouville system. It would be an interesting project to classify the global systems (if any) of the latter type which are characteristically complete in the sense of Section 1.2.3.

Now from our results so far, we see that the method of Darboux can be used to integrate the s = 0 Liouville system. In fact we may explicitly carry out the integration, as follows.

On M we use the coframing

ω^{1} = dx, ω^{2} = du − e^{v}dy, ω^{3} = dy, ω^{4} = dv − e^{u}dx.

Then on M^{(1)} we have the coframing

ω_{10} = ω^{1}, θ_{10} = ω^{2} − h_{20} ω^{1}, ω_{01} = ω^{3}, θ_{01} = ω^{4} − h_{02} ω^{3}

and

π_{20} = dh_{20} − e^{u+v}dy
π_{02} = dh_{02} − e^{u+v}dx.

Differentiating, we obtain

dθ_{10} = −π_{20} ∧ ω_{10} + e^{v}ω_{01} ∧ θ_{01}
dθ_{01} = −π_{02} ∧ ω_{01} + e^{u}ω_{10} ∧ θ_{10}

and

dπ_{20} ≡ e^{u+v}ω_{01} ∧ (θ_{10} + θ_{01}) mod ω_{10}
dπ_{02} ≡ e^{u+v}ω_{10} ∧ (θ_{10} + θ_{01}) mod ω_{01}.

By definition the characteristic systems are

Ξ_{10}^{(1)} = [ω_{10}, θ_{10}, π_{20}]
Ξ_{01}^{(1)} = [ω_{01}, θ_{01}, π_{02}],

and clearly

\begin{matrix}dθ_{10} & ≡ e^{v} & ω_{01} ∧ θ_{01} \\ dπ_{20} & ≡ e^{u+v} & ω_{01} ∧ θ_{01} \end{matrix}\} mod Ξ_{10}^{(1)}

with similar formulas holding for the other characteristic system. It follows that the first derived systems are

Ξ_{10}^{(1)⟨1⟩} = [ω_{10}, π_{20} − e^{u}θ_{10}]
= [dx, d(h_{20} − e^{u})]

and similarly

Ξ_{01}^{(1)⟨1⟩} = [dy, d(h_{02} − e^{v})].

1.5.21 Using this result we may integrate the s = 0 Liouville system, as follows: Since

d(h_{20} − e^{u}) ∧ dx = dy ∧ d(h_{02} − e^{v}) = 0

on solutions we may set

h_{20} − e^{u} = α′(x)
h_{02} − e^{v} = β′(y)

for functions α(x) and β(y). The equations

θ_{10} = θ_{01} = 0

then yield

du = (α′(x) + e^{u}) dx + e^{v}dy
dv = (β′(y) + e^{v}) dy + e^{u}dx

and so we must solve the ODEs

u_{x} = α′(x) + e^{u}
v_{y} = β′(y) + e^{v}.

For this we set

U = u − α(x) = log f
V = v − β(y) = log g

and consider functions a(x), b(x) which satisfy

\{ \begin{matrix}a′(x) & = −e^{α(x)} \\ b′(y) & = −e^{β(y)}. \end{matrix}

Then the above ODE system is

U_{x} = e^{α(x)}e^{U}
V_{y} = e^{β(y)}e^{V}

or, equivalently,

\frac{f_{x}}{f^{2}} = −a′(x)
\frac{g_{y}}{g^{2}} = −b′(y).

These may be integrated to give

e^{u}e^{−α} = f =\frac{1}{a(x) + b̃(y)}
e^{v}e^{−β} = g = \frac{1}{ã(x) + b(y)}.

1.5.22 Substituting in the expressions for du and dv allows us to set ã = a, b̃ = b and finally

e^{u} = \frac{−a′(x)}{a(x) + b(y)}
e^{v} = \frac{−b′(y)}{a(x) + b(y)}

for the general solution to the s = 0 Liouville system. We note that a(x), b(y) are arbitrary subject to a′(x) < 0 and b′(y) < 0.

Remark: To close the loop with the previous discussion at the end of Section 1.4 of the s = 1 Liouville equation, we may differentiate the first equation with respect to y to obtain

e^{u+v} = \frac{a′(x)b′(y)}{(a(x) + b(y))^{2}}

which since

(u + v)_{xy} = 2e^{u+v}

gives the general solution to the s = 1 Liouville equation mentioned in Section 1.4.3.

1.5.23 1.6 Hyperbolic Euler-Lagrange systems of class s = 0

1.6.1 Bi-symplectic structures. Among the hyperbolic exterior differential systems of class s = 0 are the Euler-Lagrange systems that we introduced as Example 4 in Section 1.1.3. In this section, we shall develop some of the special properties of these systems, culminating in a variant of Noether’s Theorem, which will describe the relationship between symmetries and classical conservation laws for these Euler-Lagrange systems. (The notion of conservation laws for general hyperbolic exterior differential systems will be developed more fully in Section 2.1.)

We begin by recalling the construction of hyperbolic Euler-Lagrange systems of class s = 0. Let M be a 4-manifold and let Φ be a symplectic 2-form on M. Associated to any other 2-form Λ on M, we want to consider the functional F_{Λ} on immersed Φ-Lagrangian surfaces S ⊂ M which are oriented and compact (possibly with boundary) which is defined by the rule

(1)  F_{Λ}(S) = ∫_{S} Λ.

In Example 4 in Section 1.1.3, we saw that the critical points S of this functional relative to variations that leave fixed the boundary of S are the integral surfaces of the exterior differential system

Φ = Ψ = 0.

Here, Ψ = dψ where ψ is defined to be the (unique) 1-form satisfying dΛ = Φ∧ψ. We shall call E(Λ) = {Φ, Ψ} the Euler-Lagrange system associated to the functional F_{Λ}.

Note that if Λ̃ = Λ + dγ + fΦ for some 1-form γ and some function f, then for any compact, oriented Φ-Lagrangian surface S, we will have

FΛ̃(S) = F_{Λ}(S) + ∫_{∂S} γ.

It follows that S is critical for FΛ̃ with respect to variations through Φ-Lagrangian surfaces fixing the boundary if and only if it is critical for F_{Λ} with respect to the same variations. Thus, it would not be surprising if Λ and Λ̃ gave rise to the same Euler-Lagrange system.

In fact, this is precisely what happens. Let I_{Φ} be the differential ideal generated by the symplectic form Φ, and recall that the characteristic cohomology group H̄^{2}(M, I_{Φ}) is by definition H^{2}(Ω^{∗}(M)/I_{Φ}) where the differential on the quotient complex Ω^{∗}(M)/I_{Φ} is induced by d. Note that Λ and Λ̃ determine the same cohomology class in H̄^{2}(M, I_{Φ}) if and only if there exist a 1-form γ and a function f so that Λ̃ = Λ + dγ + fΦ. In this case, we will have

dΛ̃ = dΛ + df ∧ Φ = (ψ + df) ∧ Φ,

so we can take ψ̃ = ψ + df, implying Ψ̃ = dψ̃ = dψ = Ψ. In particular, Ψ depends only on the characteristic cohomology class [Λ]_{Φ} ∈ H̄^{2}(M, I_{Φ}) defined by Λ. In fact, this construction can be carried further to show that, on any open set U ⊂ M satisfying H^{2}(U, ℝ) = H^{3}(U, ℝ) = 0, there is actually an isomorphism

H̄^{2}(U, I_{Φ}) ≃ { Ψ ∈ Ω^{2}(U) Φ ∧ Ψ = 0, dΨ = 0 }

defined by the obvious assignment [Φ] ↦ Ψ.

Throughout this section we shall make the assumption that the exterior differential system E(Λ) is non-degenerate in the sense that the 2-form Ψ satisfies the condition that Ψ∧Ψ is nowhere vanishing. This is equivalent to the assumption that the associated PDE system have non-degenerate symbol.

1.6.1 Indeed, taking the exterior derivative of the equation dΛ = Φ∧ψ yields

Φ ∧ Ψ = 0.

Hence, in order that {Φ, Ψ} span a hyperbolic pencil at each point one must have

Ψ ∧ Ψ = −f^{2} Φ ∧ Φ

for some non-vanishing function f (which we may take to be positive), while the condition that {Φ, Ψ} span an elliptic pencil at each point is that

Ψ ∧ Ψ = f^{2} Φ ∧ Φ

for some non-vanishing function f (which we could also take to be positive).

The properties of these pairs (Φ, Ψ) are of sufficient interest to warrant giving the separate

Definition : A bi-symplectic structure (M; Φ, Ψ) on a 4-manifold M is a pair (Φ, Ψ) of everywhere linearly independent symplectic forms. The bi-symplectic structure is said to be special in case Φ∧Ψ = 0 and is said to be hyperbolic (respectively, elliptic) if the pencil generated by Φ and Ψ is everywhere hyperbolic (respectively, elliptic).

Example 1: Prescribed Gauss curvature. In an open set M ⊂ ℝ^{4}, consider the standard symplectic form

Φ = dp ∧ dx + dq ∧ dy.

If f and g are any non-vanishing (smooth) functions of two variables, then the 2-form

Ψ = f(p, q) dp ∧ dq − g(x, y) dx ∧ dy

is closed, non-degenerate, and satisfies

Φ ∧ Ψ = 0.

On solution surfaces to Φ = Ψ = 0 of the form

(x, y) → (x, y, p(x, y), q(x, y))

we obviously have p = z_{x} and q = z_{y} for some function z(x, y) which satisfies the PDE

(2)  f(z_{x}, z_{y})(z_{xx}z_{yy} − z_{xy}^{2}) = g(x, y).

A special case is when

f = \frac{1}{(1 + p^{2} + q^{2})^{2}} .

Then we may think of (x, y, z(x, y)) as the graph of a piece of surface Σ in Euclidean space 𝔼^{3}, and the left hand side of (2) is the Gauss curvature of Σ.

Note that this defines an elliptic bi-symplectic structure if g is positive and a hyperbolic bi-symplectic structure if g is negative. Especially noteworthy is the case when g = −c^{2} < 0 for some constant c. Then this system models the equation for constant negative curvature surfaces. In this case, the decomposable 2-forms in the pencil generated by Φ and Ψ are

Ω_{10} = (\frac{dp}{1 + p^{2} + q^{2}} + c dy) ∧ (\frac{dq}{1 + p^{2} + q^{2}} − c dx)
Ω_{01} = (\frac{dp}{1 + p^{2} + q^{2}} − c dy) ∧ (\frac{dq}{1 + p^{2} + q^{2}} + c dx) .

1.6.2 It may be checked that the characteristic curves on integral surfaces are the usual asymptotic curves of elementary differential geometry.

1.6.2 A structural symmetry. For a special bi-symplectic structure the roles of Φ and Ψ are obviously symmetric, which suggests that hyperbolic Euler-Lagrange systems of class s = 0 might have a corresponding structural symmetry. We shall now see that this is indeed the case.

Proposition: Suppose that (M; Φ, Ψ) is a special bi-symplectic structure on a manifold M satisfying H^{2}(M) = H^{3}(M) = 0. Then there exist 2-forms Λ and Ω on M so that the exterior differential system generated by the equations

Φ = Ψ = 0

is the Euler-Lagrange system both of the functional F_{Λ} defined on Φ-Lagrangian surfaces and of the functional F_{Ω} defined on Ψ-Lagrangian surfaces.

Proof: Since H^{2}(M) = 0, there are 1-forms φ and ψ such that

\{ \begin{matrix}Φ = dφ \\ Ψ = dψ \end{matrix}

from which it follows that

d(Φ ∧ ψ) = 0

since Φ∧Ψ = 0. Thus, since H^{3}(M) = 0, there is a 2-form Λ so that

Φ ∧ ψ = dΛ,

and we conclude that

(3)  Φ = Ψ = 0

is the Euler-Lagrange system for the functional

F_{Λ}(S) = ∫_{S} Λ

on Φ-Lagrangian surfaces S.

Symmetrically, since d(Ψ∧φ) = 0 and H^{3}(M) = 0, there exists a 2-form Ω so that

Ψ ∧ φ = dΩ.

Then (3) is also the Euler-Lagrange system for the functional

F_{Ω}(R) = ∫_{R} Ω

on Ψ-Lagrangian surfaces R. □

The importance of this proposition is that it gives a sufficient criterion for an exterior differential system I generated by a pencil of 2-forms on a 4-manifold to be expressible as a (non-degenerate) Euler-Lagrange system; namely it must be generated by a bi-symplectic structure.

Example 2: Systems defined by conservation laws. We want to illustrate this situation. To put this example in context, recall that, by the last proposition in Section 1.1.4, a (real-analytic) hyperbolic Euler-Lagrange system is (at least, locally) the exterior differential system associated to a hyperbolic system of

1.6.3 conservation laws. In this example, we want to look at the converse situation. We will determine conditions that a translation-invariant hyperbolic system of conservation laws

\begin{aligned}u_{t} + (f(u, v))_{x} & = 0 \\ v_{t} + (g(u, v))_{x} & = 0 \end{aligned}

might satisfy in order to be an Euler-Lagrange system. Using the notation from Section 1.1, we can express this pair of PDE as the exterior differential system

\begin{matrix}Φ = −du ∧ dx + df ∧ dt = d(−u dx + f dt), \\ Ψ = −dv ∧ dx + dg ∧ dt = d(−v dx + g dt). \end{matrix}

The conditions

Φ ∧ Φ ≠ 0,  Ψ ∧ Ψ ≠ 0,  Φ ∧ Ψ = 0

are, respectively,

f_{v} ≠ 0,  g_{u} ≠ 0,  f_{u} = g_{v}.

By the last of these relations we have

f = F_{v}, g = F_{u}

for a function F (u, v), and the first two inequalities give

F_{vv} ≠ 0, F_{uu} ≠ 0.

To determine the functional F_{Λ}, we set

ψ = −v dx + F_{u} dt

so that dψ = Ψ and seek to determine a 2-form Λ so that

dΛ = ψ ∧ Φ.

The right hand side is

\begin{aligned}v dF_{v} ∧ dx ∧ dt − F_{u} du ∧ dx ∧ dt & = v dF_{v} ∧ dx ∧ dt − dF ∧ dx ∧ dt + F_{v} dv ∧ dx ∧ dt \\ & = d((vF_{v} − F)dx ∧ dt). \end{aligned}

To better understand this we may locally introduce new coordinates U and V by the relations

\begin{aligned}U & = −u \\ V & = F_{v} \end{aligned}

so that u = −U and v = H(U, V). Then we have

\begin{aligned}Φ & = dU ∧ dx + dV ∧ dt \\ Λ & = (V H(U, V) − F (−U, H(U, V)))dx ∧ dt. \end{aligned}

Then Φ-Lagrangian surfaces which are locally graphs over xt-space are given by U = W_{x}, V = W_{t} for a function W (x, t), and

Λ = (W_{t} H(W_{x}, W_{t}) − F (−W_{x}, H(W_{x}, W_{t}))) dx ∧ dt = L(W_{x}, W_{t}) dx ∧ dt

1.6.4 defines a first order functional on W (x, t).

For a familiar specific case we may take

F = \frac{1}{2}(u^{2} + v^{2})

in which case U = −u, V = v and the Lagrangian is

L = \frac{1}{2}(V_{t}^{2} − V_{x}^{2}).

The hyperbolic PDE is, not surprisingly, the s = 0 wave equation

\begin{aligned}u_{t} + v_{x} & = 0 \\ v_{t} + u_{x} & = 0. \end{aligned}

More interesting is the general case when

F = \frac{1}{λ}(u^{λ} + v^{λ}), λ ≠ 0, 1, 2.

Then the functional is a constant times

∫ (W_{x}^{λ} + (λ − 1)W_{t}^{λ/λ−1}) dx ∧ dt.

The hyperbolic PDE system is

\begin{aligned}u_{t} + (λ − 1)v^{λ−2}v_{x} & = 0 \\ v_{t} + (λ − 1)u^{λ−2}u_{x} & = 0. \end{aligned}

For λ = 3, this is a pair of coupled Burgers’-type equations that, after adjusting constants, becomes

\begin{aligned}u_{t} + vv_{x} & = 0 \\ v_{t} + uu_{x} & = 0. \end{aligned}

In these specific examples, the symmetry between Φ and Ψ is accomplished by exchanging the roles of u and v and carrying out the same computations.

Special case: The Fermi-Pasta-Ulam equation gives rise to the s = 0 hyperbolic system generated by

\begin{aligned}Φ & = dp ∧ dx + dq ∧ dy \\ Ψ & = dq ∧ dx − dh ∧ dy \end{aligned}

where h(p) = k^{2}(p) in the notation of Example 3 in Section 1.2. This system is special bi-symplectic and can be expressed as an Euler-Lagrange system by taking in the above formulae.

F (p, q) = q − H(p)

where H′(p) = h(p). Working through the above recipe, we see that the FPU equation is the equation for critical points of the functional

z(x, y) ↦ ∫ (H(z_{x}) + \frac{1}{2}z_{y}^{2})dx ∧ dy.

1.6.3 An analog of Noether’s Theorem. We will now explore the relationship between symmetries and conservation laws for (hyperbolic) Euler-Lagrange systems I. (As will be explained in Section 2.1, a

1.6.5 ‘conservation law’ for I can be thought of as a closed 2-form which lies in the ideal I and thus we will make this identification without further comment.)

The usual statement of this relationship is some version of the classical theorem of E. Noether which asserts that there is an isomorphism

\{ \begin{matrix}symmetries of \\ a variational \\ problem \end{matrix}\} ≃ \{ \begin{matrix}conservation laws \\ for the variational \\ equations \end{matrix}\} .

The result we shall give below is of this general type but differs from this particular statement in a number of important specifics.

Let (Φ, Ψ) define a hyperbolic, special bi-symplectic structure on a 4-manifold M. We have seen that Φ = Ψ = 0 gives the Euler-Lagrange system for a pair of functionals

∫_{S} Λ, ∫_{R} Ω

defined, respectively, on Φ- and Ψ-Lagrangian surfaces. Thus our formulation of Noether’s Theorem will need to treat Φ and Ψ symmetrically. In addition, because the hyperbolic system generated by Φ and Ψ always has at least these two 2-forms as independent conservation laws while there is no a priori reason that it should have symmetries, we may suspect that the correct version of the above isomorphism should instead be with a quotient space of all conservation laws. With these observations in mind, we set

𝔊 = {Lie algebra of vector fields v on M that preserve each of Φ and Ψ}
⟨Φ, Ψ⟩ = {constant linear combinations of Φ and Ψ}
C_{0} = {closed 2-forms which are linear combinations of Φ and Ψ}
C̄_{0} = C_{0}/⟨Φ, Ψ⟩.

To relate the definition of 𝔊 to symmetries of a variational problem, let us suppose that we have found 2-forms Λ and Ω as described in the Proposition of the previous subsection. We denote by (M, Φ, [Λ]_{Φ}) the data consisting of the symplectic manifold (M, Φ) together with the equivalence class [Λ]_{Φ} ∈ H̄^{2}(M, I_{Φ}). We make the corresponding interpretation of (M, Ψ, [Ω]_{Ψ}). Then the following proposition should be expected, in view of the explicit identification of H^{2}(M, I_{Φ}) made in Section 1.6.1.

Proposition: If v ∈ 𝔊, then v is an infinitesimal symmetry of each of the data (M, Φ, [Λ]_{Φ}) and (M, Ψ, [Ω]_{Ψ}). Conversely, if a vector field v is an infinitesimal symmetry of either of the data (M, Φ, [Λ]_{Φ}) or (M, Ψ, [Ω]_{Ψ}), then v ∈ 𝔊.

Proof: Let v ∈ 𝔊. From dΛ = ψ∧Φ and dψ = Ψ we infer that

L_{v}ψ = dh

for some function h, and then a short calculation gives

d(L_{v}(Λ − hΦ)) = 0.

It follows that

L_{v}Λ = hΦ + dλ

for some 1-form λ, which is the same as

L_{v}([Λ]_{Φ}) = 0.

1.6.6 Applying a similar argument to L_{v}φ, the proposition now follows. □

We can now state and prove our version of Noether’s Theorem. Proposition: There is a natural isomorphism

(*)  η : 𝔊 → C̄_{0}.

Proof: To define η, we write locally, as was done above,

\{ \begin{matrix}Φ = dφ \\ Ψ = dψ \end{matrix}

for 1-forms φ and ψ, which are well-defined up to transformations

(4)  \{ \begin{matrix}φ & → φ̄ & = φ & + dG \\ ψ & → ψ̄ & = ψ & + dH \end{matrix}

for functions G and H. For v ∈ 𝔊 we have

L_{v}Φ = L_{v}Ψ = 0

from which it follows that

(5)  \begin{aligned}L_{v}φ & = dg \\ L_{v}ψ & = dh \end{aligned}

for suitable functions g and h. We set

(6)  η(v) = (v φ)Ψ + (v ψ)Φ − (gΨ + hΦ).

First, we show that η is well-defined: The functions g and h are well-defined up to constants by (5). Thus, η is well-defined modulo ⟨Φ, Ψ⟩ once we know φ and ψ. These are in turn well-defined up to a substitution (4). Then

L_{v}φ̄ = L_{v}(φ + dG) = d(g + L_{v}G)
L_{v}ψ̄ = L_{v}(ψ + dH) = d(h + L_{v}H)

which gives

ḡ = g + L_{v}G
h̄ = h + L_{v}H

and

(v φ̄ − ḡ)Ψ = (v φ + L_{v}G − g − L_{v}G)Ψ
= (v φ − g)Ψ,

showing that η is well-defined.

1.6.7 Second, we show that η(v) = 0 is closed: We have

d[(v φ − g)Ψ + (v ψ − h)Φ] = (d(v φ − g)) ∧ Ψ + (d(v ψ − h)) ∧ Φ

= −(v Φ) ∧ Ψ − (v Ψ) ∧ Φ = 0 − v (Φ ∧ Ψ) = 0

since Φ∧Ψ = 0.

Third, we show that η is injective: If η(v) = 0 then

\{ \begin{matrix}v φ − g = a \\ v ψ − h = b \end{matrix}

for constants a and b. The exterior derivative of the first equation gives

0 = L_{v}φ − v Φ − dg
= v Φ

which implies that v = 0.

Fourth and finally, we show that η is surjective: We may choose a local coframe ω^{1}, ω^{2}, ω^{3}, ω^{4} so that

Φ = ω^{1} ∧ ω^{2} + ω^{3} ∧ ω^{4}
Ψ = ω^{1} ∧ ω^{2} − ω^{3} ∧ ω^{4}.

For any function F we set

dF = ∑_{i=1}^{4} F_{i} ω^{i}.

A conservation law of level zero is a closed 2-form

A Φ + C Ψ.

From

dA ∧ Φ + dC ∧ Ψ = 0

we infer that

(7)  \{ \begin{matrix}A_{1} − C_{1} = 0 \\ A_{2} − C_{2} = 0 \\ A_{3} + C_{3} = 0 \\ A_{4} + C_{4} = 0. \end{matrix}

We want to find a vector field v such that if we define g and h by

(8)  \{ \begin{matrix}A = v ψ − h \\ B = v φ − g \end{matrix}

then (5) holds. In fact (5) and (8) imply

(9)  \{ \begin{matrix}dA = v Ψ \\ dB = v Φ, \end{matrix}

1.6.8 and the exterior derivatives of these equation give

0 = d(v Ψ) = L_{v}Ψ
0 = d(v Φ) = L_{v}Φ

as required.

We note that for g and h defined by (8), equations (9) imply equations (5). Now there are unique vector fields v_{A} and v_{B} satisfying

dA = v_{A} Ψ
dB = v_{B} Φ.

We then have to show that v_{A} = v_{B}, and this is just a restatement of (7). □

Example 3: Suppose that

Φ = dp ∧ dx + dq ∧ dy

and that we have a translation-invariant Lagrangian

Λ = L(p, q) dx ∧ dy

as in Section 1.1 above. Then the Euler-Lagrange form is given by

Ψ = L_{pp} dp ∧ dy + L_{pq} (dq ∧ dy + dx ∧ dp) + L_{qq} dx ∧ dq

and we may work through the mapping (*) to obtain

η(∂/∂x) = pΨ − L_{q}Φ
η(∂/∂y) = qΨ − L_{p}Φ.

These conservation laws are similar to linear momenta. Example 2 (continued): Consider the hyperbolic Euler-Lagrange system generated by

\begin{aligned}Φ & = −du ∧ dx + dF_{v} ∧ dt \\ Ψ & = −dv ∧ dx + dF_{u} ∧ dt \end{aligned}

where F (u, v) is a function satisfying F_{uu}F_{vv} ≠ 0. This system admits translation symmetries and therefore has conservation laws corresponding to linear momenta. These are

\begin{aligned}η(∂/∂x) & = Ξ = u(−dv ∧ dx + dF_{u} ∧ dt) + v(−du ∧ dx + dF_{v} ∧ dt) \\ η(∂/∂t) & = Λ = −F_{v}(−dv ∧ dx + dF_{u} ∧ dt) − F_{u}(−du ∧ dx + dF_{v} ∧ dt). \end{aligned}

Suppose we assume that F is homogeneous of degree μ, i.e., that

F (αu, αv) = α^{μ}F (u, v).

Then F_{u} and F_{v} are homogeneous of degree μ − 1 and the 1-parameter group of dilation symmetries

(u, v, x, y) → (αu, αv, \frac{x}{α}, \frac{t}{α^{μ−1}} )

preserves both Φ and Ψ. The corresponding conservation law is Γ = (ux − (μ − 1)tF_{v})(−dv ∧ dx + dF_{u} ∧ dt) + (vx − (μ − 1)tF_{u})(−du ∧ dx + dF_{v} ∧ dt).

1.6.9 2. Conservation Laws for Hyperbolic Systems of Class s = 0

2.1 General form of the conservation laws for hyperbolic systems of class s = 0

2.1.1 Conservation laws. In the standard literature on evolution equations, a conservation law for a given evolution equation is a function on the configuration space of the problem which is constant under the specified evolution of states.

For example, when the configuration space is a (finite dimensional) manifold M and the evolution equation is represented by a vector field X on M, a conservation law for X is simply a function f ∈ C^{∞}(M) which is constant on the flow lines of X, i.e., which satisfies Xf = 0. Of course, one wants to ignore “trivial” conservation laws, i.e., functions f which are locally constant, so one might represent the space of conservation laws in this case as a quotient

C(X, M) = \frac{\{ f ∈ C^{∞}(M) | df(X) = 0 \}}{\{ f ∈ C^{∞}(M) | df = 0 \}} .

If the dimension of M is n and one can find n−1 independent conservation laws, say, f_{1}, …, f_{n−1} with independent differentials, then the integral curve of X through a given point x_{0} ∈ M can be described implicitly by the n−1 equations

f_{i}(x) = f_{i}(x_{0}).

Thus, knowing “enough” independent conservation laws for X describes the integral curves of X completely.

In exterior differential systems, this notion of conservation law generalizes naturally. Given an exterior differential system I on a manifold M, we are interested in studying the (p+1)-dimensional integral manifolds of I which contain a given “initial” p-dimensional integral manifold γ : L → M. A “conservation law” of degree p should then be a functional on the space of (compact) p-dimensional integral manifolds γ : L → M which gives the same value for any two integral manifolds γ_{0}, γ_{1} : L → M which bound a (p+1)-dimensional integral manifold Γ : L × [0, 1] → M of I.

Now, one source of functionals on the space of compact, oriented p-dimensional integral manifolds is the space Ω^{p}(M) of p-forms on M. For any ϕ ∈ Ω^{p}(M) and any immersion γ : L → M with L compact, we can define the functional

F_{ϕ}(γ) = ∫_{L} γ^{∗}(ϕ).

By Stokes’ Theorem, we have

F_{ϕ}(γ_{1}) − F_{ϕ}(γ_{0}) = ∫_{L×[0,1]} Γ^{∗}(dϕ).

Thus, in particular, if Γ^{∗}(dϕ) = 0 for all integral manifolds Γ : L × [0, 1] → M of I, then F_{ϕ} will be a conservation law in our intended sense. Of course, this will necessarily be true if dϕ lies in I. On the other hand, if ϕ itself lies in I, then the functional F_{ϕ} will be identically zero. Moreover, if ϕ ≡ dψ mod I for some (p−1)-form ψ, then the functional F_{ϕ} will, by Stokes’ Theorem, take the same value on any two integral manifolds γ_{0}, γ_{1} : L → M of I which are homologous, whether the homology is via an integral manifold of I or not. Clearly, we will want to regard such functionals as trivial conservation laws. (Compare this with the case of the trivial conservation laws as described above for vector fields.) This discussion leads us to make a definition of the form

C_{0}^{p}(M, I) = \frac{\{ ϕ ∈ Ω^{p}(M) | dϕ ∈ I^{p+1} \}}{\{ dψ | ψ ∈ Ω^{p−1}(M)\} + I^{p}} .

In other words, C_{0}^{p}(M, I) is the cohomology of the quotient complex ( Ω̄^{∗}(M, I), d̄) where Ω̄^{∗}(M, I) = Ω^{∗}(M)/I and d̄ = d mod I. (The subscript ‘0’ is meant to warn the reader that this is our starting point for the definition of conservation laws, the real definition will be given below.)

2.1.1

While this definition of conservation laws does properly generalize the notion of conservation laws for ordinary differential equations, i.e., the case discussed above of a vector field X on a manifold M, it has serious shortcomings as a definition for partial differential equations. To see why, consider the exterior differential system associated to the classical wave equation

u_{xx} − u_{yy} = 0.

As a Monge-Ampere system, this is usually set up as an exterior differential system on M = ℝ^{5} with coordinates (x, y, u, p, q) and an ideal I generated by θ = du − p dx − q dy and the 2-form

Υ = dp ∧ dy + dq ∧ dx.

The 1-form

φ = \frac{1}{2}(p^{2} + q^{2}) dx + pq dy

satisfies dφ = −p dθ+q Υ ∈ I, and hence is closed on all integral surfaces of I. It corresponds to the classical law of conservation of energy for the wave equation:

E(u) = ∫_{ℝ} \frac{1}{2}(u_{x}^{2} + u_{y}^{2}) dx.

However, other expressions which are also known to be conserved, for example

E(u_{x}) = ∫_{ℝ} \frac{1}{2}(u_{xx}^{2} + u_{xy}^{2}) dx.

cannot be represented by an element of C_{0}^{1}(I, M). This is because the manifold M has no coordinates which represent the second derivatives of u.

To get around this problem, we pass to the first prolongation. Now, as discussed in Section 1.3, the underlying manifold M^{(1)} on which the first prolongation is defined is a ℙ^{1} ∪ ℙ^{1} bundle over M. However, the dense open subset W ⊂ M^{(1)} consisting of those integral elements on which dx∧dy is non-vanishing is more easily parametrized: W is simply M × ℝ^{2} = ℝ^{7} with coordinates (x, y, u, p, q, r, s) and the prolonged ideal I^{(1)} is generated in W by three 1-forms

\begin{aligned}θ_{0} & = du − p & dx − q & dy, \\ θ_{1} & = dp − r & dx − s & dy, \\ θ_{2} & = dq − s & dx − r & dy. \end{aligned}

On W, the conservation law E(u_{x}) is then represented by the 1-form

φ′ = \frac{1}{2}(r^{2} + s^{2}) dx + rs dy,

which satisfies dφ′ = −r dθ_{1} − s dθ_{2} ∈ I.

This example points up two problems which must be addressed. First, while this construction captures E(u_{x}), it clearly cannot capture E(u_{xx}), which is also a conservation law. In order to capture all of what are classically known as the ‘local’ conservation laws, one must pass to the infinite prolongation (M^{(∞)}, I^{(∞)}). Second, it can be verified that the class [φ′] ∈ C_{0}^{1}(W, I^{(1)}) is not the restriction of a class in C_{0}^{1}(M^{(1)}, I^{(1)}). This latter problem is not serious if one is only interested in the integral manifolds of I which represent classical solutions of the wave equation since the lift of such an integral manifold clearly lies in W anyway. However, it does point out that one must be careful about domains of definition when one discusses conservation laws for exterior differential systems.

2.1.2

With these preliminary cautions, we are now ready to define the main object of study for the rest of the paper. Let (M, I) be an exterior differential system and let (M^{(k)}, I^{(k)}) be its sequence of prolongations (for integral manifolds of dimension n). We will assume, as usual, that these are all smooth manifolds and that each of the natural maps M^{(k+1)} → M^{(k)} is a smooth, surjective submersion. Also, as usual, we will let M^{(∞)} denote the inverse limit of this tower of submersions

M = M^{(0)} ⟵ M^{(1)} ⟵ M^{(2)} ⋯ ⟵ M^{(k)} ⟵ ⋯ ⟵ M^{(∞)},

and regard it as a sort of infinite dimensional manifold, whose smooth forms are defined as the direct limit of the inclusions

Ω^{∗}(M) = Ω^{∗}(M^{(0)}) ⟶ Ω^{∗}(M^{(1)}) ⟶ Ω^{∗}(M^{(2)}) ⋯ ⟶ Ω^{∗}(M^{(k)}) ⟶ ⋯ ⟶ Ω^{∗}(M^{(∞)}).

Frequently, it will be necessary to consider forms and other objects defined on ‘open subsets’ W^{(∞)} ⊂ M^{(∞)}, where, by definition, an open subset of M^{(∞)} is a sequence of open subsets W^{(k)} ⊂ M^{(k)} with the property that the restrictions of the natural submersions M^{(k+1)} → M^{(k)} to the various W^{(k+1)} define a sequence of surjective submersions W^{(k+1)} → W^{(k)}. To save on notation, we will generally just use I^{(∞)} to denote the differential ideal generated on W^{(∞)} by the restriction of the forms in I^{(∞)}.

The characteristic cohomology of I on W^{(∞)} is the cohomology H̄^{∗}(W^{(∞)}, I) = H^{∗}(Ω̄^{∗}(W^{(∞)}), d̄) of the quotient complex

Ω̄^{∗}(W^{(∞)}) = Ω^{∗}(W^{(∞)})/I^{(∞)},

where d̄= d mod I^{(∞)}.

In [BG_{1}] it was proved that, when I is involutive for integral manifolds of dimension n, the characteristic cohomology satisfies a local vanishing result

H̄^{q}(W^{(∞)}, I) = 0, 0 < q < n−l.

Here, l is an easily computed integer measuring what might be called the degree of “over-determination” of I and should be thought of intuitively as the codimension in an integral n-manifold of the appropriate “initial condition” submanifolds. (In most applications, l = 1.) Thus, in the absence of topological complications, the first (potentially) non-vanishing group is H̄^{n−l}(W^{(∞)}, I), and it is generally defined to be the space C(W^{(∞)}, I) of conservation laws of the exterior differential system I.^{18}

In cases where the exterior differential system I arises from a determined PDE system, it is known that this definition (with an appropriate choice of W^{(∞)}) recaptures the usual notion of the local conservation laws of the system. An important consequence of this fact and the above definition is that the local conservation laws of a PDE system form a group which is nearly invariant under contact transformations, the only dependence coming from the choice of independence conditions imposed by a choice of independent variables.

Another consequence of the definition is that C(W^{(∞)}, I), being the first non-vanishing group and therefore occupying a special place in the spectral sequence of the filtered complex F^{p}Ω^{∗}(W^{(∞)}), is given canonically as the kernel of a certain linear differential operator

D : E_{1}(W^{(∞)}) → E_{2}(W^{(∞)}).

where E_{i}(W^{(∞)}) is the space of sections of a certain vector bundle E_{i} over W^{(∞)} defined in terms of the crude structure equations of the exterior differential system.

2.1.3 In fact, what happens is this: When W^{(∞)} has trivial topology, the short exact sequence of complexes

18Although we will not use any of the attendant machinery or ideas, it would be remiss of us not to point out that what is actually going on is that the quotient complex (Ω̄^{∗}, d̄) is a sheaf of differential complexes and that our cohomology theory is actually a sheaf cohomology.

0 ⟶ I^{(∞)} ⟶ Ω^{∗}(W^{(∞)}) ⟶ Ω̄^{∗}(W^{(∞)}) ⟶ 0

gives rise to an isomorphism

H̄^{n−l}(W^{(∞)}, I) ⟶^{d} H^{n−l+1}(W^{(∞)}, I^{(∞)})

which sends the class [ϕ] ∈ H̄^{n−l}(W^{(∞)}, I) to the class [dϕ] ∈ H^{n−l+1}(W^{(∞)}, I^{(∞)}). Now the vanishing theorem described above actually allows us to find a sub-bundle E_{1} ⊂ (I^{(∞)})^{n−l+1} with the properties that, first, E_{1}(W^{(∞)}) ∩ d((I^{(∞)})^{n−l}) = 0 and, second, there is a containment

Z^{n−l+1}(I^{(∞)}) ⊂ E_{1}(W^{(∞)}) ⊕ d ((I^{(∞)})^{n−l}) .

It then follows directly that

H̄^{n−l}(W^{(∞)}, I) ≃ H^{n−l+1}(W^{(∞)}, I^{(∞)}) ≃ { Φ ∈ E_{1}(W^{(∞)}) dΦ = 0 } .

In specific examples, this gives a method of avoiding the traditional problem of factoring out trivial conservation laws, yielding a canonical representative (albeit a differential form of degree n−l+1) for each conservation law of degree n−l.^{19}

Thus, this approach suggests a systematic approach to computing the space of conservation laws, beginning with a normal form which depends only on the symbol of the system. The first step in this computation is to determine this normal form for conservation laws, i.e., the bundle E_{1}, and that is what we shall turn to now in the case of hyperbolic systems of class s = 0.

2.1.2 A normal form for conservation laws. Let (M, I) be a hyperbolic exterior differential system of class s = 0 which is non-degenerate in the sense of Section 1.5. Let (M^{(∞)}, I^{(∞)}) be the infinite prolongation of (M, I). Our calculations will take place in the domain U ⊂ M of a 1-adapted coframing η = (η^{1}, η^{2}, η^{3}, η^{4}). For simplicity, we whall assume that U is connected and that its deRham cohomology vanishes in positive degrees. Let us fix such a coframing η and let U^{(k)} ⊂ M^{(k)} denote the (open) subset of integral elements of I^{(k−1)} on which Ω = η^{1}∧η^{3} is non-zero and whose projection to M lies in U.^{20} Recall that the assumption that the coframing η be 1-adapted implies

dη^{1} ≡ 0  mod η^{1}, η^{2},  dη^{3} ≡ 0  mod η^{3}, η^{4},
dη^{2} ≡ η^{3} ∧ η^{4} mod η^{1}, η^{2},  dη^{4} ≡ η^{1} ∧ η^{2} mod η^{3}, η^{4}.

Recall now that, in Section 1.3, we showed that there exist (unique) functions h_{20} and h_{02} on U^{(1)} so that the ideal I^{(1)} on U^{(1)} was generated by {η^{2} − h_{20} η^{1}, η^{4} − h_{02} η^{3}}. Conforming to the notation established in that section, let us set

ω_{10} = η^{1},  θ_{10} = η^{2} − h_{20} η^{1},  ω_{01} = η^{3},  θ_{01} = η^{4} − h_{02} η^{3}.

Of course, this immediately gives equations of the form

(1a)  \begin{aligned}dω_{10} & ≡ 0 mod \{ω_{10}, θ_{10}\} \\ dω_{01} & ≡ 0 mod \{ω_{01}, θ_{01}\}. \end{aligned}

2.1.4 Moreover, there clearly exist 1-forms π_{20} and π_{02} satisfying

19In this paper, we will refer to this canonical representative (n−l+1)-form as the differentiated form of the conservation law that it represents and refer to a representing (n−l)-form as the undifferentiated form of the conservation law.

20In most applications, one can find a 1-adapted coframing η whose domain is all of M anyway. Then U^{(k)} is actually open and dense in M^{(k)}, consisting exactly of the integral elements tangent to integral manifolds which satisfy the ‘natural’ independence condition Ω ≠ 0.

\begin{matrix}π_{20} & ≡ dh_{20} \\ π_{02} & ≡ dh_{02} \end{matrix}\} mod ω_{10}, θ_{10}, ω_{01}, θ_{01}

so that

\begin{aligned}dθ_{10} & ≡ −π_{20} ∧ ω_{10} + ω_{01} ∧ θ_{01} mod θ_{10} \\ dθ_{01} & ≡ −π_{02} ∧ ω_{01} + ω_{10} ∧ θ_{10} mod θ_{01}. \end{aligned}

Note that these equations determine π_{20} uniquely modulo {ω_{10}, θ_{10}} and π_{02} uniquely modulo {ω_{01}, θ_{01}}. Let us fix choices for these two 1-forms. Now, again by the argument in Section 1.3, we know that there exist unique functions h_{30} and h_{03} on U^{(2)} so that I^{(2)} on U^{(2)} is generated by {θ_{10}, θ_{01}, θ_{20}, θ_{02}} where

θ_{20} = π_{20} − h_{30} ω_{10}, θ_{02} = π_{02} − h_{03} ω_{01}.

Moreover, there are structure equations of the form

\begin{aligned}dθ_{20} & ≡ −π_{30} ∧ ω_{10} + T_{20} ω_{01} ∧ θ_{01} mod \{θ_{10}, θ_{20}\} \\ dθ_{02} & ≡ −π_{03} ∧ ω_{01} + T_{02} ω_{10} ∧ θ_{01} mod \{θ_{01}, θ_{02}\} \end{aligned}

for some 1-forms π_{30} and π_{30}, congruent to dh_{30} and dh_{03}, respectively, modulo terms semi-basic for U^{(1)}. Using θ_{20} − T_{20} θ_{10} and θ_{02} − T_{02} θ_{02}, respectively, instead of our original θ_{20} and θ_{02}, we can arrange, as we shall, that T_{20} = T_{02} = 0.

Continuing on in this way, we can show that, for every k ≥ 1, there exist functions h_{k+1,0} and h_{0,k+1} defined on U^{(k)} which restrict to each fiber of U^{(k)} → U^{(k−1)} to give a coordinate system and 1-forms θ_{k0} and θ_{0k} well-defined on U^{(k)} with the following properties:

First, the ideal I^{(k)} on U^{(k)} is generated by the 1-forms {θ_{10}, …, θ_{k0}, θ_{01}, …, θ_{0k}}; second, the following structure equations hold

(1b)  \begin{aligned}dθ_{k0} & ≡ −θ_{k+1,0} ∧ ω_{10} + T_{k0} ω_{01} ∧ θ_{01} mod \{θ_{10}, …, θ_{k0}\} \\ dθ_{0k} & ≡ −θ_{0,k+1} ∧ ω_{01} + T_{0k} ω_{10} ∧ θ_{10} mod \{θ_{01}, …, θ_{0k}\} \end{aligned}

with T_{10} = T_{01} = 1 and T_{k0} = T_{0k} = 0 for k > 1; and, third, for all k > 1,

\begin{matrix}θ_{k0} & ≡ dh_{k0} \\ θ_{0k} & ≡ dh_{0k} \end{matrix}\} mod terms semi-basic to U^{(k−1)}.

Note that any 1-form well-defined on U^{(k)} is a linear combination of the forms

{ω_{10}, θ_{10}, …, θ_{k+1,0}, ω_{01}, θ_{01}, …, θ_{0,k+1}}

as these are a basis for the 1-forms semi-basic to U^{(k)}.

In this section, we shall use these equations to deduce general properties about the space C(U^{(∞)}, I), which we will henceforth denote simply by C when there is no danger of confusion. We begin with the following

Proposition : The space C is isomorphic to the space of closed 2-forms on U^{(∞)} of the form

(2)  Φ = A_{10} ω_{10} ∧ θ_{10} + A_{01} ω_{01} ∧ θ_{01} + B_{10} + C + B_{01}

2.1.5 where

B_{10} = ∑_{i<j} B_{10}^{ij} θ_{i0} ∧ θ_{j0} ,  C = ∑_{i,j} C^{ij} θ_{i0} ∧ θ_{0j} ,  B_{01} = ∑_{i<j} B_{01}^{ij} θ_{0i} ∧ θ_{0j} .

Proof : The proof is similar to—but much simpler than—the analogous Proposition in [BG_{2}]. The relevant Spencer-type complex is

Λ^{∗}[ω̄_{10}, θ̄_{10}, θ̄_{20}, …, ω̄_{01}, θ̄_{01}, θ̄_{02}, …]

with boundary operators

δω̄_{10} = δω̄_{01} = 0,  δθ̄_{k,0} = −θ̄_{k+1,0} ∧ ω̄_{10} ,  δθ̄_{0,k} = −θ̄_{0,k+1} ∧ ω̄_{01} .

A simple computation shows that H^{2} of this complex has dimension 2 with basis

Ω̄_{10} = ω̄_{10} ∧ θ̄_{10},  Ω̄_{01} = ω̄_{01} ∧ θ̄_{01}.

Now, from the general theory, C = H̄^{1} is isomorphic to

ker d_{1} : E_{1}^{1,1} → E_{1}^{2,1}

where (E_{r}^{p,q}, d_{r}) is the spectral sequence of the filtered complex F^{p}Ω^{∗}(M^{(∞)}). Moreover, again from the general theory E_{1}^{1,0} = 0 and E_{1}^{1,1} is itself the abutment of the weight spectral sequence Ē_{r}^{k,q} whose Ē_{1}-term is given by sections of a vector bundle whose fibers are cohomology groups of the above complex. Unwinding the definitions leads to (2). □

Note that, regarding the general description of normal forms we gave earlier, the above proposition serves to identify the vector bundle E_{1} over U^{(∞)} as

E_{1} = I ⊕ Λ^{2}(I^{(∞)}),

where I ⊂ Λ^{2}(T^{∗}U) is the rank 2 sub-bundle whose sections are the 2-forms in I and I^{(∞)} ⊂ T^{∗}U^{(∞)} is the sub-bundle (of infinite rank) whose sections generate I^{(∞)}. Thus the above proposition is just the statement that C consists of the closed 2-forms in E_{i}(U^{(∞)}).

We shall refer to the expression (2) as the normal form for a conservation law. Recalling that C was originally defined as a cohomology group (and hence as a quotient space), the import of the above proposition is that each cohomology class has a unique representative 2-form in the above normal form. In fact, without any danger of confusion, we may clearly identify C with the space of closed 2-forms in normal form and we shall do this from now on. We shall write Φ ∈ C as

Φ = A_{10} ω_{10} ∧ θ_{10} + A_{01} ω_{01} ∧ θ_{01} + B_{10} + C + B_{01}

and shall refer to the sum

Q = B_{10} + C + B_{01}

as the quadratic terms in Φ (quadratic refers to quadratic in the ideal I^{(∞)}). It is not difficult to see that the condition

dΦ = 0

implies, first of all, that the coefficients B_{10}^{ij}, C^{ij}, B_{01}^{ij} are expressible as certain linear differential expressions in A_{10} and A_{01} and, secondly, that dΦ = 0 then becomes a linear system of equations on A_{10} and A_{01}. This is, in fact, an overdetermined linear PDE system, and our goal is to relate the analysis of this system to the invariants of the hyperbolic system I.

2.1.6

In order to begin to carry out this program, we shall show that the quadratic terms of a closed 2-form in the normal form (2) satisfy further restrictions in form. To explain these restrictions, we need to define a certain filtration on the group C.

Definition : A conservation law Φ ∈ C has level k if its normal representative is well-defined as a 2-form on U^{(k)}. We denote by C_{k} the vector space of conservation laws of level k.

Since a 2-form Φ in normal form must be closed to represent a conservation law, we see that C_{k} consists of the 2-forms Φ ∈ C which are quadratic expressions in

ω_{10}, θ_{10}, …, θ_{k+1,0}; ω_{01}, θ_{01}, …, θ_{0,k+1}.

It is clear that C_{k} ⊆ C_{k+1} and that

⋃_{k≥0} C_{k} = C.

A somewhat subtle point is that a conservation law Φ may have level exactly equal to k and yet have another representative 2-form—albeit not in normal form—which is defined on U^{(l)} for some l ≤ k.^{21} We shall return to this point later on.

Here we want to indicate the proofs of three basic facts:

Fact 1 : We have

(3)  C_{2k} = C_{2k−1},  k ≧ 1.

That is, when we increase the level we can add new conservation laws in normal form only at odd levels.

Fact 2 : If Φ ∈ C_{2k−1} then the quadratic terms have the form

(4)  \begin{aligned}B_{10} & = B_{10}^{k} ( ∑^{2k−1} ; _{j}=0(−1)^{j}θ_{2k−j,0} ∧ θ_{j+1,0} ) & + \begin{aligned}& ∑ \\ \{ & \begin{matrix}j+i≦2k \\ i<j \end{matrix} \end{aligned} B_{10}^{ij} θ_{i0} ∧ θ_{j0} \\ C & = 0 \\ B_{01} & = B_{01}^{k} ( ∑^{2k−1} ; j=0(−1)^{j}θ_{0,2k−j} ∧ θ_{0,j+1} ) & + \begin{aligned}& ∑ \\ \{ & \begin{matrix}i+j≦2k \\ i<j \end{matrix} \end{aligned} B_{01}^{ij} θ_{0i} ∧ θ_{0j}. \end{aligned}

Since C = 0, the quadratic terms in the conservation law are “unmixed”, i.e.,, if we think of the θ_{k0} and θ_{0k} as “belonging” to the two characteristic systems, then no cross-terms θ_{k0}∧θ_{0l} occur in Φ. This vanishing is a reflection of the fact that in the structure equations the coupling between the two characteristic systems occurs only at the lowest level and only in the form of the terms ω_{10}∧θ_{10} and ω_{01}∧θ_{01}.

For a conservation law Φ of level k, we shall refer to the coefficients B_{10}^{k} and B_{01}^{k} as the highest order terms in Q. It follows from the form of B_{10} and B_{01} that if we have two conservation laws of level 2k−1, say, Φ and Φ̄, such that (in the obvious notation)

B_{10}^{k} = B̄_{10}^{k}, B_{01}^{k} = B̄_{01}^{k}

2.1.7 then Φ − Φ̄ is a conservation law of level 0 when k = 1 and is of level at most 2k−3 when k ≥ 2.

21For example, a hyperbolic system of class s = 0 may become Darboux integrable exactly at level k for k = 0, 1, 2, . . ..

At this point the system gets (at least) “two functions of two variables worth” of new conservation laws. But these new conservation laws will not in general be in normal form, and we may have to pass to a higher level to achieve this. On the other hand, if we pick up two functions of two variables worth of new conservation laws in algebraic normal form at some level 2k + 1, then the system is Darboux integrable at this level, and perhaps even at a lower one.

Before stating our final property, we recall that the characteristic systems are given by the formulae

Ξ_{10}^{(∞)} = {ω_{10}, θ_{10}, θ_{20}, … }
Ξ_{01}^{(∞)} = {ω_{01}, θ_{01}, θ_{02}, … }.

Fact 3 : For two conservation laws Φ and Φ̄ ∈ C_{2k−1}, the ratios [B_{10}^{k} : B̄_{10}^{k}] = B_{10}^{k}/B̄_{10}^{k} and [B_{01}^{k} : B̄_{01}^{k}] = B_{01}^{k}/B̄_{01}^{k} are functions that satisfy

(5)  \begin{aligned}d[B_{10}^{k} : B̄_{10}^{k}] & ∈ Ξ_{10}^{(k)} \\ d[B_{01}^{k} : B̄_{01}^{k}] & ∈ Ξ_{01}^{(k)}. \end{aligned}

To see how useful these facts are, let us note that they imply the following result:

Corollary : If neither of the characteristic systems has an integrable subsystem, then for k ≧ 1,

dim C_{2k−1}/C_{2k−2} ≦ 2.

Discussion : Fact 3 clearly links the issue of adding new conservation laws at a certain (odd) level to integrable subsystems of the characteristic systems. This corollary should be contrasted with the discussion of Darboux’ method in Section 1.4, which depended on there being integrable subsystems in the characteristic systems. In fact, if there is a rank 2 integrable subsystem in one of the characteristic systems, then we always have an infinite dimensional space of conservation laws at some finite level. Actually, this and other evidence suggests a sort of converse to this result might be true. Roughly stated, this converse should be: “If the new conservation laws that appear at a certain level ‘depend on’ at least two arbitrary functions of two variables, then the system is integrable by the method of Darboux at this level.”

Before turning to the proof of these results, we would like to make some general observations on the equations

d(A_{10} ω_{10} ∧ θ_{10} + A_{01} ω_{01} ∧ θ_{01} + B_{10} + B_{01}) = 0

for a conservation law of level k. Denote by Ξ_{10}^{(k)⊥} and Ξ_{01}^{(k)⊥} the distributions dual to the characteristic systems (recall that dim M^{(k)} = 2k + 4 and each of these distributions has dimension k + 2), and denote by ▽_{10} and ▽_{01} the operation of total differentiation relative to these distributions. Then a consequence of dΦ = 0 will be equations

B_{10}^{ij} = linear function of A_{10}, A_{01} and their derivatives
B_{01}^{ij} = linear function of A_{10}, A_{01} and their derivatives

and

(6)  \{ \begin{matrix}▽_{01}A_{10} & = L_{10}(A_{10}, A_{01}, B_{10}^{ij}, B_{01}^{ij}) \\ ▽_{10}A_{01} & = L_{01}(A_{10}, A_{01}, B_{10}^{ij}, B_{01}^{ij}) \end{matrix}

where L_{10} and L_{01} are linear in the A’s and B’s.

We will also find linear equations

(7)  \{ \begin{matrix}▽_{01}B_{10}^{k} = K_{10} · B_{10}^{k} \\ ▽_{10}B_{01}^{k} = K_{01} · B_{01}^{k} \end{matrix}

2.1.8 for the highest order terms; these equations obviously imply (5). Of course, the relation dΦ = 0 will contain much more information than (6) and (7); we simply wanted to point out once again the role of the characteristic systems.

These considerations—especially equation (6)—perhaps help to explain the following consequence of the corollary to Fact 3:

Suppose that the characteristic distributions Ξ_{10}^{(k)⊥} and Ξ_{01}^{(k)⊥} are bracket generating. Then C_{k}/C_{0} is finite dimensional.

We will only prove (3) and (4) in the cases of levels k = 1, 2, 3. In this way we believe the general pattern should be clear and the argument more easily understood than with a notationally complicated inductive procedure.

We will, however, first list some general principles that would apply to make the calculations at any level. We will use the notation Ω^{(∞)} for the differential forms on M^{(∞)} and F^{p} for the image

Λ^{p}I^{(∞)} ⊗ Ω^{(∞)} → Ω^{(∞)}.

The calculations will use only the structure equations (1). For a 2-form in normal form giving a conservation law we have

Φ ≡ A_{10} ω_{10} ∧ θ_{10} + A_{01} ω_{01} ∧ θ_{01} mod F^{2}

and

dΦ ≡ 0 mod F^{2}.

Thus dΦ ∈ F^{2} and the calculation will only use the assumption that

dΦ ≡ 0 mod F^{3},

which appears to be weaker than assuming that dΦ = 0. However, because the ideal I^{(∞)} contains no integrable subsystems, it is easy to see that any closed 3-form in Λ^{3}(I^{(∞)}) must be identically zero anyway.

For the general principles, we will use the notation

\begin{aligned}Θ_{k0} & = θ_{k0} ∧ … ∧ θ_{10} \\ Θ_{0k} & = θ_{0k} ∧ … ∧ θ_{01}. \end{aligned}

Then we have from (1)

(i)  \begin{aligned}d(ω_{10} ∧ θ_{10}) ∧ Λ^{2}[θ_{10}, ω_{10}, θ_{01}, ω_{01}] & = 0 \\ d(ω_{01} ∧ θ_{01}) ∧ Λ^{2}[θ_{10}, ω_{10}, θ_{01}, ω_{01}] & = 0. \end{aligned}

We note that (i) implies

(ii)  d(A_{10} ω_{10} ∧ θ_{10} + A_{01} ω_{01} ∧ θ_{01}) ∧ Λ^{3}[θ_{10}, ω_{10}, θ_{01}, ω_{01}] = 0.

Next, for any function F and 1-forms α and β, we have the identity

(iii)  d(F α ∧ β) ∧ α ∧ β = 0.

Finally, again from (i)

(iv)  \begin{aligned}dθ_{k0} ∧ Θ_{k0} ∧ ω_{10} ∧ ω_{01} & = dθ_{k0} ∧ Θ_{k0} ∧ ω_{10} ∧ θ_{01} & = 0 \\ dθ_{0k} ∧ Θ_{0k} ∧ ω_{01} ∧ ω_{10} & = dθ_{0k} ∧ Θ_{0k} ∧ ω_{01} ∧ θ_{10} & = 0. \end{aligned}

2.1.9 Level k = 1 : A priori we have

Φ = A_{10}ω_{10} ∧ θ_{10} + A_{01}ω_{01} ∧ θ_{01} + B_{10}θ_{20} ∧ θ_{10} + B_{01}θ_{01} ∧ θ_{02}

+ C_{22}θ_{20} ∧ θ_{02} + C_{21}θ_{20} ∧ θ_{01} + C_{12}θ_{10} ∧ θ_{02} + C_{11}θ_{10} ∧ θ_{01},

and we want to show that all C_{ij} = 0. By (1), (ii), (iii), (iv)

0 = dΦ ∧ (Θ_{20} ∧ θ_{01} ∧ ω_{01}) = −C_{22}θ_{30} ∧ ω_{10} ∧ (Θ_{20} ∧ θ_{01} ∧ ω_{01})

which gives C_{22} = 0. Once C_{22} = 0 we have

0 = dΦ ∧ (Θ_{20} ∧ θ_{02} ∧ θ_{03} ∧ ω_{01}) = −C_{21}θ_{30} ∧ ω_{10} ∧ θ_{01} ∧ (Θ_{20} ∧ θ_{02} ∧ θ_{03} ∧ ω_{01})

which gives C_{21} = 0, and then by symmetry C_{12} = 0. Finally

0 = dΦ ∧ (θ_{10} ∧ θ_{02} ∧ ω_{10}) = −C_{11}θ_{20} ∧ ω_{10} ∧ θ_{01} ∧ (θ_{10} ∧ θ_{02} ∧ ω_{10})

which gives C_{11} = 0. Level k = 2 : We are given a closed 2-form

(8)  Φ = A + B_{10} + C + B_{01}

where

A = A_{10} ω_{10} ∧ θ_{10} + A_{01} ω_{01} ∧ θ_{01}

and

\begin{aligned}B_{10} & = ∑_{j<i≤3} B_{10}^{ij} & θ_{i0} & ∧ θ_{j0} \\ C & = ∑_{i,j≤3} C^{ij} & θ_{i0} & ∧ θ_{0j} \\ B_{01} & = ∑_{i<j≤3} B_{01}^{ij} & θ_{0i} & ∧ θ_{0j} \end{aligned}

and we want to show that Φ must already of level 1, i.e. all terms with a θ_{30} or θ_{03} should drop out. By (1) and our general principles

0 = dΦ ∧ (θ_{30} ∧ θ_{10} ∧ Θ_{03} ∧ ω_{01}) = −B_{10}^{32}θ_{40} ∧ ω_{10} ∧ θ_{20} ∧ (θ_{30} ∧ θ_{10} ∧ Θ_{03} ∧ ω_{01})

so that B_{10}^{32} = 0. Next

0 = dΦ ∧ (θ_{30} ∧ θ_{20} ∧ Θ_{03} ∧ ω_{01}) = −B_{10}^{31}θ_{40} ∧ ω_{10} ∧ θ_{10} ∧ (θ_{30} ∧ θ_{20} ∧ Θ_{03} ∧ ω_{01})

gives B_{10}^{31} = 0. By symmetry B_{01}^{32} = B_{01}^{31} = 0. Turning to the cross-terms we have successively

0 = dΦ ∧ (Θ_{30} ∧ Θ_{02} ∧ ω_{01})  = −C^{33}θ_{40} ∧ ω_{10} ∧ θ_{03} ∧ (Θ_{30} ∧ Θ_{02} ∧ ω_{01})
0 = dΦ ∧ (Θ_{30} ∧ θ_{01} ∧ θ_{03} ∧ ω_{01}) = −C^{32}θ_{40} ∧ ω_{10} ∧ θ_{02} ∧ (Θ_{30} ∧ θ_{01} ∧ θ_{03} ∧ ω_{01})
0 = dΦ ∧ (Θ_{30} ∧ θ_{02} ∧ θ_{03} ∧ ω_{01}) = −C^{31}θ_{40} ∧ ω_{10} ∧ θ_{01} ∧ (Θ_{30} ∧ θ_{02} ∧ θ_{03} ∧ ω_{01})

which gives C^{33} = C^{32} = C^{31} = 0, and by symmetry all C^{ij} = 0.

2.1.10 Level k = 3 : We again have a closed 2-form (8) where

\begin{aligned}B_{10} & = ∑_{j<i≤4} B_{10}^{ij} & θ_{i0} & ∧ θ_{j0} \\ C & = ∑_{i,j≤4} C^{ij} & θ_{i0} & ∧ θ_{0j} \\ B_{01} & = ∑_{i<j≤4} B_{01}^{ij} & θ_{0i} & ∧ θ_{0j}. \end{aligned}

We have

\begin{aligned}0 & = dΦ ∧ (θ_{40} ∧ θ_{20} ∧ θ_{10} ∧ Θ_{04} ∧ ω_{01}) & = −B_{10}^{43}θ_{50} ∧ ω_{10} ∧ θ_{30} ∧ (θ_{40} ∧ θ_{20} ∧ θ_{10} ∧ Θ_{04} ∧ ω_{01}) \\ 0 & = dΦ ∧ (θ_{40} ∧ θ_{30} ∧ θ_{10} ∧ Θ_{04} ∧ ω_{01}) & = −B_{10}^{42}θ_{50} ∧ ω_{10} ∧ θ_{20} ∧ (θ_{40} ∧ θ_{30} ∧ θ_{10} ∧ Θ_{04} ∧ ω_{01}) \end{aligned}

which gives B^{43} = B^{42} = 0. More interestingly

0 = dΦ ∧ (θ_{30} ∧ θ_{10} ∧ Θ_{04} ∧ ω_{01}) = (B_{10}^{41} + B_{10}^{32}) θ_{40} ∧ θ_{20} ∧ ω_{10} ∧ (θ_{30} ∧ θ_{10} ∧ Θ_{04} ∧ ω_{01})

which gives

B_{10} = B_{10}^{41}(θ_{40} ∧ θ_{10} − θ_{30} ∧ θ_{20}) + (lower order terms)

and similarly for B_{01}.

An argument similar to those for levels k = 1, 2 shows that the cross terms C are identically zero.

This completes our discussions of (3) and (4), and we now turn to (5) which again we shall do only for levels 1 and 3. We consider a conservation law

Φ = A_{10} ω_{10} ∧ θ_{10} + A_{01} ω_{01} ∧ θ_{01} + B_{10} θ_{20} ∧ θ_{10} + B_{01} θ_{01} ∧ θ_{02}

of level 1. We will derive a relation for dB_{10}, and for the calculation it is notationally convenient to set θ_{00} = ω_{01}.

From the structure equations we will have a relation

0 = dΦ ∧ ω_{10} = (dB_{10} + χB_{10}) ∧ θ_{20} ∧ θ_{10} ∧ ω_{10} + ∑_{ν≥0} ρ^{ν} ∧ θ_{0ν} + ∑_{0≦μ<ν} ρ^{μν} ∧ θ_{0μ} ∧ θ_{0ν}

where χ does not depend on Φ and where

ρ^{ν} ≡ 0 mod Λ^{3}[θ_{10}, ω_{10}, ω_{01}, θ_{01}, θ_{02}].

In particular ρ^{ν} does not contain a θ_{20} term, and it follows that

dB_{10} + χB_{10} ≡ 0 mod Ξ_{10}^{(1)}.

If Φ̄ is another conservation law, then this equation implies that

B̄_{10}dB_{10} − B_{10}dB̄_{10} ≡ 0 mod Ξ_{10}^{(1)},

which was to be proved.

2.1.11 Turning now to a level 3 conservation law, which we now write as

Φ = Φ_{10} + Φ_{01}

where

Φ_{10} = A_{10}ω_{10} ∧ θ_{10} + B_{10}^{21}θ_{20} ∧ θ_{10} + B_{10}^{31}θ_{30} ∧ θ_{10} + B_{10}(θ_{40} ∧ θ_{10} − θ_{30} ∧ θ_{20})
Φ_{01} = A_{01}ω_{01} ∧ θ_{01} + B_{01}^{12}θ_{01} ∧ θ_{02} + B_{01}^{13}θ_{01} ∧ θ_{03} + B_{01}(θ_{01} ∧ θ_{04} − θ_{02} ∧ θ_{03})

we have as before

0 = dΦ ∧ θ_{30} ∧ θ_{20} ∧ ω_{10} = (dB_{10} + χB_{10}) ∧ θ_{40} ∧ θ_{30} ∧ θ_{20} ∧ θ_{10} ∧ ω_{10} + ∑_{0≦ν} ρ^{ν} ∧ θ_{0ν} + ∑_{0≦μ<ν} ρ^{μν} ∧ θ_{0μ} ∧ θ_{0ν}.

Inspection of the structure equations shows that

ρ^{ν} ≡ 0 mod Λ^{5}[θ_{30}, θ_{20}, θ_{10}, ω_{01}, ω_{01}, θ_{01}, θ_{02}, …]

i.e., ρ^{ν} does not contain a θ_{40}-term. It follows that

dB_{10} + χB_{10} ≡ 0 mod Ξ_{10}^{(3)},

and the argument proceeds as before.

2.1.3 An example of higher level conservation laws. In the remaining sections of this paper we shall be concerned with level 0 or, as we shall call them, ‘classical’ conservation laws. Before going on, however, we would like to conclude this section with a subsection in which we give one calculation of higher level conservation laws in an interesting example.

Example: We consider the EDS associated to the (slightly modified) s = 0 sine-Gordon system

u_{y} = sin v
v_{x} = sin u.

As coframing we take

ω^{1} = dx, ω^{2} = du − sin v dy, ω^{3} = dy, ω^{4} = dv − sin u dx

with structure equations

dω^{1} = dω^{3} = 0

and

dω^{2} = cos v (− sin u ω^{1} ∧ ω^{3} + ω^{3} ∧ ω^{4})
dω^{4} = cos u (− sin v ω^{3} ∧ ω^{1} + ω^{1} ∧ ω^{2}).

We will show that for this system

dim C_{0} = 1  and  dim C_{1}/C_{0} = 3.

Remark: For the s = 0 sine-Gordon system we will see that both Ξ_{10}^{(1)} and Ξ_{01}^{(1)} have rank one, spanned by dx and dy respectively. Thus, the corollary following Fact 3 does not apply.

2.1.12 First, we compute the level 0 conservation laws. For Φ ∈ C_{0}, we have

Φ = A ω^{1} ∧ ω^{2} + B ω^{3} ∧ ω^{4}

the condition dΦ = 0 gives

(9)  \begin{aligned}A_{4} & = 0 & B_{2} = 0 \\ A_{3} & = B cos u & B_{1} = A cos v \end{aligned}

where we have set df = f_{i} ω^{i} for any function f. Explicitly, in terms of the classical partials notation,

\begin{aligned}f_{2} & = f_{u} & f_{4} = f_{v} \\ f_{1} & = f_{v} sin u + f_{x} & f_{3} = f_{u} sin v + f_{y}. \end{aligned}

Thus, (9) expands to

A_{v} = 0  B_{u} = 0
A_{u} sin v + A_{y} = B cos u  B_{v} sin u + B_{x} = A cos v.

The top two equations imply that A does not depend on v and that B does not depend on u. Freezing the value of u in the lower left hand equation at some value u_{0} for which cos u_{0} is non-zero then shows that B must be of the form B = b_{0} + b_{1} sin v for some functions b_{0} and b_{1} of x and y. Similarly, from the lower right hand equation, we see that A = a_{0} + a_{1} sin u for some functions a_{0} and a_{1} of x and y. Substituting these expressions into the lower two equations yields

a_{1} cos u sin v + (a_{0})_{y} + (a_{1})_{y} sin u = b_{0} cos u + b_{1} sin v cos u
b_{1} cos v sin u + (b_{0})_{x} + (b_{1})_{x} sin v = a_{0} cos v + a_{1} sin u cos v .

Now comparing coefficients in these equations shows that we must have a_{0} = b_{0} = 0 while a_{1} = b_{1} and (a_{1})_{y} = (b_{1})_{x} = 0. These latter equations clearly imply that a_{1} = b_{1} = c for some constant c. Thus, dim C_{0} = 1 and the space of classical (i.e., level 0) conservation laws of the s = 0 sine-Gordon system is spanned by

Φ = sin u dx ∧ du + sin v dy ∧ dv.

The corresponding undifferentiated conservation law has the form

φ = cos u dx + cos v dy.

We now turn to the analysis of level 1 conservation laws. On M^{(1)} = M ×ℝ^{2}, where ℝ^{2} has coordinates (h_{20}, h_{02}), we take as coframing

\begin{aligned}ω_{10} & = ω^{1}  θ_{10} & = ω^{2} − h_{20} ω^{1} \\ ω_{01} & = ω^{3}  θ_{01} & = ω^{4} − h_{02} ω^{3} \end{aligned}

and

π_{20} = dh_{20} − cos v sin u ω_{01}
π_{02} = dh_{02} − cos u sin v ω_{10}.

With this coframing, we have

\begin{aligned}dθ_{10} & = −π_{20} ∧ ω_{10} − cos v θ_{01} ∧ ω_{01} \\ dθ_{01} & = −π_{02} ∧ ω_{01} − cos u θ_{10} ∧ ω_{10} \end{aligned}

2.1.13 conforming to the general form of the structure equations as discussed above.

According to Fact 2 of the general theory, any Φ ∈ C_{1} has the general form

Φ = A′ θ_{10} ∧ ω_{10} + B′ θ_{01} ∧ ω_{01} + C θ_{10} ∧ θ_{20} + D θ_{01} ∧ θ_{02}.

By modifying A′ and B′ in the obvious way, we may rewrite Φ as

Φ = A θ_{10} ∧ ω_{10} + B θ_{01} ∧ ω_{01} + C θ_{10} ∧ π_{20} + D θ_{01} ∧ π_{02}.

For notational convenience, we will rename the 1-forms in the coframing of M^{(1)} as follows:

\begin{aligned}ψ_{1} & = ω_{10}  ψ_{3} & = θ_{10}  ψ_{5} & = π_{20} \\ ψ_{2} & = ω_{01}  ψ_{4} & = θ_{01}  ψ_{6} & = π_{02}. \end{aligned}

Thus, Ξ_{10}^{(1)} = {ψ_{1}, ψ_{3}, ψ_{5}} and Ξ_{01}^{(1)} = {ψ_{2}, ψ_{4}, ψ_{6}}. For any function F on M^{(1)}, we define the functions F_{i} by the rule

dF = F_{1} ψ_{1} + F_{2} ψ_{2} + ⋯ + F_{6} ψ_{6}.

Now, expanding out the equation dΦ = 0 and collecting coefficients yields the following equations

C_{2} = C_{4} = C_{6} = 0, D_{1} = D_{3} = D_{5} = 0

as well as the equations

A_{2} = B cos u + C (sin^{2} u sin v − h_{20} cos v cos u) B_{1} = A cos v + D (sin^{2} v sin u − h_{02} cos u cos v)
A_{4} = D sin u sin v  B_{3} = C sin v sin u
A_{6} = D cos u  B_{5} = C cos v

and

A_{5} = C_{1}  B_{6} = D_{2} .

Now the first set of equations imply that

dC ∈ Ξ_{10}^{(1)} dD ∈ Ξ_{01}^{(1)}.

However, it may be verified that the largest integrable subsystem of Ξ_{10}^{(1)} (respectively, Ξ_{01}^{(1)}) is of rank one and is spanned by ω_{10} (respectively, ω_{01}). This implies the equations

C_{3} = C_{5} = D_{4} = D_{6} = 0.

In particular, since ω_{10} = dx, it follows that C must be a function of x alone and, for a similar reason, D must be a function of y alone. Of course, this implies that C_{1} = C_{x} and D_{2} = D_{y}.

Now, from the definition of the 1-forms ψ_{i} it follows that, in the coordinate system (x, y, u, v, h_{20}, h_{02}), for any function F on M^{(1)}, the expression F_{5} is the partial of F with respect to h_{20} and F_{6} is the partial of F with respect to h_{02}. The above equations for A_{5}, A_{6}, B_{5} and B_{6} then suggest that the functions Ã and B̃ defined by

Ã = A − C_{1} h_{20} − D cos u h_{02}  B̃ = B − C cos v h_{20} − D_{2} h_{02}

2.1.14 should depend only on x, y, u, and v. In fact, if we redefine our notation so as to rewrite the conservation law in the form

Φ = (A + C_{x} h_{20} + Dh_{02} cos u) θ_{10} ∧ ω_{10} + (B + Ch_{20} cos v + D_{y} h_{02}) θ_{01} ∧ ω_{01} + C θ_{10} ∧ π_{20} + D θ_{01} ∧ π_{02}

where C (respectively, D) is understood to be a function only of x (respectively, y), then the closure conditions for Φ simplify to

A_{5} = B_{5} = A_{6} = B_{6} = 0

together with

A_{2} = B cos u + C sin^{2} u sin v − C_{x} sin u cos v + Dh_{20} sin v sin u
B_{1} = A cos v + D sin^{2} v sin u − D_{y} sin v cos u − Ch_{02} cos u cos v
A_{4} = D sin u sin v
B_{3} = C sin u sin v.

These last two equations suggest that, in fact, A + D sin u cos v is a function of x, y, and u only and that B + C sin v cos u is a function of x, y, and v only. Thus, if we redefine our notation once again so that

Φ = (A − D sin u cos v + C_{x} h_{20} + D cos u h_{02}) θ_{10} ∧ ω_{10}
+ (B − C sin v cos u + C cos v h_{20} + D_{y} h_{02}) θ_{01} ∧ ω_{01}
+ C θ_{10} ∧ π_{20} + D θ_{01} ∧ π_{02}

we find that the closure conditions simplify yet again. They simply become (assuming, as usual, that C is a function of x alone and D is a function of y alone)

A_{4} = A_{5} = A_{6} = 0, B_{3} = B_{5} = B_{6} = 0

plus the relations

A_{2} = B cos u + (D_{y} − C_{x}) sin u cos v + (D cos v − C cos u) cos u sin v
B_{1} = A cos v + (C_{x} − D_{y}) sin v cos u + (C cos u − D cos v) cos v sin u.

Moreover, since A is a function of x, y, and u alone while B is a function of x, y, and v alone, it easily follows from our definitions that

A_{2} = A_{y} + A_{u} sin v  and  B_{1} = B_{x} + B_{v} sin u.

Let us substitute these expressions into the above equations and take note of the fact that A does not depend on v while B does not depend on u. In the second equation, freezing v at some value v_{0} for which cos v_{0} is non-zero, we see that A is a linear combination of {1, cos u, sin u, cos u sin u} with coefficients which are functions of x and y. In a similar manner, the first equation implies that B is a linear combination of {1, cos v, sin v, cos v sin v} with coefficients which are functions of x and y. Thus, let us write

A = a_{0} + a_{1} cos u + a_{2} sin u + a_{3} cos u sin u, B = b_{0} + b_{1} cos v + b_{2} sin v + b_{3} cos v sin v

where the a_{i} and b_{i} are functions of x and y alone. Expanding the above relations out and setting equal the various coefficients of the resulting trigonometric polynomials then yields the relation C_{x} − D_{y} = 0, the relations

a_{0} = a_{1} = b_{0} = b_{1} = 0,

2.1.15 the relations a_{3} = −C and b_{3} = −D, and finally equations which imply that a_{2} = b_{2} = c where c is a constant. Since C is a function of x alone and D is a function of y alone, it follows that the common value C_{x} = D_{y} must be a constant. Thus, we finally get

A = c sin u − (E_{0}x + C_{0}) cos u sin u
B = c sin v − (E_{0}y + D_{0}) cos v sin v
C = E_{0}x + C_{0}
D = E_{0}y + D_{0}

where c, C_{0}, D_{0}, and E_{0} are constants. Moreover, these values of A, B, C, and D are easily seen to yield a closed 2-form Φ. Thus, we have shown that dim C_{3} = 4, as we wanted.

In fact, with a little inspection, we can write

Φ = d (cϕ_{0} + C_{0} ϕ_{1} + D_{0} ϕ_{2} + E_{0} ϕ_{3})

where the ϕ_{i} are the undifferentiated conservation laws and are given by formulae of the form

ϕ_{0} = cos u dx + cos v dy
ϕ_{1} = −h_{20} θ_{10} − \frac{1}{2}(h_{20}^{2} + sin^{2} u) dx + cos u cos v dy
ϕ_{2} = −h_{02} θ_{01} − \frac{1}{2}(h_{02}^{2} + sin^{2} v) dy + cos u cos v dx
ϕ_{3} = −xh_{20} θ_{10} − yh_{02} θ_{01} − \frac{1}{2}(xh_{20}^{2} − x cos u − 2y cos u cos v) dx − \frac{1}{2}(yh_{02}^{2} − y cos v − 2x cos u cos v) dy.

2.1.16 2.2 Determination of hyperbolic systems having the maximum number of classical conserva-

tion laws: first steps

Let (M, I) be a hyperbolic exterior differential system of class s = 0 and denote by C_{0} the space of conservation laws of level zero or, as we shall call them, classical conservation laws. Assuming symmetric behavior in the two characteristic systems in this section we will prove that:

If dim C_{0} ≧ 7, then both characteristic systems have an integrable subsystem.

Later on we will see that if dim C_{0} ≧ 7, then dim C_{0} = ∞ and there is a local normal form for such hyperbolic systems.

Our technique is to use the equivalence method to express the structure equations of (M, I) in an invariant manner, which has been done in Section 1.5 above, and then to simply compute. The calculations are generally straightforward but a couple of them are somewhat lengthy. For these we have used MAPLE. The main issue of course is to interpret the calculations as we go along, and we shall give these steps without always reproducing the full printout of MAPLE calculations. We now proceed to the calculations.

Recall that a classical conservation law has the form

(1)  Φ = A Ω_{10} + C Ω_{01}

where

Ω_{10} = ω^{1} ∧ ω^{2}  and  Ω_{01} = ω^{3} ∧ ω^{4}.

Using the structure equations (5) in Section 1.5, we easily deduce that the closure condition dΦ = 0 is equivalent to the following differential relations on the functions A and C

(2)  \begin{aligned}dA & = A ϕ_{44} + A_{i} ω^{i} \\ dC & = C ϕ_{22} + C_{i} ω^{i} \end{aligned}

where

(3)  \{ \begin{matrix}A_{3} & = C & A_{4} & = 0 \\ C_{1} & = A & C_{2} & = 0. \end{matrix}

Since the 2-plane distributions defined by Ω_{10} and Ω_{01} are both non-integrable, we know that by taking commutators we may “capture” at least one more derivative of each of A and C. In this regard the main initial step in the computation is given by the

Lemma: With the above notations

A_{2} − C_{4} = Cq_{3} − Ap_{1}.

Proof : This follows directly from the identities

\begin{aligned}d(dA) & ≡ 0 mod ω^{1}, ω^{2}, \\ d(dC) & ≡ 0 mod ω^{3}, ω^{4}. \end{aligned}

□

Using the lemma, if we define B by

A_{2} + C_{4} = 2B

2.2.1 then we have the relations

(4)  \{ \begin{matrix}A_{2} & = B − \frac{1}{2}(Ap_{1} & − Cq_{3}) \\ C_{4} & = B − \frac{1}{2}(Cq_{3} & − Ap_{1}). \end{matrix}

Lemma: We have

(5)  dB = B(ϕ_{22} + ϕ_{44}) + \frac{1}{2}(Cp_{4} ϕ_{21} + Aq_{2} ϕ_{43}) + B_{i} ω^{i}

where^{22}

(6)  B_{i} ≡ 0 mod A, C, B, A_{1}, C_{3}.

Proof : The identities

d(dA) ≡ 0 mod ω^{1}
d(dC) ≡ 0 mod ω^{3}

give, using (3) and (4), relations of the form (6). The exact formula (6) will result from an explicit computation that we shall give below. □

Knowing the values of A and C at a point x gives us A_{3}, A_{4}, C_{1}, C_{2} at that point. Knowing B(x) gives us A_{2}(x) and C_{4}(x). By the above lemma, however, we may determine all the A_{2i}(x) and C_{4i}(x) in terms of A, C, B, A_{1}, C_{3} at that point. Iterating this gives us all higher derivatives at x of A_{2}, A_{3}, A_{4} and C_{1}, C_{2}, C_{3} in terms of A, C, B, A_{1}, C_{3} at the point. Thus in the formal Taylor’s series of A and C we are only free to specify the derivatives A_{1} and C_{3}. This suggests that

The space of classical conservation laws can at most depend on two arbitrary functions of one variable.

One way of thinking about this “function count” is this: Solutions to a determined set of two equations for two unknowns are uniquely determined by initial values along a 3-dimensional hypersurface. Adding one more equation will in general cut the initial value manifold down to 2 dimensions, and adding a fourth equation will in general cut the initial value set down to a curve. Since (3) is a set of 4 equations for 2 unknowns we expect that the solutions will at most be given by arbitrary initial data along a curve. We say “at most” because this function count will only work if the system is involutive, otherwise there are even fewer solutions. Our main result will be to classify systems for which this upper bound is achieved.

Now, we have already developed linear expressions for

A_{2}, A_{3}, A_{4}; C_{1}, C_{2}, C_{4}

in terms of A, C, B, A_{1}, C_{3}; now we shall turn to A_{1} and C_{3}. From the identities d(dA) = d(dC) = 0 we will show that:

(7)  \begin{aligned}dA_{1} & = A_{1}(2ϕ_{44} − ϕ_{22}) & + (B − \frac{1}{2}(Ap_{1} − Cq_{3}))ϕ_{21} & + Aq_{1} ϕ_{43} & + A_{1i} ω^{i} \\ dC_{3} & = C_{3}(2ϕ_{22} − ϕ_{44}) & + (B − \frac{1}{2}(Cq_{3} − Ap_{1}))ϕ_{43} & + Cp_{3} ϕ_{21} & + C_{3i} ω^{i} \end{aligned}

2.2.2

22In this and similar situations to be encountered below, the expression F ≡ 0 mod G_{1}, …, G_{n} will mean “there exists a formula or identity expressing F as a linear combination of the functions G_{1}, …, G_{n} where the coefficients are known combinations of coframing invariants and their derivatives”. In this specific instance, these formulae for B_{i} are given explictly below. However, we will not always write out these formulae for it frequently suffices to know just that there exists such formulae in order to complete a proof. Frequently, the explicit formulae are quite ungainly.

where

\begin{matrix}A_{12} & ≡ A_{13} & ≡ A_{14} & ≡ 0 \\ C_{34} & ≡ C_{31} & ≡ C_{32} & ≡ 0 \end{matrix}\} mod A, C, B, A_{1}, C_{3}.

In particular this implies

\begin{matrix}dA_{1} & ≡ A_{11} ω^{1} \\ dC_{3} & ≡ C_{33} ω^{3} \end{matrix}\} mod A, C, B, A_{1}, C_{3}.

As expected, there is at most one free derivative of the quantities A_{1} and C_{3} respectively. Iterating this, in the Taylor’s series of A and C we find that we are at most free to assign the terms A_{1...1} and C_{3...3}, confirming the statement above.

Proof of (7) : The identities d(dA) ≡ 0 mod ω^{1} and d(dC) ≡ 0 mod ω^{3} give

\begin{aligned}B_{1} & = −\frac{1}{2}A(p_{11} − q_{3}) + \frac{1}{2}C(q_{31} − k_{14}) − \frac{1}{2}A_{1}p_{1} + C_{3}q_{1} \\ B_{3} & = −\frac{1}{2}C(q_{33} − p_{1}) + \frac{1}{2}A(p_{13} − k_{32}) − \frac{1}{2}C_{3}q_{3} + A_{1}p_{3} \end{aligned}

and

\begin{aligned}B_{2} & = −A(\frac{1}{2}p_{12} − \frac{1}{4}p_{1}^{2} + p_{4}) & + C(\frac{1}{2}q_{32} − k_{24} − \frac{1}{2}p_{3}q_{1} − \frac{1}{4}p_{1}q_{3}) & − \frac{1}{2}Bp_{1} & + C_{3}q_{2} \\ B_{4} & = −C(\frac{1}{2}q_{34} − \frac{1}{4}q_{3}^{2} + q_{2}) & + A(\frac{1}{2}p_{14} − k_{42} − \frac{1}{2}q_{1}p_{3} − \frac{1}{4}q_{3}p_{1}) & − \frac{1}{2}Bq_{3} & + A_{1}p_{4}. \end{aligned}

From

0 = d(dA) = d(A ϕ_{44} + A_{i} ω^{i}) = dA_{1} ∧ ω^{1} + (other terms),

and the formulas for A_{2}, A_{3}, A_{4} and B_{i} which give the “(other terms)”, we obtain the expression for dA_{1} where A_{12}, A_{13}, A_{14} ≡ 0 mod A, C, B, A_{1}, C_{3}. The argument for dC_{3} is similar. □

The issue now is: What conditions are imposed on the invariants of the system in order that we may arbitrarily assign the coefficients A_{1...1} and C_{3...3}?

To begin developing the answer to this we shall first show that

(8)  \begin{matrix}p_{4}A_{11} & ≡ 0 \\ q_{2}C_{33} & ≡ 0 \end{matrix}\} mod A, C, B, A_{1}, C_{3}.

Proof : Above we have given formulae for B_{i} as explicit linear combinations of A, C, B, A_{1}, C_{3}. In turn, except for A_{11} and C_{33} the exterior derivatives of A, C, B, A_{1}, C_{3} are again explicit linear combinations of these five quantities. Thus, the condition

d(dB) = 0

may impose conditions on A_{11} and C_{33}. The explicit formula for d(dB) is lengthy, but when we compute it on MAPLE and then set A = C = B = A_{1} = C_{3} = 0, the expression reduces to

(9)  (p_{3}A_{11} − q_{1}C_{33}) ω^{1} ∧ ω^{3} + p_{4}A_{11} ω^{1} ∧ ω^{4} + q_{2}C_{33} ω^{3} ∧ ω^{2}.

The last two terms come from A_{1}p_{4} in B_{4} and C_{3}q_{2} in B_{2} respectively. It is clear that setting this expression equal to zero gives (8). □

The following is an immediate consequence of (8).

Proposition : If p_{4}q_{2} ≠ 0, then dim C_{0} ≦ 5.

Proof : If p_{4}q_{2} ≠ 0, then we may inductively determine each term in the formal Taylor’s series of A and C as a linear combination of A, C, B, A_{1}, C_{3}. Thus, the PDE system for A and C is at best a completely integrable system whose solutions are specified by five constants. □

2.2.3

It follows that if dim C_{0} ≧ 6, then (keeping in mind our assumption of symmetry) we must have p_{4} = q_{2} = 0, i.e., the rank 2 Pfaffian system Θ = Ξ_{10}^{⟨1⟩} ∪ Ξ_{01}^{⟨1⟩} is integrable. Assuming this, it follows from (9) that

p_{3}A_{11} − q_{1}C_{33} ≡ 0 mod A, C, B, A_{1}, C_{3}.

The exterior derivative of this relation gives

\begin{matrix}p_{3}A_{111} & ≡ 0 \\ q_{1}C_{333} & ≡ 0 \end{matrix}\} mod A, C, B, A_{1}, C_{3}, A_{11}, C_{33} .

By reasoning similar to that above, this immediately implies the following result. Proposition: If p_{4} = q_{2} = 0 but p_{3}q_{1} ≠ 0, then dim C_{0} ≦ 6.

It follows that if dim C_{0} ≧ 7, then we must have p_{4} = q_{2} = 0 and p_{3} = q_{1} = 0, i.e., the Pfaffian systems Ξ_{10}^{⟨1⟩} and Ξ_{01}^{⟨1⟩} are integrable.

For systems with

(10)  p_{4} = q_{2} = 0 and p_{3} = q_{1} = 0

the formulas for B_{i} simplify to give

\begin{aligned}B_{1} & = −\frac{1}{2}A(p_{11} − q_{3}) + \frac{1}{2}C(q_{31} − k_{14}) − \frac{1}{2}A_{1}p_{1} \\ B_{3} & = −\frac{1}{2}C(q_{33} − p_{1}) + \frac{1}{2}A(p_{13} − k_{32}) − \frac{1}{2}C_{3}q_{3} \end{aligned}

and

\begin{aligned}B_{2} & = −\frac{1}{2}A(p_{12} − \frac{1}{2}p_{1}^{2}) & + C(\frac{1}{2}q_{32} − k_{24} − \frac{1}{4}p_{1}q_{3}) & − \frac{1}{2}Bp_{1} \\ B_{4} & = −\frac{1}{2}C(q_{34} − \frac{1}{2}q_{3}^{2}) & + A(\frac{1}{2}p_{14} − k_{42} − \frac{1}{4}q_{3}p_{1}) & − \frac{1}{2}Bq_{3}. \end{aligned}

The condition

d(dB) ≡ 0 mod A, C, B

then gives, after a MAPLE calculation using the formulas obtained so far,

(11)  0 = (A_{1}k_{23} − C_{3}k_{41}) ω^{1} ∧ ω^{3} + A_{1}k_{24} ω^{1} ∧ ω^{4} + C_{3}k_{42} ω^{3} ∧ ω^{2}.

Thus, if k_{24}k_{42} ≠ 0 this implies as before that dim C_{0} ≦ 3.

We may summarize our results as the following Proposition:

(i) If Θ = Ξ_{10}^{⟨1⟩} ∪ Ξ_{01}^{⟨1⟩} is not integrable, then

dim C_{0} ≦ 5;

(ii) Assuming that Θ is integrable, if Ξ_{10}^{⟨1⟩} and Ξ_{01}^{⟨1⟩} are not separately integrable, then dim C_{0} ≦ 6;

(iii) Assuming that Ξ_{10}^{⟨1⟩} and Ξ_{01}^{⟨1⟩} are separately integrable, if k_{24}k_{42} ≠ 0 then

dim C_{0} ≦ 3.

To compute the conservation laws for a given example we may either compute the invariants of the given system and see how these fit with the general analysis given in this and the following section, or else we may compute directly using the general theory to guide the calculation. At the end of the preceeding section, we gave an example of the latter.

2.2.4 2.3 The two classes of hyperbolic systems having the maximum number of classical conser-

vation laws

2.3.1 Symmetric systems for which dim C_{0} ≧ 7. In this section, our main goal is to prove the following theorem

Theorem: If (M, I) is a non-degenerate, symmetric hyperbolic system with s = 0 on a connected manifold M for which dim C_{0}(U) ≧ 7 for all open sets U ⊂ M, then, either (M, I) is locally linearizable, or else (M, I) is locally isomorphic to the s = 0 Liouville system.

As we shall show in the next subsection, systems of this type actually have dim C_{0}(U) = ∞ for all open sets U ⊂ M.

Proof: By the results of the previous section, we know that we can now restrict attention to hyperbolic systems whose characteristic systems each have an integrable subsystem; i.e., whose invariants satisfy

(1)  p_{4} = q_{2} = 0 and p_{3} = q_{1} = 0

and which in addition have

(2)  k_{24} = k_{42} = 0.

By the discussion in Section 2.2, these conditions hold for any symmetric hyperbolic system which has dim C_{0} ≧ 7.

The structure equations (from Section 1.5) of systems satisfying (1) and (2) now simplify to

d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}ϕ_{44} − ϕ_{22} & 0 & 0 & 0 \\ ϕ_{21} & ϕ_{22} & 0 & 0 \\ 0 & 0 & ϕ_{22} − ϕ_{44} & 0 \\ 0 & 0 & ϕ_{43} & ϕ_{44} \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}−p_{1} ω^{1}∧ω^{2} \\ ω^{3}∧ω^{4} \\ −q_{3} ω^{3}∧ω^{4} \\ ω^{1}∧ω^{2} \end{matrix} )

with

\begin{aligned}dϕ_{22} & = q_{3} ω^{1} ∧ ω^{2} & − κ_{1} ∧ ω^{1} & − κ_{2} ∧ ω^{2} \\ dϕ_{44} & = p_{1} ω^{3} ∧ ω^{4} & − κ_{3} ∧ ω^{3} & − κ_{4} ∧ ω^{4} \end{aligned}

and

\begin{aligned}dp_{1} & = p_{1} ϕ_{22} + ∇p_{1} \\ dq_{3} & = q_{3} ϕ_{44} + ∇q_{3}. \end{aligned}

With these assumptions, the condition d(dB) = 0 may then be expanded to give (cf. (11) of the preceeding section)

A_{1}k_{23} − C_{3}k_{41} ≡ 0 mod A, C, B.

Differentiating this gives

\begin{matrix}A_{11}k_{23} & ≡ 0 \\ C_{33}k_{41} & ≡ 0 \end{matrix} \} mod A, C, B, A_{1}, C_{3}.

It follows that if k_{23}k_{41} ≠ 0 then we must have dim C_{0} ≦ 4. Consequently, (since we are only considering symmetric systems) we shall assume that

(3)  k_{23} = 0  and  k_{41} = 0.

2.3.1

Using (1), (2) and (3) in the identities d(dω^{i}) = 0 we obtain

(4)  \{ \begin{matrix}p_{13} & = k_{32} \\ q_{31} & = k_{14}. \end{matrix}

With (1)–(4) satisfied, the formula for dB now simplifies to

(5)  \begin{aligned}dB = B(ϕ_{22} + ϕ_{44}) & − \frac{1}{2}(A(p_{11} − q_{3}) + A_{1}p_{1})ω^{1} & − \frac{1}{2}(A(p_{12} − \frac{1}{2}p_{1}^{2}) − \frac{1}{2}Cp_{1}q_{3} + Bp_{1})ω^{2} \\ & − \frac{1}{2}(C(q_{33} − p_{1}) + C_{3}q_{3})ω^{3} & − \frac{1}{2}(C(q_{34} − \frac{1}{2}q_{3}^{2}) − \frac{1}{2}Aq_{3}p_{1} + Bq_{3})ω^{4}. \end{aligned}

From d(dB) = 0 we obtain

(6)  \begin{aligned}(i) & (B + \frac{1}{2}(Ap_{1} + Cq_{3}))k_{14} & − Aq_{34} & = 0 \\ (ii) & (B + \frac{1}{2}(Ap_{1} + Cq_{3}))k_{32} & − Cp_{12} & = 0. \end{aligned}

Taking the derivative of (i) in the ω^{1}-direction, we see from (5) that dB has a −\frac{1}{2}A_{1}p_{1} term so that cancellation occurs and we are left with

A_{1}q_{34} ≡ 0 mod A, C, B

and similarly

C_{3}p_{12} ≡ 0 mod A, C, B.

It follows that if p_{12}q_{34} ≠ 0 then dim C_{0} ≦ 3. Thus, we can assume that the relations

(7)  p_{12} = 0  and  q_{34} = 0

are satisfied.

From (6) and (7) we obtain

(8)  \{ \begin{matrix}(i) & (B + \frac{1}{2}(Ap_{1} + Cq_{3}))k_{14} & = 0 \\ (ii) & (B + \frac{1}{2}(Ap_{1} + Cq_{3}))k_{32} & = 0. \end{matrix}

We are now going to show that k_{14} = k_{32} = 0. Suppose not, then (8) implies

(9)  B = −\frac{1}{2}(Ap_{1} + Cq_{3}).

Computing dB in two ways—using (5) and using (9)—we find

Aq_{3} = 0  and  Cp_{1} = 0.

Now, since AC ≠ 0, we necessarily have p_{1} = q_{3} = 0 and so ∇p_{1} = ∇q_{3} = 0. Equation (4) now gives k_{14} = k_{32} = 0 anyway.

Using k_{14} = k_{32} = 0, the identity d(dB) = 0 yields
(11)  (B + \frac{1}{2}(Ap_{1} + Cq_{3}))(k_{13} − k_{31}) − A(q_{33} − p_{1}) + C(p_{11} − q_{3}) = 0.

2.3.2 The ω^{1} and ω^{3}-derivatives of this relation give

\begin{matrix}A_{1}(q_{33} − p_{1}) & ≡ 0 \\ C_{3}(p_{11} − q_{3}) & ≡ 0 \end{matrix}\} mod A, C, B

implying that if (p_{11} − q_{3})(q_{33} − p_{1}) ≠ 0 then dim C_{0} ≦ 3. We thus can assume that

(12)  \begin{aligned}p_{11} − q_{3} & = 0 \\ q_{33} − p_{1} & = 0 \end{aligned}

in which case, (11) reduces to

(B + \frac{1}{2}(Ap_{1} + Cq_{3}))(k_{13} − k_{31}) = 0.

At this point the analysis splits into two cases:

(13)  (i) k_{13} − k_{31} = 0  or  (ii) B + \frac{1}{2}(Ap_{1} + Cq_{3}) = 0.

Suppose (13.ii) holds. This is equation (9) above and we get the same conclusion as in that case, namely that p_{1} = q_{3} = 0. Moreover, we now have, from the structure equations and identities found so far that dϕ_{22} = α_{1}∧ω^{1} and dϕ_{44} = α_{3}∧ω^{3} for some 1-forms α_{i}. Also, differentiating the equations

dω^{1} = −(ϕ_{44} − ϕ_{22}) ∧ ω^{1}, dω^{3} = −(ϕ_{22} − ϕ_{44}) ∧ ω^{3}

yields that d(ϕ_{22} − ϕ_{44})∧ω^{1} = d(ϕ_{22} − ϕ_{44})∧ω^{3} = 0. Of course, this implies that α_{1}∧ω^{1}∧ω^{3} = α_{3}∧ω^{1}∧ω^{3} = 0, so we finally conclude that

dϕ_{22} = k_{13} ω^{1} ∧ ω^{3}, dϕ_{44} = k_{31} ω^{3} ∧ ω^{1}.

We now see that the structure equations satisfy the conditions given by the Proposition in Section 1.5.6 which characterize linear systems. It follows that the systems which satisfy (13.ii) (as well as the previously derived conditions) are precisely the linear systems.

We shall next analyze the situation when (13.i) holds, and for this we set

k_{13} = k_{31} = k_{00}.

If we now differentiate (12) we obtain the relations

(14)  \{ \begin{matrix}p_{1}(k_{00} − 1) & = 0 \\ q_{3}(k_{00} − 1) & = 0. \end{matrix}

Thus, there are two possibilities: First, we could have k_{00} ≠ 1, in which case we would have to have p_{1} = q_{3} = 0, which, as we have seen, leads to the linear case. Thus, we may set this aside and assume, as we shall, that k_{00} = 1, i.e., k_{13} = k_{31} = 1.

The structure equations of the system are

(15)  d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}ϕ_{44} − ϕ_{22} & 0 & 0 & 0 \\ ϕ_{21} & ϕ_{22} & 0 & 0 \\ 0 & 0 & ϕ_{22} − ϕ_{44} & 0 \\ 0 & 0 & ϕ_{43} & ϕ_{44} \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}−p_{1} ω^{1}∧ω^{2} \\ ω^{3}∧ω^{4} \\ −q_{3} ω^{3}∧ω^{4} \\ ω^{1}∧ω^{2} \end{matrix} )

2.3.3

(16)  \begin{aligned}dϕ_{22} & = q_{3} ω^{1} ∧ ω^{2} + ω^{1} ∧ ω^{3} \\ dϕ_{44} & = p_{1} ω^{3} ∧ ω^{4} + ω^{3} ∧ ω^{1} \end{aligned}

and

(17)  \begin{aligned}dp_{1} & = p_{1} ϕ_{22} + q_{3} ω^{1} \\ dq_{3} & = q_{3} ϕ_{44} + p_{1} ω^{3}. \end{aligned}

The eidetic reader will recognize these equations as (11-13) of Section 1.5.7, which characterize the systems whose first prolongations are integrable by the method of Darboux. In that subsection, we proved that, on the domain in M where p_{1} and q_{3} were both non-zero (the only case we are concerned with due to our hypothesis of symmetry), the ideal I was diffeomorphic to that of the s = 0 Liouville system. This completes our proof. □

2.3.2 Explicit conservation laws in the case dim C_{0} = ∞. We are now going to explicitly describe the conservation laws in the two cases found in the previous section.

The first case is that of linear systems. According to Section 1.5.6, such a system is locally of the form

I = { (du − P v dy) ∧ dx, (dv − Qu dx) ∧ dy }

where P and Q are positive functions of x and y only. Tracing through all of the above calculations (or just doing it directly, we see that all of the conservation laws are of the form

Φ = A (du − P v dy) ∧ dx + C (dv − Qu dx) ∧ dy

where A and C are functions of x and y alone satisfying the equations

\begin{aligned}A_{y} & = QC \\ C_{x} & = P A. \end{aligned}

Thus, the space of local conservation laws is clearly of infinite dimension.

Consider now the s = 0 Liouville system described in Section 1.5. This is described by the ideal

I = { (du − e^{v} dy) ∧ dx, (dv − e^{u} dx) ∧ dy }

and a short calculation using the above formulae shows that the conservation laws are of the form

Φ = [(g(y) + f(x))e^{u} + f′(x)] (du − e^{v}dy) ∧ dx + [(f(x) + g(y))e^{v} + g′(y)] (dv − e^{u}dx) ∧ dy

where f and g are arbitrary functions of one variable. Thus, again, we see that the space of conservation laws is of infinite dimension. The undifferentiated conservation law corresponding to such a Φ is given by

\begin{aligned}φ & = (f(x) + g(y))(du − e^{v}dy) − g(y)du + (f(x) + g(y))(dv − e^{u}dx) − f(x)dv \\ & = f(x)(du − e^{v} dy − e^{u} dx) + g(y)(dv − e^{u} dx − e^{v} dy). \end{aligned}

(The first form of φ enables us to easily check that dφ = −Φ). On classical solution surfaces

(x, y) → (x, y, u(x, y), v(x, y))

2.3.4 to s = 0 Liouville we have

φ = f(x)(u_{x} − e^{u})dx + g(y)(v_{y} − e^{v})dy;

clearly dφ = 0 on any such solution surface.

Our calculations have thus proved the following result:

Proposition: If a symmetric hyperbolic system (M, I) has dim C_{0} ≧ 7, then it has dim C_{0} = ∞.

We will now show how the conservation laws of s = 0 Liouville may be used to analyze the singularities of the solutions. This discussion will be heuristic and is merely intended to illustrate how conservation laws might be used to gain analytical insight into the solutions of an equation. Of course, in the particular case of the s = 0 Liouville equation, the precise results may be either derived analytically or verified from the explicit form of the solutions to the s = 0 Liouville system given in Section 1.5 above.

The initial value problem is typically posed by prescribing initial data on the line x+y = 0 and seeking a solution in the half-plane x + y ≥ 0. Thus, setting

t = \frac{1}{2}(x + y)  ξ = \frac{1}{2}(x − y)

or equivalently

x = t + ξ  y = t − ξ

we are given initial data u(ξ, 0), v(ξ, 0) and we seek to find a solution u(ξ, t), v(ξ, t) defined for t ≥ 0. Since

\{ \begin{matrix}u_{x} & − e^{u} & = & u_{ξ} & + e^{v} & − e^{u} \\ v_{y} & − e^{v} & = − & v_{ξ} & + e^{u} & − e^{v} \end{matrix}

the conservation laws give that for any functions f, g the integrals

\begin{matrix}∫ f(t + ξ) & ( & u_{ξ} & + e^{v} & − e^{u}) & dξ = C_{1} \\ ∫ g(t − ξ) & (− & v_{ξ} & + e^{u} & − e^{v}) & dξ = C_{2} \end{matrix}

are formally independent of t. More precisely, if we assume that the initial data is smooth then u(ξ, t), v(ξ, t) will remain smooth for 0 ≤ t < t_{0}. If we now restrict ξ, t to the rectangle |ξ| ≤ a, 0 ≤ t < t_{0} then for all smooth compactly supported f, g with uniformly bounded L^{2} norm we will have

\begin{matrix}∫ f(t + ξ) & ( & u_{ξ} & + e^{v} & − e^{u}) & dξ = O(1) \\ ∫ g(t − ξ) & (− & v_{ξ} & + e^{u} & − e^{v}) & dξ = O(1). \end{matrix}

By taking f, g to be sequences tending to δ-functions we will thus have

(i)  u_{ξ} + e^{v} − e^{u} = O(1)
(ii)  −v_{ξ} + e^{u} − e^{v} = O(1)

2.3.5 on rectangles as above. We will use these relations to analyze the possible singularities that u and v can develop.

For t = t_{0} we assume that u(ξ, t_{0}), v(ξ, t_{0}) are of class C^{1} in a neighborhood 0 < ξ < ϵ and seek to determine what sort of singularity may develop as we approach the origin.^{23} For this we assume that u_{ξ} and v_{ξ} have asymptotic expansions

\begin{aligned}u_{ξ} & ∼ −\frac{c}{ξ^{λ}} & + r, & c ≠ 0 \\ v_{ξ} & ∼ −\frac{c′}{ξ^{λ′}} & + s, & c′ ≠ 0 \end{aligned}

where ξ^{λ}r → 0 and ξ^{λ′}s → 0 as ξ ↓ 0. By adding (i) and (ii) we see that

λ = λ′, c = c′

and we shall consider the case c > 0.

If λ > 1 we then have

u = (\frac{c}{λ − 1}) \frac{1}{ξ^{λ−1}} + R

where ξ^{λ−1}R → 0 as ξ ↓ 0, and similarly for v. By (i) we have up to a bounded term

−\frac{c}{ξ^{λ}} + r = e^{^{(\frac{c}{λ−1})}^{(\frac{1}{ξ^{λ−1}} )}} (e^{R} − e^{S}) ,

which is impossible unless u = v, a case that may be handled directly.^{24} Thus we must have λ ≤ 1. If 0 < λ < 1 then both e^{u} and e^{v} are O(1) as ξ ↓ 0 and this contradicts (i). Thus we must have λ = 1 which gives the expansions

\begin{aligned}u_{ξ} & ∼ −\frac{c}{ξ} & + O(1) \\ v_{ξ} & ∼ −\frac{c}{ξ} & + O(1) \end{aligned}

as ξ ↓ 0. If the initial data is real-analytic then it may be shown that this heuristic reasoning is justified leading to the result:

There is a real analytic curve Γ in the half-plane x + y ≥ 0 such that u, v are bounded and piecewise real analytic away from Γ, but where singularities of u, v occur at the points of intersection Γ ∩ {t = t_{0}}. Moreover, c is a positive integer giving the order of contact of Γ with the line t = t_{0} at the point of intersection.^{25}

We offer the above discussion not as a definitive analysis of the singularity structure of the solutions of the s = 0 Liouville system but rather to illustrate how conservation laws may be used to infer analytic behavior.

2.3.6 2.4 Moment conditions

23To be more precise, we should consider u(ξ, t), v(ξ, t) as t ↑ t_{0} and ξ ↓ 0, and assuming that u, v are of class C^{1} in ξ for t < t_{0} see what type of singularity may develop at t = t_{0}, ξ = 0.

24When u = v, the s = 0 Liouville system reduces to a simple Ricatti f′ = f^{2} with dependence on a parameter.

25If t = t_{0} is the first time a singularity is encountered, then for t slightly larger than t_{0} the singularity will split into “simple” singularities for each of which the asymptotic expansion holds with c = 1.

Given a differential ideal I on a manifold M we may speak of the complex of piecewise smooth singular I-chains and the resulting I-homology, denoted by H_{∗,I}(M) (cf. Section 6 in [BG_{1}]). Of particular interest are the first non-vanishing local I-homology groups and their relationship to conservation laws. In the situation of a hyperbolic exterior differential system on a 4-manifold, we work locally in a contractible open set and the issue is this:

Let γ ⊂ M be a closed curve. Then γ = ∂S is the boundary of a (piecewise smooth) surface, and we ask when S may be taken to be an integral surface of I?

Necessary conditions are given by the moment conditions coming from the space C_{0} of classical conservation laws, as follows: For Φ ∈ C_{0} we have

Φ = dφ

for some 1-form φ, and by Stokes’ theorem

∫_{γ} φ = ∫_{S} Φ.

The right hand side vanishes in case S is an integral surface of I, and by definition the moment conditions are

∫_{γ} φ = 0, Φ ∈ C_{0}.

The question is whether they are sufficient as well as necessary in order to be able to fill in γ with an integral surface, or at least with a piecewise smooth integral chain whose simplices are integral surfaces.

We shall discuss this question in the following variational form. Let γ_{0} ⊂ M be a non-characteristic smooth curve given as the image of a smooth immersion

f : [0, 1] → M,

and for some δ > 0, let γ_{t}, 0 ≦ t ≦ δ be a variation of γ_{0} given by a smooth mapping

F : [0, 1] × [0, δ] → M

where γ_{t} = f_{t}([0, 1]) with f_{t} = F |_{[0,1]×{t}}.

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γ_{δ} γ_{0}

2.4.1 For δ sufficiently small f_{t} will be an immersion and γ_{t} will be non-characteristic. For 0 ≤ T ≤ δ we set

S_{T} = ∪_{0≦t≦T}γ_{t}

with

S = S_{δ}

and we make the assumption

For all T and for all Φ ∈ C_{0} we have

(1)  ∫_{∂S_{T}} φ = 0,  0 ≤ T ≤ δ.

We then have the following

Proposition: If dim C_{0} = ∞, then the moment conditions (1) are necessary and sufficient that the S be an integral surface of (M, I).

Proof: We must show that for each point p ∈ S_{ϵ} the tangent plane T_{p}S_{ϵ} is an integral element of I. Clearly it will suffice to do this for p ∈ γ_{0}. Let τ be the tangent vector field to γ_{0} and ν the variation vector field for the family of curves γ_{t}. Note that the vector field ν is only defined modulo τ. We want to show that for each p ∈ γ_{0} the 2-plane ν(p)∧τ(p) is an integral element.

Suppose now that Φ ∈ C_{0} is compactly supported along each γ_{t}—i.e., Φ vanishes near the endpoints of the γ_{t}. Then a standard calculation gives

(2)  0 = \frac{d}{dt} (∫_{S_{t}} Φ)_{t=0} = ∫_{γ_{0}} ν Φ.

Now Φ ∈ C_{0} has the form

Φ = A Ω_{10} + C Ω_{01}

and depends upon “two arbitrary functions of one variable”. In fact, as is clear from the discussion in Section 2.3.2, we may specify A and C arbitrarily along a non-characteristic curve γ_{0}. Thus, the only way that we can have (2) for all Φ is that ν Ω_{10} and ν Ω_{01} both restrict to zero on γ_{0}. This is equivalent to

⟨Ω_{10}, ν ∧ τ⟩ = ⟨Ω_{01}, ν ∧ τ⟩ = 0

along γ_{0}, which is what was to be proved. □

2.4.2 3. Symplectic Hyperbolic Systems

3.1 Definition and structure equations; characterization of Euler-Lagrange systems

A symplectic hyperbolic exterior differential system is given by the data (M, Φ, I) where M is a 4-dimensional manifold, Φ is a symplectic form on M, and I is a differential ideal that defines a hyperbolic system with Φ ∈ I. Integral surfaces of I are thus necessarily Lagrangian surfaces for Φ. Examples of symplectic hyperbolic systems include the Euler-Lagrange systems introduced in Section 1.6.

The automorphisms of a symplectic hyperbolic system (M, Φ, I) are diffeomorphisms of M that preserve both Φ and I. As we shall see, inequivalent symplectic hyperbolic systems may well become equivalent when considered as hyperbolic systems. In particular, a non-linear (M, Φ, I) may linearize when we consider only (M, I).

We shall quickly work through the equivalence problem for symplectic hyperbolic systems. As before, we shall assume that the hyperbolic system itself is symmetric and non-degenerate. This leads us to consider coframes ω^{1}, ω^{2}, ω^{3}, ω^{4} such that

(1)  \{ \begin{matrix}Ω_{10} = ω^{1}∧ω^{2} \\ Ω_{01} = ω^{3}∧ω^{4} \end{matrix}

and with the additional condition that the symplectic form is

(2)  Φ = ω^{1} ∧ ω^{2} + ω^{3} ∧ ω^{4}.

As in Section 1.5 we may use the assumed non-integrability of Ω_{10} and Ω_{01} to assume further that ω^{1} and ω^{3} generate the first derived systems of Ξ_{10} and Ξ_{01} and that

(3)  \{ \begin{matrix}dω^{2} ≡ ω^{3}∧ω^{4} mod \{ω^{1}, ω^{2}\} \\ dω^{4} ≡ ω^{1}∧ω^{2} mod \{ω^{3}, ω^{4}\}. \end{matrix}

Coframes satisfying (1)–(3) will be called symplectic coframes.

Keeping the notation of Section 1.5, diagonal transformations that preserve the conditions (2), (3) must satisfy

ac^{−1} · a^{−1} = 1
a^{−1}c · c^{−1} = 1,

and hence must all be 1. Thus the structure group of the G-structure B_{G} → M is now 2-dimensional consisting of matrices of the form

( \begin{matrix}1 & 0 & 0 & 0 \\ μ & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & ν & 1 \end{matrix} ) .

Being smaller than our previous group we may expect more invariants, although the condition dΦ = 0 will tend to impose relations among them.

The equivalence problem proceeds much as in Section 1.5. The structure equations have the form

(4)  d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}0 & 0 & 0 & 0 \\ φ_{21} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & φ_{43} & 0 \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}−φ_{11}∧ω^{1} & −φ_{12}∧ω^{2} − ω^{1}∧ω^{3} \\ φ_{11}∧ω^{2} & + ω^{3}∧ω^{4} \\ −φ_{33}∧ω^{3} & −φ_{34}∧ω^{4} − ω^{3}∧ω^{1} \\ φ_{33}∧ω^{4} & + ω^{1}∧ω^{2} \end{matrix} )

3.1.1 By modifying the pseudo-connection terms in the obvious way we may assume that the torsion is

(5)  ( \begin{matrix}−k_{0} ω^{1} ∧ ω^{2}− & (a_{3} ω^{3} & + a_{4} ω^{4}) ∧ ω^{1} & − (p_{3} ω^{3} & + p_{4} ω^{4}) ∧ ω^{2} & − ω^{1} ∧ ω^{3} \\ & (a_{3} ω^{3} & + a_{4} ω^{4}) ∧ ω^{2} & & & + ω^{3} ∧ ω^{4} \\ −k_{0} ω^{3} ∧ ω^{4}− & (c_{1} ω^{1} & + c_{2} ω^{2}) ∧ ω^{3} & − (q_{1} ω^{1} & + q_{2} ω^{2}) ∧ ω^{4} & − ω^{3} ∧ ω^{1} \\ & (c_{1} ω^{1} & + c_{2} ω^{2}) ∧ ω^{4} & & & + ω^{1} ∧ ω^{2} \end{matrix} ) .

The fiber variation of the individual terms in the torsion is given by

δk_{0} = δp_{4} = δq_{2} = 0

and

\{ \begin{matrix}δp_{3} & = p_{4} φ_{43} \\ δa_{3} & = a_{4} φ_{43} & + p_{3} φ_{21} \\ δa_{4} & = p_{4} φ_{21} \\ δq_{1} & = q_{2} φ_{21} \\ δc_{1} & = c_{2} φ_{21} & + q_{1} φ_{43} \\ δc_{2} & = q_{2} φ_{43} & . \end{matrix}

Thus k_{0}, p_{4}, q_{2} are (absolute) invariants of the G-structure. If p_{4} = q_{2} = 0 then p_{3}, q_{1} and a_{4}, c_{2} become invariants, and if in addition these vanish then a_{3} and c_{1} become invariants. The interpretations of these quantities may be given in an analogous manner to what was done in Section 1.5.

As an application of these structure equations, we shall answer the following interesting question: When is a symplectic hyperbolic exterior differential system an Euler-Lagrange system? This is a special case of the more general question: When is a hyperbolic exterior differential system an Euler-Lagrange system? We shall not attempt to answer this latter question in this paper, but, in principle, the techniques we use could do so.

Our answer is summarized in the following

Proposition : The necessary and sufficient conditions that a hyperbolic system with torsion given by (5) be Euler-Lagrange is that the invariants satisfy

(6)  k_{0} = 0, p_{4} = q_{2}, a_{4} = q_{1}, c_{2} = p_{3}, a_{3} = c_{1}.

Proof : We seek the conditions that there exist

Ψ = A ω^{1} ∧ ω^{2} + C ω^{3} ∧ ω^{4}

satisfying

(i)  Φ ∧ Ψ = 0
(ii)  dΨ = 0.

Clearly (i) gives that

Ψ = B(ω^{1} ∧ ω^{2} − ω^{3} ∧ ω^{4})

for some function B. Then (ii) gives

dB = −2B(ω^{1} + ω^{3}),

and then the identity d(dB) = 0 implies the conditions (6). □

This result has a curious

3.1.2 Corollary : Let (M, I) be a hyperbolic Euler-Lagrange exterior differential system, and assume that each characteristic system Ξ_{10}, Ξ_{01} has an integrable subsystem.^{26} Then (M, I) is linear.

This corollary stands in strong contrast to the exterior differential system associated to f-Gordon equations z_{xy} = f(z), all of which are Euler-Lagrange but which are non-linear unless f is linear.

For the proof we note that under the above assumptions we have

k_{0} = p_{4} = q_{2} = p_{3} = q_{1} = a_{4} = c_{2} = 0 and a_{3} = c_{1} = s_{0}

and hence the structure equations reduce to

dω^{1} = (s_{0} − 1) ω^{1} ∧ ω^{3}
dω^{3} = (s_{0} − 1) ω^{3} ∧ ω^{1}

and

dω^{2} ≡ s_{0} ω^{3} ∧ ω^{2} + ω^{3} ∧ ω^{4} mod ω^{1}
dω^{4} ≡ s_{0} ω^{1} ∧ ω^{4} + ω^{1} ∧ ω^{2} mod ω^{3}

where

ds_{0} ≡ 0 mod ω^{1}, ω^{3}.

The last equation results from the identities d^{2}ω^{1} = d^{2}ω^{3} = 0. These structure equations can now be applied to compute the invariants of the ideal I as described in Section 1.5. The result of comparing these structure equations to those is that the system satisfies the conditions to be a linear system set forth in Section 1.5.6.

Another interesting consequence is the:

Corollary: The s = 0 Liouville exterior differential system is not an Euler-Lagrange system.

Proof : We consider the above situation of a symplectic manifold (M, Φ) and a hyperbolic Euler-Lagrange exterior differential system I such that Φ ∈ I. The above proposition implies that we may write the structure equations in the form

d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}ϕ_{44} − ϕ_{22} & 0 & 0 & 0 \\ ϕ_{21} & ϕ_{22} & 0 & 0 \\ 0 & 0 & ϕ_{22} − ϕ_{44} & 0 \\ 0 & 0 & ϕ_{43} & ϕ_{44} \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}ω^{2}∧(−p_{3} ω^{1} + p_{3} ω^{3} + t_{0} ω^{4}) \\ ω^{3}∧ω^{4} \\ ω^{4}∧(−q_{1} ω^{3} + q_{1} ω^{1} + t_{0} ω^{2}) \\ ω^{1}∧ω^{2} \end{matrix} )

where

ϕ_{22} = −(s_{0} ω^{3} + q_{1} ω^{4}) − ω^{1}
ϕ_{44} = −(s_{0} ω^{1} + p_{3} ω^{2}) − ω^{3}

and

ϕ_{21} ≡ φ_{21} − ω^{2} mod ω^{1}
ϕ_{43} ≡ φ_{43} − ω^{4} mod ω^{3}.

Comparing this with the structure equations (5) in Section 1.5 we deduce that

If a hyperbolic EDS has the properties that (i) each characteristic system contains an integrable subsystem and (ii) p_{1}q_{3} ≠ 0, then the system is not Euler-Lagrange.

By the computation at the end of Section 1.5, the s = 0 Liouville system has these properties and hence cannot be Euler-Lagrange. □

\{ \begin{matrix}u_{y} & = f(x, y, u, v) \\ v_{x} & = g(x, y, u, v). \end{matrix}

3.1.3 3.2 The three classes of symplectic hyperbolic systems having an infinite number of classical

26This is equivalent to (M, I) being locally the exterior differential system associated to a PDE system

conservation laws

In this section we shall discuss the conditions imposed on a (symmetric) symplectic hyperbolic exterior differential system (M, Φ, I) to have dim C_{0} = ∞. Writing

Ψ = A Ω_{10} + C Ω_{01}

the condition

dΨ = 0

is equivalent to the overdetermined system

(1)  \{ \begin{matrix}A_{3} & = C & − A & A_{4} = 0 \\ C_{1} & = A & − C & C_{2} = 0. \end{matrix}

The analysis of the solutions to these equations proceeds in exactly the same way as that given in sections 2.2 and 2.3, and we shall only give the conclusions.

It turns out that there are three classes of symplectic hyperbolic systems with dim C_{0} = ∞ and we shall discuss these.

Class A: The structure equations have the form

(2)  d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}0 & 0 & 0 & 0 \\ φ_{21} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & φ_{43} & 0 \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}(a_{3} − 1) ω^{1}∧ω^{3} \\ −a_{3} ω^{2}∧ω^{3} + ω^{3}∧ω^{4} \\ (c_{1} − 1) ω^{3}∧ω^{1} \\ −c_{1} ω^{4}∧ω^{1} + ω^{1}∧ω^{2} \end{matrix} )

where

(3)  \{ \begin{matrix}da_{3} & = a_{31} ω^{1} & + a_{33} ω^{3} \\ dc_{1} & = c_{13} ω^{3} & + c_{11} ω^{1}. \end{matrix}

We shall show that any such structure is linear. More precisely, we shall say that (M, Φ, I) is symplectic linearizable in case there are local coordinates (x, y, u, v) together with a 2-form Ψ ∈ I such that Φ and Ψ generate I and each is linear in u and v. Such a system is clearly linearizable in our previous sense, but not conversely. We shall prove that symplectic hyperbolic systems satisfying (2) and (3) are symplectic linearizable.

Proof : Since ω^{1} and ω^{3} are separately integrable we may find local coordinates x, y such that

ω^{1} = A dx
ω^{3} = C dy.

These coordinates are determined up to

(x, y) = (X(x̄), Y (ȳ)).

From the structure equations we infer that A, C and a_{3}, c_{1} are all functions of x, y and furthermore that the hyperbolic PDE system

(4)  \{ \begin{matrix}A_{y} & = −(a_{3} & − 1)AC \\ C_{x} & = −(c_{1} & − 1)AC \end{matrix}

3.2.1 is satisfied.

Next we may find functions u, v such that

\begin{matrix}ω^{2} & ≡ B du \\ ω^{4} & ≡ D dv \end{matrix}\} mod ω^{1}, ω^{3}.

From the structure equations we have that

B_{v} = D_{u} = 0.

Thus we may introduce new coordinates u, v so that

ω^{2} ≡ du + E dy mod ω^{1}
ω^{4} ≡ dv + F dx mod ω^{3}.

From the structure equations we obtain

\begin{aligned}E_{u} & = −a_{3}C & E_{v} & = −C \\ F_{v} & = −c_{1}A & F_{u} & = −A. \end{aligned}

Integrating these gives

\begin{aligned}E & = −C(x, y)(a_{3} & (x, y)u & + v) & − E_{0}(x, y) \\ F & = −A(x, y)(c_{1} & (x, y)v & + u) & − F_{0}(x, y). \end{aligned}

Thus the exterior differential system

(5)  \begin{aligned}ω^{1} ∧ ω^{2} & = A dx & ∧ (du & + E dy) & = 0 \\ ω^{3} ∧ ω^{4} & = C dy & ∧ (dv & + F dx) & = 0 \end{aligned}

models the affine linear PDE system

\begin{aligned}u_{y} & = C(x, y) & (a_{3}(x, y)u & + v) & + E_{0}(x, y) \\ v_{x} & = A(x, y) & (c_{1}(x, y)v & + u) & + F_{0}(x, y). \end{aligned}

By subtracting from u, v a particular solution we may assume that E_{0} = F_{0} = 0. Clearly (5) is linear in u, v as is Φ = ω^{1}∧ω^{2} + ω^{3}∧ω^{4}. □

Of particular interest is the case when a_{3} = a_{0} and c_{1} = c_{0} are constant. Then the group of symmetries of (M, Φ, I) is an infinite-dimensional transitive pseudogroup whose general element depends on two functions of one variable. Note that the PDE system (4) is then a constant scaling in x and y of the s = 0 Liouville system for the quantities log A, log C.

Assuming that (a_{0} − 1)(c_{0} − 1) ≠ 0, the general solution is given by

A = \frac{−a′(x)}{(c_{0} − 1)(c(y) − a(x))}
C = \frac{−c′(y)}{(a_{0} − 1)(a(x) − c(y))}

and so (5) models the interesting linear PDE system

\{ \begin{matrix}u_{y} & = −\frac{c′(y)(a_{0}u + v)}{(a_{0} − 1)(a(x) − c(y))} \\ v_{x} & = −\frac{a′(x)(c_{0}v + u)}{(c_{0} − 1)(c(y) − a(x))}. \end{matrix}

3.2.2 In case a_{0} = c_{0} = 1, (4) gives

\begin{aligned}A_{y} & = 0 \\ C_{x} & = 0 \end{aligned}

and so

\begin{aligned}A & = A(x) \\ C & = C(y). \end{aligned}

Thus (5) yields the following linear system

\{ \begin{matrix}u_{y} & = \frac{u + v}{C(y)} \\ v_{x} & = \frac{u + v}{A(x)} . \end{matrix}

Class B: The structure equations have the form

(6) d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}0 & 0 & 0 & 0 \\ φ_{21} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & φ_{43} & 0 \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}−k_{0} ω^{1}∧ω^{2} − & (a_{3} ω^{3} + a_{4} ω^{4})∧ω^{1} − ω^{1}∧ω^{3} \\ & (a_{3} ω^{3} + a_{4} ω^{4})∧ω^{2} + ω^{3}∧ω^{4} \\ −k_{0} ω^{3}∧ω^{4} − & (c_{1} ω^{1} + c_{2} ω^{2}) ∧ω^{3} − ω^{3}∧ω^{1} \\ & (c_{1} ω^{1} + c_{2} ω^{2}) ∧ω^{4} + ω^{1}∧ω^{2} \end{matrix} )

where k_{0} ≠ 0 and

(7)  \{ \begin{matrix}dk_{0} & = & − (a_{4} + c_{1}k_{0}) ω^{1} & − c_{2}k_{0} ω^{2} & − (c_{2} + a_{3}k_{0}) ω^{3} & − a_{4}k_{0} ω^{4} \\ da_{3} & = a_{4} φ_{43} & + a_{3}(c_{1} − 2) ω^{1} & + a_{3}c_{2} ω^{2} & + a_{33} ω^{3} & − (2a_{3}k_{0} + a_{4}) ω^{4} \\ da_{4} & = & − a_{4}(c_{1} + 1) ω^{1} & − a_{4}c_{2} ω^{2} & − (a_{4} + a_{3}k_{0}) ω^{3} & − a_{4}k_{0} ω^{4} \\ dc_{1} & = c_{2} φ_{21} & + c_{1}(a_{3} − 2) ω^{3} & + c_{1}a_{4} ω^{4} & + c_{11} ω^{1} & − (2c_{1}k_{0} + c_{2}) ω^{2} \\ dc_{2} & = & − c_{2}(a_{3} + 1) ω^{3} & − c_{2}a_{4} ω^{4} & − (c_{2} + c_{1}k_{0}) ω^{1} & − c_{2}k_{0} ω^{2}. \end{matrix}

Comparing (6), (7) with the proposition in Section 1.5 we obtain the following

Proposition: Any system satisfying (6) and (7) is Darboux integrable on the first prolongation.

In order to give a normal form we need to separate into the subcases

(8)  a_{4} = c_{2} = 0
(9)  a_{4}c_{2} ≠ 0.

We shall deal mainly with the first case and shall show that Proposition: A hyperbolic exterior differential system satisfying (6),(7) and (8) has k_{0} = constant and is equivalent to the system modeled on the s = 0 Liouville system

\{ \begin{matrix}u_{y} & = e^{k_{0}v} \\ v_{x} & = e^{k_{0}u}. \end{matrix}

Proof: From (7) we have a_{3} = c_{1} = 0 and so k_{0} = constant. Thus (6) reduces to

(10)  d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}0 & 0 & 0 & 0 \\ φ_{21} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & φ_{43} & 0 \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}−k_{0} ω^{1}∧ω^{2} − ω^{1}∧ω^{3} \\ ω^{3}∧ω^{4} \\ −k_{0} ω^{3}∧ω^{4} − ω^{3}∧ω^{1} \\ ω^{1}∧ω^{2} \end{matrix} )

3.2.3 where k_{0} ≠ 0 parametrizes the equivalence classes of these structures. Since ω^{1} and ω^{3} are integrable we may introduce local coordinates (x, y, u, v) such that

\{ \begin{matrix}ω^{1} & = A dx \\ ω^{3} & = C dy \end{matrix}

and

\{ \begin{matrix}ω^{2} & ≡ B du & + E dy mod ω^{1} \\ ω^{4} & ≡ D dv & + F dx mod ω^{3}. \end{matrix}

From the structure equations for dω^{2} and dω^{4} we have

B_{v} = D_{u} = 0.

By changing u, v appropriately we may assume that

\{ \begin{matrix}ω^{2} & = du & + E dy mod ω^{1} \\ ω^{4} & = dv & + F dx mod ω^{3}. \end{matrix}

Again from the structure equations we have

\{ \begin{matrix}(i) E_{u} & = 0 & E_{v} = −C \\ (ii) F_{v} & = 0 & F_{u} = −A \end{matrix}

and

\{ \begin{matrix}(iii) & A_{u} & = k_{0}A & A_{v} & = 0 \\ (iv) & C_{v} & = k_{0}C & C_{u} & = 0. \end{matrix}

Integrating (iii) and (iv) gives

A = k_{0}A_{0}(x, y)e^{k_{0}u} C = k_{0}C_{0}(x, y)e^{k_{0}v},

while from (i) and (ii) we obtain

E = −C_{0}e^{k_{0}v} + E_{0}(x, y)
F = −A_{0}e^{k_{0}u} + F_{0}(x, y).

Thus we now have

\{ \begin{matrix}ω^{2} & ≡ du & + (−C_{0}e^{k_{0}v} & + E_{0}) dy mod dx \\ ω^{4} & ≡ dv & + (−A_{0}e^{k_{0}u} & + F_{0}) dx mod dy. \end{matrix}

Clearly we may make a substitution

\begin{aligned}u & → u & + ∫ E_{0} dy \\ v & → v & + ∫ F_{0} dx \end{aligned}

to set E_{0} = F_{0} = 0 (with a new A_{0} and C_{0}). The structure equations for dω^{1} and dω^{3} then give

(A_{0})_{y} = (C_{0})_{x} = 0.

3.2.4 By choosing new coordinates x and y we may assume that

\{ \begin{matrix}ω^{1} & = e^{k_{0}u}dx \\ ω^{3} & = e^{k_{0}v}dy \end{matrix}

and

\{ \begin{matrix}ω^{2} & = du & − e^{k_{0}v}dy mod dx \\ ω^{4} & = dv & − e^{k_{0}u}dx mod dy \end{matrix}

which gives the s = 0 Liouville system. □

Remark: These systems with parameter k_{0} are inequivalent as symplectic hyperbolic systems but equivalent as hyperbolic systems.

For symplectic hyperbolic systems satisfying (6), (7) and (9) we may restrict to the sub-bundle of coframes that satisfy

a_{3} = c_{1} = 0,

in which case the structure equations reduce to

(11)  d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = ( \begin{matrix}−k_{0} ω^{1}∧ω^{2} − a_{4} ω^{4}∧ω^{1} − ω^{1}∧ω^{3} \\ − φ_{21}∧ω^{1} + a_{4} ω^{4}∧ω^{2} + ω^{3}∧ω^{4} \\ −k_{0} ω^{3}∧ω^{4} − c_{2} ω^{2}∧ω^{3} − ω^{3}∧ω^{1} \\ − φ_{43}∧ω^{3} + c_{2} ω^{2}∧ω^{4} + ω^{1}∧ω^{2} \end{matrix} )

where

(12)  \{ \begin{matrix}dk_{0} & = −a_{4} ω^{1} & − c_{2}k_{0} ω^{2} & − c_{2} ω^{3} & − a_{4}k_{0} ω^{4} \\ da_{4} & = −a_{4} ω^{1} & − a_{4}c_{2} ω^{2} & − a_{4} ω^{3} & − a_{4}k_{0} ω^{4} \\ dc_{2} & = −c_{2} ω^{1} & − c_{2}k_{0} ω^{2} & − c_{2} ω^{3} & − c_{2}a_{4} ω^{4}. \end{matrix}

In addition, from (7) we have

(13)  \{ \begin{matrix}φ_{21} & ≡ ω^{2} mod ω^{1} \\ φ_{43} & ≡ ω^{4} mod ω^{3}. \end{matrix}

An example of such a system is provided by the hyperbolic system associated to the s = 0 Goursat equations

\begin{aligned}u_{y} & = \frac{2\sqrt{uv}}{(x + y)}, \\ v_{x} & = \frac{2\sqrt{uv}}{(x + y)}. \end{aligned}

This system is Darboux integrable and is in fact equivalent, as hyperbolic systems, to the linear system I_{1,0} given in Section 1.5.

Since every system of Class B is Darboux integrable, from the discussion in Section 1.5, we see that as hyperbolic exterior differential systems (though not as symplectic hyperbolic EDS) there are exactly two equivalence classes, corresponding to the s = 0 Liouville system and the s = 0 Goursat system.

Class C: The structure equations have the form

d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}0 & 0 & 0 & 0 \\ φ_{21} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & φ_{43} & 0 \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}−k_{0} ω^{1}∧ω^{2} − (a_{3} ω^{3} + k_{0} ω^{4})∧ω^{1} − ω^{1}∧ω^{3} \\ (a_{3} ω^{3} + k_{0} ω^{4})∧ω^{2} + ω^{3}∧ω^{4} \\ −k_{0} ω^{3}∧ω^{4} − (c_{1} ω^{1} + k_{0} ω^{2})∧ω^{3} − ω^{3}∧ω^{1} \\ (c_{1} ω^{1} + k_{0} ω^{2})∧ω^{4} + ω^{1}∧ω^{2} \end{matrix} )

3.2.5

dk_{0} = −k_{0}(c_{1} + 1) ω^{1} − k_{0}^{2} ω^{2} − k_{0}(a_{3} + 1) ω^{3} − k_{0}^{2} ω^{4}

and

\begin{aligned}da_{3} & = k_{0} φ_{43} & + (b_{00} + c_{1} − a_{3}) ω^{1} & + a_{3}k_{0} ω^{2} & + a_{33} ω^{3} & − k_{0}(2a_{3} + 1) ω^{4} \\ dc_{1} & = k_{0} φ_{21} & + (b_{00} + a_{3} − c_{1}) ω^{3} & + c_{1}k_{0} ω^{4} & + c_{11} ω^{1} & − k_{0}(2c_{1} + 1) ω^{2}. \end{aligned}

In addition k_{0} ≠ 0 and so from the last two equations we see that we may restrict to a sub-bundle defined by a_{3} = c_{1} = 0. After further computation the structure equations then become

(14)  d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = ( \begin{matrix}−  k_{0} ω^{1}∧ω^{2} + k_{0} ω^{1}∧ω^{4} − ω^{1}∧ω^{3} \\ −(ω^{2} + f_{0} ω^{3})∧ω^{1} − k_{0} ω^{2}∧ω^{4} + ω^{3}∧ω^{4} \\ −  k_{0} ω^{3}∧ω^{4} + k_{0} ω^{3}∧ω^{2} − ω^{3}∧ω^{1} \\ −(ω^{4} + f_{0} ω^{1})∧ω^{3} − k_{0} ω^{4}∧ω^{2} + ω^{1}∧ω^{2} \end{matrix} )

where

(15)  dk_{0} = −k_{0} (ω^{1} + k_{0} ω^{2} + ω^{3} + k_{0} ω^{4})

and

(16)  df_{0} ≡ k_{0}f_{0}(ω^{2} + ω^{4}) mod ω^{1}, ω^{3}.

Remark: Since k_{0} ≠ 0 the structures of Class C cannot be homogeneous as symplectic hyperbolic systems. The structure equations may be proved to be involutive with Cartan characters s_{1} = s_{2} = 1, s_{3} = s_{4} = 0 (the ω^{1} and ω^{3} derivatives of f_{0} may be specified arbitrarily); hence structures of Class C depend on one function of two variables.

We will prove that

Proposition: (i) No structure satisfying (14)–(16) is symplectic linearizable. (ii) On the other hand, every such structure is linearizable as a hyperbolic exterior differential system.

Proof: For the proof, since ω^{1} and ω^{3} are integrable we may introduce local coordinates x, y, u, v such that

\{ \begin{matrix}ω^{1} & = A dx \\ ω^{3} & = C dy \end{matrix}

and

\{ \begin{matrix}ω^{2} & = (B du & + P dx & + Q dy)/A \\ ω^{4} & = (D dv & + S dx & + T dy)/C. \end{matrix}

From the structure equations we deduce that

B_{v} = D_{u} = 0,

so we may introduce new coordinates so that

\{ \begin{matrix}ω^{2} & = (du & + P dx & + Q dy)/A \\ ω^{4} & = (dv & + S dx & + T dy)/C. \end{matrix}

3.2.6 Our exterior differential system thus models the PDE system

(17)  \{ \begin{matrix}u_{y} & + Q(x, y, u, v) & = 0 \\ v_{x} & + S(x, y, u, v) & = 0. \end{matrix}

The coordinates x, y are determined up to

(x, y) = (X(x̄), Y (ȳ)),

and A, C then undergo the transformation

(A, C) → (X′(x̄)A, Y′(ȳ)C)

while u, v undergo

(u, v) → (X′(x̄)u, Y′(ȳ)v).

Noting that

Φ = (du + Q dy) ∧ dx + (dv + S dx) ∧ dy

it follows that the condition that Q(x, y, u, v) and S(x, y, u, v) be non-linear in u, v has intrinsic meaning under symplectic equivalence and measures the non-linearity of the original system.

From the dω^{2} and dω^{4} equations we have

Q_{v} + A = 0
S_{u} + C = 0

and

k_{0}(AT − CQ) + C(A_{y} + AQ_{u}) = 0
−k_{0}(AS − CP) + A(C_{x} + CS_{v}) = 0,

while the dω^{1} and dω^{3} equations give

Q_{u} + C = 0
S_{v} + A = 0.

Thus

\begin{aligned}(Q − S)_{u} & = 0 \\ (Q − S)_{v} & = 0 \end{aligned}

and so

\begin{aligned}Q & = R & + f(x, y) \\ S & = R & − f(x, y) \end{aligned}

for some function f. We may then choose new coordinates such that

\{ \begin{matrix}ω^{2} & = (du & + P dx & + R dy)/A \\ ω^{4} & = (dv & + R dx & + T dy)/C. \end{matrix}

The PDE system (17) becomes

(18)  \{ \begin{matrix}u_{y} & + R(x, y, u, v) & = 0 \\ v_{x} & + R(x, y, u, v) & = 0. \end{matrix}

3.2.7 We will show that: R is necessarily non-linear in u, v.

From the dω^{2} and dω^{4} equations we obtain

\begin{aligned}R_{u} & = −C \\ R_{v} & = −A \end{aligned}

while from the dω^{1} and dω^{3} equations we find

(19)  \begin{aligned}A_{u} & = k_{0}  A_{v} = −\frac{k_{0}A}{C} \\ C_{v} & = k_{0}  C_{u} = −\frac{k_{0}C}{A} . \end{aligned}

Thus

R_{uv} = −k_{0} ≠ 0

so that R cannot be linear in u, v.

To complete the argument we need to show that systems satisfying (14) – (16) linearize as hyperbolic systems. But in Section 2.2 we have seen that there is only one class of non-linear hyperbolic system having an infinite number of classical conservation laws, namely the class of the s = 0 Liouville system. We have already seen that this system is of Class B. □

We may summarize the situation by the following schematic

Symplectic equivalence classes of hyperbolic systems having an infinite number of classical conservation laws ⟼ Equivalence classes of hyperbolic systems having an infinite number of classical conservation laws

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s = 0 Goursat s = 0 Liouville  I (linear) II (non-linear) C A B

The notation means that B contains exactly two classes, one of which maps to I and the other of which maps to II.

This illustrates again in a rather dramatic way just how an exterior differential system may linearize when we increase its symmetry group.

To complete the story we shall derive a symplectic normal form for equations of Class C. 3.2.8 Proposition: Any hyperbolic exterior differential system of Class C is symplectically equivalent to the exterior differential system generated by^{27}

(20)  \{ \begin{matrix}u_{y} & = F (x, y)\sqrt{uv} \\ v_{x} & = F (x, y)\sqrt{uv}. \end{matrix}

Proof: From the expression for dk_{0} in (15) we obtain

(k_{0})_{u} = −\frac{k_{0}^{2}}{A} (k_{0})_{v} = −\frac{k_{0}^{2}}{C} .

Combining these equations with (19) yields the relations

(i)  (AA_{u})_{u} = 0  (CC_{v})_{v} = 0
(ii)  (AC)_{u} = 0  (AC)_{v} = 0.

From (i) it follows that

(R_{v}^{2})_{uu} = 0 (R_{u}^{2})_{vv} = 0

and from (ii) it follows that

(R_{u}R_{v})_{u} = (R_{u}R_{v})_{v} = 0.

Holding x, y fixed for the moment we shall prove the following

Lemma: Let R(u, v) be a function that satisfies

(i) R_{u}^{2} is linear in v and R_{v}^{2} is linear in u;
(ii) R_{u}R_{v} = C is constant;

then

R = γ\sqrt{u + β}\sqrt{v + α} + δ

where α, β, γ, δ are constants.

Proof: By (i)

\begin{aligned}R_{u} & = a(u) & \sqrt{v + α(u)} \\ R_{v} & = b(v) & \sqrt{u + β(v)} \end{aligned}

for suitable functions a(u), α(u), b(v), β(v). In the following argument we shall assume that a(u)α(u) ≠ 0 and b(v)β(v) ≠ 0; the cases where these conditions are not satisfied may be handled by (easier) special arguments. Differentiating

\begin{aligned}R_{u}^{2} & = a(u)^{2}(v & + α(u)) \\ R_{v}^{2} & = b(v)^{2}(u & + β(v)) \end{aligned}

Φ = ω^{1} ∧ ω^{2} + ω^{3} ∧ ω^{4}

\begin{aligned}ω^{1} & = − \frac{1}{2} F (x, y)\sqrt{\frac{u}{v}} dx & ω^{2} = − \frac{2}{F (x, y)} \sqrt{\frac{v}{u}} (du − F (x, y)\sqrt{uv} dy) \\ ω^{3} & = − \frac{1}{2} F (x, y)\sqrt{\frac{u}{v}} dy & ω^{4} = − \frac{2}{F (x, y)} \sqrt{\frac{u}{v}} (dv − F (x, y)\sqrt{uv} dx) . \end{aligned}

3.2.9 with respect to v and u respectively gives

27The symplectic form of the EDS that models (20) is where

\begin{aligned}2R_{u}R_{uv} & = a^{2}(u) \\ 2R_{v}R_{vu} & = b^{2}(v). \end{aligned}

Multiplying these equations and using (ii) gives

4CR_{uv}^{2} = a^{2}(u)b^{2}(v).

Thus

\frac{a^{2}(u)b^{2}(v)}{4C} = R_{uv}^{2} = \frac{a^{4}(u)}{4R_{u}^{2}} = \frac{a^{2}(u)}{4(v + α(u))}

which implies that

b^{2}(v) = \frac{C}{v + α(u)}

and therefore α(u) = α is a constant.

Similarly, β(v) = β is a constant and

R_{v} = \sqrt{C}\sqrt{\frac{u + β}{v + α}}

which integrates to give

R = γ\sqrt{(u + β)(v + α)} + δ(u), γ = 2\sqrt{C}.

But then

R_{u} = \sqrt{C}\sqrt{\frac{v + α}{u + β}} + δ′(u)

and so δ′(u) = 0, i.e., δ(u) = δ is a constant. □

Applying the lemma to R(x, y, u, v) with x, y variable gives

(21)  R = γ(x, y)\sqrt{u + β(x, y)}\sqrt{v + α(x, y)} + δ(x, y).

Hence we can now introduce new coordinates

\begin{aligned}ũ & = u & + β(x, y) \\ ṽ & = v & + α(x, y) \end{aligned}

such that (dropping the tildes)

\begin{aligned}ω^{2} & = (du & + P dx & + (R_{0} & + μ(x, y))dy)/A \\ ω^{4} & = (dv & + T dy & + (R_{0} & + ν(x, y))dx)/C \end{aligned}

where R_{0} = γ(x, y)\sqrt{uv}.

To complete the proof we need to show that μ(x, y) = ν(x, y) = 0. To establish this, we return to the structure equation (14). Expanding out the dω^{2}-equation and using (19), the coefficient of the [dy∧dv]-term yields the relation

(22)  A = −\frac{1}{2}γ(x, y)\sqrt{\frac{u}{v}}

3.2.10 while the [dx∧dv]-term implies

P_{v} = 0.

Now the [dx∧du]-term gives

\begin{aligned}−P_{u} & = A + \frac{A_{x} − A_{u}P}{A} + \frac{k_{0}(R_{0} + ν)}{C} \\ & = A + \frac{A_{x} − A_{u}P − A_{v}(R_{0} + ν)}{A}. \end{aligned}

Differentiating with respect to v, keeping in mind (22), yields

ν(x, y) = 0.

Similarly, from the dω^{4}-equation we obtain

μ(x, y) = 0.
□

Remark: The explicit linearization of the F -Goursat system (20) alluded to above is achieved by the mapping

u = p^{2}, v = q^{2}

which transform the F -Goursat system into the following linear PDE system

\begin{aligned}p_{y} & = \frac{1}{2}F (x, y)q, \\ q_{x} & = \frac{1}{2}F (x, y)p. \end{aligned}

The above result stands in interesting contrast to the fact that the usual F -Goursat equation

z_{xy} = F (x, y)\sqrt{z_{x}z_{y}}

cannot be linearized by a contact transformation in (x, y, z, z_{x}, z_{y})-space.

3.2.11 3.3 Euler-Lagrange representations of hyperbolic systems

In this final section, we shall discuss the question: What is the maximum number of ways in which a given hyperbolic exterior differential system of class s = 0 may be realized as an Euler-Lagrange system? Given (M, I) what we are asking for is the maximum dimension of the space Σ of orthogonal pairs (Φ, Ψ) of conservation laws. We shall establish the following

Proposition : For hyperbolic systems I of class s = 0, the dimension of the space Σ is at most 4, with equality holding if and only if I is locally diffeomorphic to one of the linear systems I_{K,0} discussed in Section 1.5.6

Proof: What we shall actually prove is that the maximum dimension of Σ is achieved exactly when the structure equations of (M, I) are

d ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) = − ( \begin{matrix}ϕ_{44} − ϕ_{22} & 0 & 0 & 0 \\ ϕ_{21} & ϕ_{22} & 0 & 0 \\ 0 & 0 & ϕ_{22} − ϕ_{44} & 0 \\ 0 & 0 & ϕ_{43} & ϕ_{44} \end{matrix} ) ∧ ( \begin{matrix}ω^{1} \\ ω^{2} \\ ω^{3} \\ ω^{4} \end{matrix} ) + ( \begin{matrix}0 \\ ω^{3}∧ω^{4} \\ 0 \\ ω^{1}∧ω^{2} \end{matrix} )

where

\{ \begin{matrix}dϕ_{22} & = K ω^{1} ∧ ω^{3} \\ dϕ_{44} & = K ω^{3} ∧ ω^{1} \end{matrix}

where K is an arbitrary constant. Of course, comparing this with the structure equations at the end of Section 1.5.6 will then yield the result.

The proof will proceed by calculations very similar to those used in determining hyperbolic systems with infinitely many classical conservation laws. We shall only write down the main steps in the computation.

A pair of conservation laws may be written as

\begin{aligned}Φ & = A ω^{1} ∧ ω^{2} & + C ω^{3} ∧ ω^{4} \\ Ψ & = B ω^{1} ∧ ω^{2} & + D ω^{3} ∧ ω^{4}, \end{aligned}

and the orthogonality condition

Φ ∧ Ψ = 0

is

AD + BC = 0.

Thus we have

B = \frac{H}{C} , D = −\frac{H}{A}

for some function H. The closure conditions

dΦ = dΨ = 0

give

\begin{aligned}dA & = A ϕ_{44} & + A_{i} ω^{i} \\ dC & = C ϕ_{22} & + C_{i} ω^{i} \end{aligned}

with

(1)  \{ \begin{matrix}A_{3} & = C & A_{4} = 0 \\ C_{1} & = A & C_{2} = 0 \end{matrix}

3.3.1 and

dH = H(ϕ_{22} + ϕ_{44}) + H_{i} ω^{i}

where

(2)  \{ \begin{matrix}H_{1} & = H(CA_{1} − A^{2})/AC \\ H_{2} & = HA_{2}/A \\ H_{3} & = H(AC_{3} − C^{2})/AC \\ H_{4} & = HC_{4}/C. \end{matrix}

As in Section 2.2 above, from the identities d(dA) ≡ 0 mod ω^{1}, ω^{2} and d(dC) ≡ 0 mod ω^{3}, ω^{4} we obtain

(3)  \{ \begin{matrix}A_{2} = K_{0} − \frac{1}{2}(Ap_{1} − Cq_{3}) \\ C_{4} = K_{0} − \frac{1}{2}(Cq_{3} − Ap_{1}). \end{matrix}

Then d(dA) ≡ 0 mod ω^{1} and d(dC) ≡ 0 mod ω^{3} gives

(4)  dK_{0} = K_{0}(ϕ_{22} + ϕ_{44}) + \frac{1}{2}(Cp_{4} ϕ_{21} + Aq_{2} ϕ_{43}) + K_{0i} ω^{i}

where

K_{0i} ≡ 0 mod A, C, K_{0}, A_{1}, C_{3}.
From the identity d(dH) ≡ 0 mod ω^{1}, ω^{3} we obtain
H(p_{4}A^{2} − q_{2}C^{2}) = 0.

If p_{4}q_{2} ≠ 0, then we may normalize to have p_{4} = q_{2} = 1 and therefore A = ± C. This imposes conditions beyond (1) – (3) and cuts down the dimension of the solution space. Thus we assume that

p_{4} = q_{2} = 0.

From the identities d(dA) = d(dC) = 0 we find that

dA_{1} = A_{1}(ϕ_{22} − 2 ϕ_{44}) + (K_{0} − \frac{1}{2}(Ap_{1} − Cq_{3})) ϕ_{21} + Aq_{1} ϕ_{43} + A_{1i} ω^{i}
dC_{3} = C_{3}(ϕ_{44} − 2 ϕ_{22}) + (K_{0} − \frac{1}{2}(Cq_{3} − Ap_{1})) ϕ_{43} + Cp_{3} ϕ_{21} + C_{3i} ω^{i}

where

A_{12} ≡ A_{13} ≡ A_{14} ≡ C_{31} ≡ C_{32} ≡ C_{34} ≡ 0 mod A, C, K_{0}, A_{1}, C_{3}.

This is just as we found in Section 2.2 above. New conditions arise from d(dH) = 0, which gives

(5)  K_{0} = −\frac{1}{2}(Ap_{1} + Cq_{3})

and

(6)  \{ \begin{matrix}A_{1} & = \frac{L_{0}}{C^{3}} \\ C_{3} & = \frac{L_{0}}{A^{3}} \end{matrix}

for some function L_{0}. From (4) and dK_{0} = −d(\frac{1}{2}(Ap_{1} + Cq_{3})) we obtain

k_{24} = k_{42} = 0.

3.3.2 Plugging back into d(dH) = 0 then gives

\{ \begin{matrix}A^{2}p_{3} + C^{2}p_{1} = 0 \\ C^{2}q_{1} + A^{2}q_{3} = 0. \end{matrix}

Again, if p_{3}q_{1} ≠ 0 we impose further conditions on the system. Thus we assume that

p_{3} = q_{1} = 0,

which then gives also that

p_{1} = q_{3} = 0.

Substituting these conditions back into dK_{0} and comparing with (4) gives

k_{14} = k_{32} = 0.

Now we use the expressions for A_{1} and C_{3} in (6) to obtain

dL_{0} = 2L_{0}(ϕ_{22} + ϕ_{44}) + L_{0i} ω^{i}

where

L_{02} = L_{04} = 0

and

L_{01} ≡ L_{03} ≡ 0 mod A, C.

Moreover, the conditions

A_{11} ≡ C_{33} ≡ 0 mod A, C

are also forced. Noting (3), (5) and (6), at this point we have proved that Σ is finite dimensional.

From d(dϕ_{22}) = d(dϕ_{44}) = 0 we infer that
dk_{13} = k_{131} ω^{1} + k_{133} ω^{3}
dk_{31} = k_{311} ω^{1} + k_{313} ω^{3}.

Then d(dL_{0}) = 0 gives

(7)  3(k_{13} − k_{31})(L_{0} − A^{2}C^{2}) + AC(A^{2}k_{133} − C^{2}k_{311}) = 0.

If k_{13} ≠ k_{31} then the above relation impose further conditions, cutting down Σ. Thus we need to assume that

k_{13} = k_{31} = k_{00}.

We then have

dk_{00} = k_{00i} ω^{i}

where

k_{002} = k_{004} = 0.

Moreover, (7) gives

A^{2}k_{003} − C^{2}k_{001} = 0.

Again, if k_{003}k_{001} ≠ 0 we impose additional conditions, so we may assume that

k_{001} = k_{003} = 0.

3.3.3 In other words, k_{00} is a constant, say K. At this point we have

dA = A ϕ_{44} + \frac{L_{0}}{C^{3}} ω^{1} + C ω^{3}
dC = C ϕ_{22} + \frac{L_{0}}{A^{3}} ω^{3} + A ω^{1}

and

\begin{aligned}dH & = H(ϕ_{22} + ϕ_{44}) & − \frac{H(A^{2}C^{2} − L_{0})}{AC} & (\frac{ω^{1}}{C^{2}} + \frac{ω^{3}}{A^{2}} ) \\ dL_{0} & = L_{0}(ϕ_{22} + ϕ_{44}) & − \frac{(A^{4}C^{4}(k_{00} − 1) − 3L_{0}^{2})}{AC} & (\frac{ω^{1}}{C^{2}} + \frac{ω^{3}}{A^{2}} ) . \end{aligned}

This is a (non-linear) integrable Frobenius system whose solution depends on four constants. Moreover, the structure equations of the exterior differential system are the same as the structure equations of the ideal I_{K,0} described in Section 1.5.6. This proves the desired equivalence. □

3.3.4

References

[AK] I. Anderson and N. Kamran, The variational bicomplex for second order partial differential

equations in the plane I: generalized Laplace invariants and their applications, preprint, Fall 1993. [BCG^{3}] R. Bryant, S.-S. Chern, R. Gardner, H. Goldschmidt, and P. Griffiths, Exterior differential sys-

tems, Mathematical Sciences Research Institute Publications #18, Springer-Verlag (1991). [BG_{1}] R. Bryant and P. Griffiths, Characteristic cohomology of differential systems (I): General theory,

IAS preprint (1993). [BG_{2}] R. Bryant and P. Griffiths, Characteristic cohomology of differential systems (II): Conservation

Laws for a Class of Parabolic Equations, IAS preprint (1993).

[CH] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume I, Interscience, New York (1953), Volume II, Interscience, New York (1962).

[Da] G. Darboux, Leçons sur la theorie general des surfaces, Tome II, Hermann, Paris (1915).

[FX] Ph. Le Floch and Z. Xin, Uniqueness via the adoint problems for systems of conservation laws, Comm. Pure and Applied Math. 46 (1993), 1499–1533.

[Go] E. Goursat, Leçons sur l’intégration des équations aux dérivées partielles du second ordre à deux variables indépendantes, Tome 1, Hermann, Paris (1896), Tome 2, Hermann, Paris (1898).

[Ka] K. Kakié, A fundamental property of Monge characteristics in involutive systems of non-linear partial differential equations and its application, Math. Ann. 273 (1985), 89–114.

[La] P. Lax, Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations, J. Math. Phys. 5 (1964), 611–613.

[Li] S. Lie, Diskussion der Differential Gleichung s = F (z), Arc. Math. 6 (1881), 112–124.

[Ya] D. Yang, Involutive hyperbolic differential systems, Memoirs of the American Math. Society 68, no. 370, (1987).

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