Superposition Formulas for Darboux Integrable Exterior Differential Systems
aa r X i v : . [ m a t h . DG ] J un Superposition Formulas forExterior Differential Systems
Ian AndersonDept of Math. and Stat.Utah State University Mark FelsDept of Math. and Stat.Utah State UniversityPeter VassiliouDept. of Math and Stat.University of CanberraJuly 3, 2018 ntroduction In this paper we use the method of symmetry reduction for exterior differ-ential systems to obtain a far-reaching generalization of Vessiot’s integrationmethod [27], [28] for Darboux integrable, partial differential equations. Thisgroup-theoretic approach provides deep insights into this classical method; un-covers the fundamental geometric invariants of Darboux integrable systems;provides for their algorithmic integration; and has applications well beyondthose currently found in the literature. In particular, our integration methodis applicable to systems of hyperbolic PDE such as the Toda lattice equations,[19], [20], [22], 2-dimensional wave maps [3] and systems of overdetermined PDEsuch as those studied by Cartan [8].Central to our generalization of Vessiot’s work is the novel concept of a superposition formula for an exterior differential system I on a manifold M .A superposition formula for I is a pair of differential systems ˆ W , ˇ W , defined onmanifolds M and M , and a mappingΣ : M × M → M (1.1)such that Σ ∗ ( I ) ⊂ π ∗ ( ˆ W ) + π ∗ ( ˇ W ) . (1.2)Here π ∗ ( ˆ W + π ∗ ( ˇ W ) is the differential system generated by the pullbacks of ˆ W and ˇ W to the product manifold M × M by the canonical projection maps π and π . It is then clear that if ˆ φ : N → M and ˇ φ : N → M are integralmanifolds for ˆ W and ˇ W , then φ = Σ ◦ ( ˆ φ, ˇ φ ) : N × N → M (1.3)is an (possibly non-immersed) integral manifold of I .In this article we shall [i] establish general sufficiency conditions (in terms of geometric invariants ofthe differential system I ) for the existence of a superposition formula; [ii] establish general sufficiency conditions under which the superposition for-mula gives all local integral manifolds of I in terms of the integral manifoldsof π ∗ ( ˆ W )+ π ∗ ( ˇ W ) on M × M (in which case we say that the superpositionformula is surjective); [iii] provide an algorithmic procedure for finding the superposition formula; and ntroduction [iv] demonstrate the effectiveness of our approach with an extensive number ofexamples and applications.Differential systems admitting superposition formula are easily constructedby symmetry reduction. To briefly describe this construction, let G be a sym-metry group of a differential system W on a manifold N . We assume that thequotient space M = N/G of N by the orbits of G has a smooth manifold struc-ture for which the projection map q : N → M is smooth. We then define the G reduction of W or quotient of W by G as the differential system on M given by W /G = { ω ∈ Ω ∗ ( M ) | q ∗ ( ω ) ∈ W } . (1.4)The traditional application of symmetry reduction has been to integrate W byintegrating W /G . See, for example, [1].But now suppose that differential systems ˆ W and ˇ W on manifolds M and M have a common symmetry group G . Define the differential system W = π ∗ ( ˆ W ) + π ∗ ( ˇ W ) on M × M and let G act on M × M by the diagonal action.Then, by definition, the quotient map q : M × M → M = ( M × M ) /G defines a superposition formula for the quotient differential system W /G on M .In this paper we discover the means by which the inverse process to symmetryreduction is possible, that is, we show how certain general classes of differentialsystems I on M can be identified with a quotient system W /G in which casethe integral manifolds of I can then be found from those of ˆ W and ˇ W . To intrinsically describe the class of differential systems for which we shallconstruct superposition formulas, we first introduce the definition of a decom-posable differential system.
Definition 1.1.
An exterior differential system I on M is decomposable oftype [ p, q ] , where p, q ≥ , if about each point x ∈ M there is a coframe ˜ θ , . . . , ˜ θ r , ˆ σ , . . . , ˆ σ p , ˇ σ , . . . , ˇ σ q , (1.5) such that I is algebraically generated by 1-forms and 2-forms I = { ˜ θ , . . . , ˜ θ r , ˆ Ω , . . . , ˆ Ω s , ˇ Ω , . . . , ˇ Ω t } , (1.6) where s, t ≥ , ˆ Ω a ∈ Ω ( ˆ σ , . . . , ˆ σ p ) , and ˇ Ω α ∈ Ω ( ˇ σ , . . . , ˇ σ q ) . The differentialsystems algebraically generated by ˆ V = { ˜ θ i , ˆ σ a , ˇ Ω α } and ˇ V = { ˜ θ i , ˇ σ α , ˆ Ω a } (1.7) ntroduction are called the associated singular differential systems for I with respect tothe decomposition (1.6) . With the goal of constructing superposition formulas, we have found it mostnatural to focus on the case where I is decomposable (but not necessarily Pfaf-fian) and ˆ V and ˇ V are (constant rank) Pfaffian . All of the examples we considerare of this type. Note that any class r hyperbolic differential system, as definedin [6], is a decomposable differential system and that the associated character-istic Pfaffian systems coincide, for r >
0, with the singular systems(1.7).The definition of a Darboux integrable, decomposable differential system isgiven in terms of its singular systems. For any Pfaffian system V , let V ( ∞ ) denoted the largest integrable subbundle of V . The rank of V ( ∞ ) gives thenumber of functionally independent first integrals for V . By definition, a scalarsecond order partial differential equation in the plane is Darboux integrable ifthe associated singular Pfaffian systems ˆ V and ˇ V each admit at least 2 (func-tionally independent) first integrals. Thus, in order to generalize the definitionof Darboux integrablity, we must determine the required number of function-ally independent first integrals necessary to integrate a general decomposablePfaffian system. We do this with the following definitions. Definition 1.2.
A pair of Pfaffian systems ˆ V and ˇ V define a Darboux pair if the following conditions hold. [i] ˆ V + ˇ V ( ∞ ) = T ∗ M and ˇ V + ˆ V ( ∞ ) = T ∗ M. (1.8) [ii] ˆ V ( ∞ ) ∩ ˇ V ∞ = { } . (1.9) [iii] d ω ∈ Ω ( ˆ V ) + Ω ( ˇ V ) for all ω ∈ Ω ( ˆ V ∩ ˇ V ) . (1.10) Definition 1.3.
Let I be a decomposable differential system and assume thatthe associated singular systems ˆ V and ˇ V are Pfaffian. Then I is said to be Darboux integrable if { ˆ V , ˇ V } define a Darboux pair. Property [i] of Definition 1.2 is the critical one – it will insure that thereare a sufficient number of first integrals to construct a superposition formula.Property [ii] is a technical condition which states simply that ˆ V and ˇ V share no We use the term Pfaffian system to designate either a constant rank subbundle V of T ∗ M or the differential system, denoted by the corresponding calligraphic letter V , generated bythe sections of V . ntroduction ˆ V and ˇ V to a level set of any common integrals. The form of the structure equationsfor ˆ V ∩ ˇ V required by property [iii] is always satisfied when ˆ V and ˇ V are thesingular Pfaffian systems for a decomposable differential system I .Our main result can now be stated. Theorem 1.4.
Let I be a decomposable differential system on M whose asso-ciated singular Pfaffian systems { ˆ V , ˇ V } define a Darboux pair. Then there arePfaffian systems W and W on manifolds M and M which admit a commonLie group G of symmetries and such that [i] the manifold M can be identified (at least locally) as the quotient of M × M by the diagonal action of the group G ; [ii] I = ( π ∗ ( ˆ W + π ∗ ( ˇ W )) /G ; and (1.11) [iii] the quotient map q : M × M → M defines a surjective superposition for-mula for I . The manifolds M and M in Theorem 1.4 are simply any maximal integralmanifolds for ˆ V ( ∞ ) and ˇ V ( ∞ ) and the Pfaffian systems W and W are just therestrictions of ˇ V and ˆ V to these manifolds. But the proper identification of theLie group G and its action on M and M is not so easy to uncover. This is donethrough a sequence of non-trivial coframe adaptations and represents the prin-ciple technical achievement of the paper (See Theorem 4.1 and Definition 5.7)We call G the Vessiot group for the Darboux integrable, differential system I . The paper is organized as follows. In Section 2 we obtain some simplesufficiency conditions for a differential system to be decomposable and we givenecessary and sufficient conditions for a Pfaffian system to be decomposable(Theorem 2.3). We also introduce the initial adapted coframes for a Darbouxpair (Theorem 2.9). In Section 3 we answer the question of when the symmetryreduction of a Darboux pair is also a Darboux pair (Theorem 3.2) and weuse this result to give a general method for constructing Darboux integrabledifferential systems (Theorem 3.3). Section 4 establishes the sequence of coframeadaptations leading to the definition of the Vessiot group G and its action on M and hence on M and M (Theorem 4.1). In Section 5 we construct thesuperposition formula (Theorem 5.10) and prove that it may be identified with ntroduction u xx = f ( u yy ) (withsecond order invariants) and relate the integration of these equations to Car-tan’s classification of rank 3 Pfaffian systems in 5 variables [7]. This connectionbetween the method of Darboux and Cartan’s classification is apparent fromour interpretation of the method of Darboux in terms of symmetry reduction ofdifferential systems.In Examples 6.3 and 6.4 we present a number of examples of Darboux in-tegrable equations where the unknown function takes values in a group or in anon-commutative algebra. Example 6.3 provides us with a system whose Vessiotgroup is an arbitrary Lie group G .Some of the simplest examples of systems of Darboux integrable partial dif-ferential equations can be constructed by the coupling of a nonlinear Darbouxintegrable scalar equation to a linear or Moutard-type equation. These are pre-sented in Example 6.5. It is noteworthy that for these equations the Vessiotgroup is a semi-direct product of the Vessiot group for the non-linear equa-tion with an Abelian group. The representation theoretical implications of thisobservation will be further explored elsewhere.In Example 6.6 we illustrate the computational power of our methods byexplicitly integrating the B Toda lattice system. While the general solutionto the A n Toda systems were known to Darboux, this is, to the best of ourknowledge, the first time explicit general solutions to other Toda systems havebeen given. Based on this work we conjecture that the Vessiot group for the g Toda lattice system, where g is any semi-simple Lie group, is the associated Liegroup G itself. In Example 6.7 we integrate a wave map system – this systemadmits a number of interesting geometric properties which will be explored ntroduction reliminaries In this section we give some simple necessary conditions, in terms of thenotion of singular vectors, for a differential system I to be decomposable (seeDefinition 1.1); we give sufficient conditions for a Pfaffian system to be decom-posable; and we address the problem (see Theorem 2.6) of re-constructing adecomposable differential system from its associated singular systems.Let I be a differential system on M . Fix a point x in M and let E x ( I ) = { X ∈ T x M | θ ( X ) = 0 for all 1-forms θ ∈ I } . The polar equations determined by a non-zero vector X ∈ E x ( I ) are, bydefinition, the linear system of equations for Y ∈ E x ( I ) given by θ ( X, Y ) = 0 for all 2-forms θ ∈ I . Then X ∈ E x ( I ) is said to be regular if the rank of its polar equations ismaximal and singular otherwise. Lemma 2.1.
Let I be a decomposable differential system of type [ p, q ] . Then E x ( I ) decomposes into a direct sum of p and q dimensional subspaces E x ( I ) = S ⊕ S (2.1) such that [i] every vector in S and every vector in S is singular; and [ii] every 2 plane spanned by any pair of vectors X ∈ S and Y ∈ S is anintegral 2-plane for I .Proof. Let { ∂ ˜ θ i , ∂ ˆ σ a , ∂ ˇ σ α } denote the dual frame to (1.5). Then E x ( I ) =span { ∂ ˆ σ a , ∂ ˇ σ α } and the polar equations for X = X + X , where X = t a ∂ ˆ σ a and X = s α ∂ ˇ σ α , are ˆ Ω a ( X , Y ) = 0 and ˇ Ω α ( X , Y ) = 0 . Then, clearly, the vectors X and X are singular vectors and, moreover, theplane spanned by X , X is an integral 2-plane. We assume that all exterior differential systems are of constant rank. .1 Decomposable Differential Systems Remark 2.2.
The differential system ˆ V defined by (1.7) is the smallest differ-ential system containing I and Ann( S ) and it is for this reason that we haveopted to called ˆ V (and ˇ V ) the singular differential systems associated to thedecomposition of I .The necessary conditions for decomposability given in Lemma 2.1 are seldomsufficient. However, if I is a Pfaffian system and properties [i] and [ii] of Lemma2.1 are satisfied, then there exists a local coframe θ , . . . , θ r , ˆ σ , . . . , ˆ σ p , ˇ σ , . . . , ˇ σ q on M with I = { θ i } and with structure equations d θ i ≡ A iab ˆ σ a ∧ ˆ σ b + B iαβ ˇ σ α ∧ ˇ σ β mod I. (2.2)At each point x of M define linear maps A = [ A iab ( x )] : R r → Λ ( R p ) and B = [ B iαβ ( x )] : R r → Λ ( R q ) . (2.3) Theorem 2.3.
Let I be a Pfaffian system. Then I is decomposable if properties [i] and [ii] of Lemma 2.1 are satisfied; the matrices A iab and B iαβ given by (2.2) ,are non-zero, constant rank; and dim (cid:0) ker( A ) + ker( B ) (cid:1) = rank I. (2.4) Proof.
Equation (2.4) implies that we can find r linearly independent r dimen-sional column vectors T ,. . . , T r ,. . . , T r , . . . , T r such thatker( A ) ∩ ker( B ) = span { T , . . . , T r } , ker( B ) = span { T , . . . , T r , T r +1 , . . . , T r } , andker( A ) = span { T , . . . , T r , T r +1 , . . . , T r } . The 1-forms ˜ θ i = t ij θ j , where T i = [ t ij ], then satisfy d ˜ θ i ≡ i = 1 . . . r , d ˜ θ i ≡ t ij A jab ˆ σ a ∧ ˆ σ b for i = r + 1 . . . r , d ˜ θ i ≡ t ij B jαβ ˇ σ α ∧ ˇ σ β for i = r + 1 . . . r , (2.5)and the decomposability of I follows by taking ˆ Ω i = t ij A jab ˆ σ a ∧ ˆ σ b for i = r + 1 . . . r and ˇ Ω i = t ij B jαβ ˇ σ a ∧ ˇ σ b for i = r + 1 . . . r . That the integers s and t in Definition 1.1 satisfy s, t ≥ A iab and B iαβ arenon-zero and that [ t ij ] is invertible. .1 Decomposable Differential Systems Corollary 2.4. If I is a decomposable Pfaffian system, then the singular sys-tems ˆ V and ˇ V are also Pfaffian.Proof. This is immediate from (1.7) and (2.5).To each decomposable exterior differential system I we have associated apair of differential systems ˆ V and ˇ V and, when these are Pfaffian, we havedefined what it means for { ˆ V , ˇ V } to define a Darboux pair. In our subsequentanalysis, the Darboux pair { ˆ V , ˇ V } will be taken as the fundamental object ofstudy. From this viewpoint, it becomes important to address the problem ofreconstructing the EDS I from its singular systems. To this end, we introducethe following novel construction. Definition 2.5.
Let ˆ V and ˇ V be two differential systems on a manifold M .Define a new differential system K = ˆ V ∩ + ˇ V to be the exterior differential systemwhose integral elements are precisely the integral elements of ˆ V , the integralelements of ˇ V , and the sum of integral elements of ˆ V with integral elements of ˇ V . This definition is motivated by two observations. First we note that if I is adecomposable exterior differential system with singular Pfaffian systems ˆ V and ˇ V then in general I properly contains the Pfaffian system with generators ˆ V ∩ ˇ V and is properly contained in the EDS ˆ V ∩ ˇ V so that neither of these set-theoreticconstructions reproduce I .Secondly, one can interprete the EDS K in Definition 2.5 as satisfying an infinitesimal superposition principle with respect to { ˆ V , ˇ V } at the levelof integral elements. Of course, this by itself does not imply that K admits asuperposition principle for its integral manifolds. The main results of this articlecan then be re-formulated as follows: if the EDS K admits a superpositionprinciple for its integral elements with respect to a Darboux pair { ˆ V , ˇ V } , then K admits a superposition principle for its integral manifolds. Theorem 2.6. If I is a decomposable exterior differential system with singulardifferential systems ˆ V and ˇ V , then I = ˆ V ∩ + ˇ V .Proof. We first remark that the differential system ˆ V ∩ + ˇ V is always contained in ˆ V ∩ ˇ V . Let { ˜ θ i , ˆ σ a , ˇ σ α } be the coframe given by Definition 1.1. By definition,the algebraic generators for ˆ V and ˇ V are ˆ V = { ˜ θ i , ˆ σ a , ˇ Ω β } and ˇ V = { ˜ θ, ˇ σ α , ˆ Ω b } .2 The first adapted coframes for a Darboux pair ˆ V ∩ ˇ V = { ˜ θ i , ˆ Ω b , ˇ Ω β , ˆ σ a ∧ ˇ σ α } . Since ∂ ˇ σ α defines a 1-integral element for ˆ V and ∂ ˆ σ α defines a 1-integral elementfor ˇ V , the 2-plane spanned by { ∂ ˇ σ α , ∂ ˆ σ a } must be a integral element for ˆ V ∩ + ˇ V and therefore the forms ˆ σ a ∧ ˇ σ α / ∈ ˆ V ∩ + ˇ V . In view of (1.6), we conclude that ˆ V ∩ + ˇ V ⊂ I .To complete the proof, it suffices to check that every ( r + s )-form ω ∈ I vanishes on every plane ˆ E r + ˇ E s , where ˆ E r is an r -integral plane of ˆ V and ˇ E s is an s -integral plane of ˇ V . The form ω is a linear combination of the wedgeproduct of the 1-forms in (1.6) with arbitrary ( r + s − ρ and the wedgeproduct of the 2-forms in (1.6) with arbitrary ( r + s − τ . It is clearthat (˜ θ i ∧ ρ ) | ( ˆ E r + ˇ E s ) = 0. The values of ( ˆ Ω b ∧ τ ) | ( ˆ E r + ˇ E s ) can be calculated aslinear combinations of terms involving ˆ Ω b ( ˆ X, ˆ Y ) , ˆ Ω b ( ˆ X, ˇ Y ) , ˆ Ω b ( ˇ X, ˆ Y ) , ˆ Ω b ( ˇ X, ˇ Y ) , where ˆ X, ˆ Y ∈ ˆ E r and ˇ X, ˇ Y ∈ ˇ E s . These terms all vanish and in this way wemay complete the proof of the theorem. Let { ˆ V , ˇ V } be a Darboux pair on a manifold M . In addition to the prop-erties listed in Definition 1.2, we shall assume that the derived systems ˆ V ( ∞ ) and ˇ V ( ∞ ) , as well as the intersections ˆ V ( ∞ ) ∩ ˇ V , ˆ V ∩ ˇ V ( ∞ ) and ˆ V ∩ ˇ V are all(constant rank) subbundles of T ∗ M .To define our initial adapted coframe for the Darboux pair { ˆ V , ˇ V } , we firstchose tuples of independent 1-forms ˆ η and ˇ η which satisfy ˆ V ( ∞ ) ∩ ˇ V = span { ˆ η } and ˆ V ∩ ˇ V ( ∞ ) = span { ˇ η } . (2.6a)Property (1.9) implies that the forms { ˆ η , ˇ η } are linearly independent. Thenchose tuples of 1-forms θ , ˆ σ and ˇ σ such that ˆ V ( ∞ ) = span { ˆ σ , ˆ η } , ˇ V ( ∞ ) = span { ˇ σ , ˇ η } , ˆ V ∩ ˇ V = span { θ , ˆ η , ˇ η } (2.6b)and such that the sets of 1-forms { ˆ σ , ˆ η } , { ˇ σ , ˇ η } and { θ , ˆ η , ˇ η } are linearlyindependent. .2 The first adapted coframes for a Darboux pair Lemma 2.7.
Let { ˆ V , ˇ V } be a Darboux pair on a manifold M . Then the 1-forms { θ , ˆ σ , ˆ η , ˇ σ , ˇ η } satisfying (2.6) define a local coframe for T ∗ M and ˆ V = span { θ , ˆ σ , ˆ η , ˇ η } and ˇ V = span { θ , ˆ η , ˇ σ , ˇ η } . (2.6c) Proof.
From the definitions (2.6a) and (2.6b) it is clear that span { θ , ˆ σ , ˆ η , ˇ η } ⊂ ˆ V and span { θ , ˆ η , ˇ σ , ˇ η } ⊂ ˇ V . Property (1.8) then implies that the forms { θ , ˆ σ , ˆ η , ˇ σ , ˇ η } span T ∗ M .To prove the first of (2.6c), let ω = a θ + b ˆ σ + c ˆ η + d ˇ σ + e ˇ η . If ω ∈ ˆ V ,then the 1-forms d ˇ σ ∈ ˆ V and therefore, by (2.6b), d ˇ σ ∈ ˆ V ∩ ˇ V ( ∞ ) . Since theforms ˇ η and ˇ σ are independent, we must have d = 0 and (2.6c) is established.To prove the independence of the forms { θ , ˆ σ , ˆ η , ˇ σ , ˇ η } , set ω = 0. Thenthe argument just given implies that d = 0 so that bˆ σ ∈ ˆ V ∩ ˇ V . Therefore bˆ σ ∈ ˆ V ( ∞ ) ∩ ˇ V and this forces b = 0. The independence of the forms { θ , ˆ η , ˇ η } then gives a = c = e = 0 and the lemma is proved.Any local coframe { θ , ˆ σ , ˆ η , ˇ σ , ˇ η } satisfying (2.6) is called a { ˆ V , ˇ V } . We shall also need the definition of a 0-adapted frame as the dual of the0-adapted coframe. Let ˆ H = ann ˆ V , ˇ H = ann ˇ V , ˆ H ( ∞ ) = ann ˆ V ( ∞ ) , ˇ H ( ∞ ) = ann ˇ V ( ∞ ) , and K = ˆ H ( ∞ ) ∩ ˇ H ( ∞ ) . (2.7)The definition of Darboux integrability can be re-formulated in terms of thedistributions ˆ H, ˇ H ([26]). If we introduce the dual basis { ∂ θ , ∂ ˆ σ , ∂ ˆ η , ∂ ˇ σ , ∂ ˇ η } to the 0-adapted coframe { θ , ˆ σ , ˆ η , ˇ σ , ˇ η } , that is, θ ( ∂ θ ) = , ˆ σ ( ∂ ˆ σ ) = , ˆ η ( ∂ ˆ η ) = , ˇ σ ( ∂ ˇ σ ) = , ˇ η ( ∂ ˇ η ) = , (2.8)with all others pairings yielding 0, then it readily follows that ˆ H = span { ∂ ˇ σ } , ˆ H ( ∞ ) = span { ∂ θ , ∂ ˇ σ , ∂ ˇ η } , ˇ H = span { ∂ ˆ σ } , ˇ H ( ∞ ) = span { ∂ θ , ∂ ˆ σ , ∂ ˆ η } , and K = span { ∂ θ } . (2.9)Any local frame { ∂ θ , ∂ ˆ σ , ∂ ˆ η , ∂ ˇ σ , ∂ ˇ η } satisfying (2.9) is called a 0-adaptedframe.We shall use the following lemma repeatedly. .2 The first adapted coframes for a Darboux pair Lemma 2.8.
Let f be a real-valued function on M . If X ( f ) = 0 for all vectorfields in ˆ H , then df ∈ ˆ V ( ∞ ) . Likewise, if X ( f ) = 0 for all vector fields in ˇ H ,then df ∈ ˇ V ( ∞ ) .Proof. It suffices to note that if X ( f ) = 0 for all vector fields X ∈ ˆ H , then Z ( f ) = Z df = 0 for all vector fields Z ∈ ˆ H ( ∞ ) and therefore df ∈ ann ˆ H ( ∞ ) = ˆ V ( ∞ ) .Real-valued functions on M with df ∈ V (or V ( ∞ ) ) are called first integrals for V . We denote the algebra of all first integrals for V by Int( V ).Our 1-adapted coframe for a Darboux pair is easily constructed from com-plete sets of functionally independent first integrals for ˆ V ( ∞ ) and ˇ V ( ∞ ) , that is,locally defined functions { ˆ I } and { ˇ I } such that ˆ V ( ∞ ) = span { d ˆ I } and ˇ V ( ∞ ) = span { d ˇ I } . Complete the 1-forms { ˆ η } in (2.6a) to the local basis (2.6b) for ˆ V ( ∞ ) usingforms ˆ σ = d ˆ I (2.10)chosen from the { d ˆ I } . Next let { ˆ I } be a complementary set of invariants tothe set { ˆ I } , that is, { ˆ I } = { ˆ I , ˆ I } . Because the forms ˆ η belong to ˆ V ∞ wecan write ˆ η = ˆR d ˆ I + ˆ S d ˆ I . Since the 1-forms ˆ η are independent of the 1-forms ˆ σ , the coefficient matrix ˆ S must be invertible and we can therefore adjust our local basis of 1-forms { ˆ η } for ˆ V ( ∞ ) ∩ ˇ V by setting ˆ η = d ˆ I + ˆR d ˆ I = d ˆ I + ˆR ˆ σ . (2.11)The exterior derivatives of these forms are d ˆ η = d ˆR ∧ ˆ σ = ∂ ˇ σ ( ˆR ) ˇ σ ∧ ˆ σ + · · · and therefore, on account of (1.10), the fact that ˆ η ∈ ˆ V ( ∞ ) ∩ ˇ V , and (2.6c), wemust have ∂ ˇ σ ( ˆR ) = 0. Lemma 2.8 implies that d ( ˆR ) ∈ ˆ V ( ∞ ) and consequently d ˆ η = ˆ E ˆ η ∧ ˆ σ + ˆ F ˆ σ ∧ ˆ σ . (2.12)Similar arguments are used to modify the forms ˇ σ and ˇ η . These adaptationsare summarized in the following theorem. .2 The first adapted coframes for a Darboux pair Theorem 2.9.
Let { ˆ V , ˇ V } be a Darboux pair on a manifold M . Then abouteach point of M there exists a 0-adapted coframe { θ , ˆ σ , ˆ η , ˇ σ , ˇ η } with structureequations d ˆ σ = 0 , d ˆ η = ˆ E ˆ η ∧ ˆ σ + ˆ F ˆ σ ∧ ˆ σ ,d ˇ σ = 0 , d ˇ η = ˇ E ˇ η ∧ ˇ σ + ˇ F ˇ σ ∧ ˇ σ , (2.13) d θ ≡ A ˆ σ ∧ ˆ σ + B ˇ σ ∧ ˇ σ mod { θ , ˆ η , ˇ η } . (2.14)We remark that the structure equations for the 1-forms θ ∈ ˆ V ∩ ˇ V are a conse-quence of (1.10) and the properties (2.6) of a 0-adapted coframe. A for the Darboux pair { ˆ V , ˇ V } is a 0-adapted coframe { θ , ˆ σ , ˆ η , ˇ σ , ˇ η } satisfying the structure equations (2.13) and (2.14). Corollary 2.10. If { ˆ V , ˇ V } define a Darboux pair, then ˆ V ∩ + ˇ V is always a de-composable exterior differential system for which ˆ V and ˇ V are singular Pfaffiansystems.Proof. It follows from Theorem 2.9 that the algebraic generators for ˆ V and ˇ V are ˆ V = { θ , ˆ η , ˇ η , ˆ σ , ˇ F ˇ σ ∧ ˇ σ , B ˇ σ ∧ ˇ σ } and ˇ V = { θ , ˆ η , ˇ η , ˇ σ , ˆ F ˆ σ ∧ ˆ σ , A ˆ σ ∧ ˆ σ } . From these equations it is not difficult to argue that the differential system ˆ V ∩ ˇ V has generators ˆ V ∩ ˇ V = { θ , ˆ η , ˇ η , ˆ σ ∧ ˇ σ , ˆ F ˆ σ ∧ ˆ σ , A ˆ σ ∧ ˆ σ , ˇ F ˇ σ ∧ ˇ σ , B ˇ σ ∧ ˇ σ } . A repetition of the arguments used in the proof of Theorem 2.6 shows that ˆ V ∩ + ˇ V = { θ , ˆ η , ˇ η , ˆ F ˆ σ ∧ ˆ σ , A ˆ σ ∧ ˆ σ , ˇ F ˇ σ ∧ ˇ σ , B ˇ σ ∧ ˇ σ } . (2.15)It is then clear that ˆ V ∩ + ˇ V is a decomposable differential system (Definition 1.1)and that ˆ V and ˇ V are singular Pfaffian systems. Example 2.11.
There are many examples of differential equations which canbe described either by Pfaffian differential systems or by differential systemsgenerated by 1-forms and 2-forms. Our definition of Darboux integrability interms of decomposable differential system is such that theseequations will be .2 The first adapted coframes for a Darboux pair u xy = 0. The wave equation can be encoded as adifferential system onmanifold of dimensions 5, 6 and 7 by I = { θ, dp ∧ dx, dq ∧ dy } , ˆ V = { θ, dp, dx } , ˇ V = { θ, dq, dy } , I = { θ, θ x , dq ∧ dy } , ˆ V = { θ, θ x , dp, dx } , ˇ V = { θ, θ x , dq, dy } , I = { θ, θ x , θ y } , ˆ V = { θ, θ x , θ y , dp, dx } , ˇ V = { θ, θ x , θ y , dq, dy } , where θ = du − p dx − q dy , θ x = dp − r dx and θ y = dq − t dy . Each of thesesatisfy the definition of a decomposable, Darboux integrable differential system.Consistent with Corollary 2.6, one easily checks that I i = ˆ V i ∩ + ˇ V i .We remark that for the standard contact system on J ( R , R ), that is, I = { θ, dp ∧ dx + dq ∧ dy } , the Pfaffian systems ˆ V = { θ, dp, dx } and ˇ V = { θ, dq, dy } , are singular systems which form a Darboux pair but ˆ V ∩ + ˇ V = I .Equality fails because I is not decomposable. Remark 2.12.
In very special cases, such as Liouville’s equation u xy = e u ,one finds that ˆ V ∩ ˇ V ( ∞ ) = ˆ V ( ∞ ) ∩ ˇ V = { } so that the vector space sumsappearing in (1.8) are direct sum decompositions. Under these circumstances,our 0-adapted coframe is simply { θ , ˆ σ , ˇ σ } . This initial coframe is then auto-matically 2-adapted (see Theorem 4.3) and the sequence of coframe adaptationsneeded to prove Theorem 4.1 can begin with the third coframe adaptation (Sec-tion 4.2). Remark 2.13.
We define an involution of a Darboux pair { ˆ V , ˇ V } on M to be a diffeomorphism Φ : M → M such that Φ = id M and Φ ∗ ( ˆ V ) = ˇ V . Forsuch maps Φ ∗ ( ˆ V ( ∞ ) ) = ˇ V ( ∞ ) and therefore, given 1 adapted co-frame elements ˆ η and ˆ σ , one may take ˇ η = Φ ∗ ( ˆ η ) and ˇ σ = Φ ∗ ( ˆ σ ) . Such involutions are quite common, especially for differential equations arisingin geometric and physical applications, where the diffeomorphism Φ arises froma symmetry of the differential equations in the independent variables. he group theoretic construction of Darboux pairs Let W and W be differential systems on manifolds M and M . Then the direct sum W + W is the EDS on M × M generated by the pullbacks of W and W by the canonical projection maps π i : M × M → M i . The followingtheorem shows that if W and W are Pfaffian, then W + W is Darbouxintegrable in the sense of Definition 1.3. Theorem 3.1.
Let W and W be Pfaffian systems on manifolds M and M such that W ( ∞ )1 = W ( ∞ )2 = 0 . Then the direct sum J = W + W on M × M is a decomposable Pfaffian system. The associated singular Pfaffian systems ˆ W = W ⊕ Λ ( M ) and ˇ W = Λ ( M ) ⊕ W (3.1) form a Darboux pair and W + W = ˆ W ∩ + ˇ W , (3.2) where W , W , ˆ W , ˇ W are the differential systems generated by W , W , ˆ W , ˇ W .Proof. It is easy to check that J is decomposable with singular systems (3.1).Properties [i] and [ii] of Definition 1.2 follow directly from (3.1) and the simpleobservations that ˆ W ( ∞ ) = Λ ( M ) and ˇ W ( ∞ ) = Λ ( M ) . (3.3)Let ω = P i ( f i θ i + g i θ i ), where the coefficients f i , g i ∈ C ∞ ( M × M ) andthe θ iℓ ∈ W ℓ are 1-forms in J . Then to check property [iii] , it suffices to notethat every summand in the exterior derivative d ω = X i d f i ∧ θ i + d f i ∧ θ i + f i d θ i + d g j ∧ θ j + d g j ∧ θ j + g i d θ i , where d i = d M i , lies either in Ω ( ˆ W ) or Ω ( ˇ W ). Equation (3.2) is a directconsequence of Theorem 2.6.Darboux pairs of the form (3.1) are said to be trivial. .2 Symmetry Reduction of Darboux Pairs A Lie group G is a regular symmetry group of a differential system I on a manifold M if there is a regular action of G on M which preserves I . Bythe definition of regularity, the quotient space M/G of M by the orbits of G hasa smooth manifold structure such that [i] the projection map q : M → M/G isa smooth submersion, and [ii] local, smooth cross-sections to q exist on someopen neighborhood of each point of M/G . The G reduction of I or thequotient of I by G is the differential system on M = M/G given by I /G = { ω ∈ Ω ∗ ( M ) | q ∗ ( ω ) ∈ I } . (3.4)Details of this construction and basis facts regarding the G reduction of differ-ential systems are given in [1].To obtain the main results of this section, we shall require that G be trans-verse to I in the sense which we now define. Let I be any G invariant sub-bundle of T ∗ M and let Γ be the infinitesimal generators for G . We say that G is transverse to I if for each x ∈ M Γ x ∩ ann I x = { } . (3.5)We say that G is transverse to the differential system I if G is transverse to thesub-bundle of T ∗ M whose sections are 1-forms in I . With this transversalitycondition, I /G is assured to be a constant rank differential system whose localsections can be determined as follows. If A is any sub-bundle of Λ k ( M ) or sub-algebra of Λ ∗ ( M ), then the sub-bundle or sub-algebra of semi-basic forms in A is defined as A sb = { ω ∈ A | X ω = 0 for all X ∈ Γ } . (3.6)Denote by I sb the sections of I which are semi-basic, that is, which take valuesin Λ ∗ sb ( M ). Then one can show (see Lemma 4.1 of [1]) that( I /G ) | U = ξ ∗ ( I sb | U ) , (3.7)where U is any (sufficiently small) open set of M , ξ : U → M is a smoothcross-section of q , and U = q − ( U ). We remark that the G reduction I /G of aPfaffian system I need not be Pfaffian (but see [c] below).For the proof of the next theorem on the G reduction of Darboux pairs, weshall require the following elementary facts regarding the reduction of Pfaffian .2 Symmetry Reduction of Darboux Pairs I and J are Pfaffian systems with regular symmetry group G , weassume that G is transverse to I , and the G bundle reduction of I is defined as I/G = { ω ∈ Λ ( M ) | q ∗ ( ω ) ∈ I } . (3.8)By transversality, this is a sub-bundle of T ∗ M of dimension dim I − q , where q is the dimension of the orbits of G . [a] If I ⊂ J , then G is transverse to J . [b] If G is transverse to I ∩ J , then ( I/G ) ∩ ( J/G ) = ( I ∩ J ) /G . [c] If G is transverse to the derived system I ′ , then the quotient differen-tial system I /G is the constant rank Pfaffian system determined by I/G and(
I/G ) ′ = I ′ /G . [d] If G is transverse to I ( ∞ ) , then ( I/G ) ( ∞ ) = I ( ∞ ) /G .Facts [a] and [b] are trivial, while [c] is Theorem 5.1 in [1]. Fact [d] followsfrom [a] and [c] by induction.Finally, we shall also use the following technical observation. If Z is a sub-bundle of I , then the canonical pairing ( X, ω ) = ω ( X ) on Γ x × Z x , where x ∈ M ,is non-degenerate if and only if I = q ∗ ( I/G ) ⊕ Z. (3.9)Such bundles Z can always be constructed locally on G invariant open subsetsof M . Theorem 3.2.
Let { ˆ W , ˇ W } be a Darboux pair on M . Suppose that G is aregular symmetry group of ˆ W and ˇ W and that, in addition, G is transverse to ˆ W ∩ ˇ W ( ∞ ) and ˆ W ( ∞ ) ∩ ˇ W . Then the quotient differential systems { ˆ W /G, ˇ W /G } on M are (constant rank) Pfaffian systems and form a Darboux pair.Proof of Theorem 3.2. Since ˆ W ( ∞ ) ∩ ˇ W ⊂ ˆ W ( ∞ ) ⊂ ˆ W ′ ⊂ ˆ W and G is transverseto ˆ W ( ∞ ) ∩ ˇ W , it follows that G is transverse to ˆ W ( ∞ ) , ˆ W ′ and ˆ W . Thus, by [c] , ˆ V = ˆ W /G and, similarly, ˇ V = ˇ W /G are the Pfaffian systems for ˆ V = ˆ W /G and ˇ V = ˇ W /G . By [d] ( ˆ W /G ) ( ∞ ) = ˆ W ( ∞ ) /G and ( ˇ W /G ) ( ∞ ) = ˇ W ( ∞ ) /G. (3.10) .2 Symmetry Reduction of Darboux Pairs { ˆ V , ˇ V } to be aDarboux pair. From (3.8), (3.10) and fact [b] we deduce thatdim( ˆ V ) = dim ˆ W − q, dim( ˆ V ( ∞ ) ) = dim( ˆ W ( ∞ ) /G ) = dim ˆ W ( ∞ ) − q, anddim( ˆ V ∩ ˆ V ( ∞ ) ) = dim (cid:0) ( ˆ W /G ) ∩ ( ˇ W ( ∞ ) /G ) (cid:1) = dim (cid:0) ( ˆ W ∩ ˇ W ( ∞ ) ) /G (cid:1) = dim( ˆ W ∩ ˇ W ( ∞ ) ) − q. From these equations and property [i] in the definition of Darboux pair (appliedto { ˆ W , ˇ W } ) we calculatedim (cid:0) ˆ V + ˇ V ( ∞ ) (cid:1) = dim( ˆ V ) + dim( ˇ V ( ∞ ) ) − dim( ˆ V ∩ ˇ V ( ∞ ) )= dim ˆ W + dim ˇ W ( ∞ ) + dim( ˆ W ∩ ˇ W ( ∞ ) ) − q = dim M − q = dim M .
This proves that ˆ V + ˇ V ( ∞ ) = T ∗ M . Similar arguments yield ˆ V ( ∞ ) + ˇ V = T ∗ M and so { ˆ V , ˇ V } satisfy property [i] of a Darboux pair.To check that { ˆ V , ˇ V } satisfy property [ii] of a Darboux pair, we use [b] and(3.10) to calculate ˆ V ( ∞ ) ∩ ˇ V ( ∞ ) = ( ˆ W ( ∞ ) /G ) ∩ ( ˇ W ( ∞ ) /G ) = ( ˆ W ( ∞ ) ∩ ˇ W ( ∞ ) ) /G = 0 . (3.11)To prove property [iii] of a Darboux pair, we first prove that[Λ ( ˆ W ) + Λ ( ˇ W )] sb = Λ ( ˆ W sb ) + Λ ( ˇ W sb ) . (3.12)It suffices to check this locally. Since G acts transversally to ˆ W ( ∞ ) ∩ ˇ W , thereis a locally defined sub-bundle Z of ˆ W ( ∞ ) ∩ ˇ W such that the canonical pairingon Γ × Z is pointwise non-degenerate. For any such sub-bundle Z we have that ˆ W = ˆ W sb ⊕ Z and ˇ W = ˇ W sb ⊕ Z and thereforeΛ ( ˆ W ) + Λ ( ˇ W ) = Λ ( ˆ W sb ) + Λ ( ˇ W sb ) + T ∗ M ∧ Z. This (local) decomposition immediately leads to (3.12). To complete the proof ofproperty [iii] , let ω ∈ ˆ V ∩ ˇ V . Then q ∗ ( ω ) ∈ ˆ W ∩ ˇ W and hence, by the propertiesof the Darboux pair { ˆ W , ˇ W } and (3.12), q ∗ ( d ω ) ∈ Ω ( ˆ W sb ) + Ω ( ˇ W sb ). Wepullback this last equation by a local cross-section of q and invoke (3.7) toconclude that d ω ∈ Ω ( ˆ V ) + Ω ( ˇ V ). .2 Symmetry Reduction of Darboux Pairs Theorem 3.3.
Let { ˆ W , ˇ W } be a Darboux pair on M . Suppose that G is aregular symmetry group of ˆ W and ˇ W and that, in addition, G is transverse to ˆ W ( ∞ ) ∩ ˇ W and ˆ W ∩ ˇ W ( ∞ ) . Then ( ˆ W ∩ + ˇ W ) /G = ( ˆ W /G ) ∩ + ( ˇ W /G ) . (3.13) Proof.
We first remark that Theorem 3.2 implies that ˆ V = ˆ W /G and ˇ V = ˇ W /G define a Darboux pair and, second, that it suffices to check (3.13) locally.Transversality implies that there are locally defined sub-bundles ˆ Z ⊂ ˆ W ( ∞ ) ∩ ˇ W and ˇ Z ⊂ ˆ W ∩ ˇ W ( ∞ ) on which the natural pairing with Γ (the infinitesimalgenerators for G ) is non-degenerate at each point x ∈ M . It follows from (3.9)that ˆ W ( ∞ ) ∩ ˇ W = q ∗ ( ˆ V ( ∞ ) ∩ ˇ V ) ⊕ ˆ Z, ˆ W ( ∞ ) = q ∗ ( ˆ V ( ∞ ) ) ⊕ ˆ Z, (3.14a) ˆ W ∩ ˇ W ( ∞ ) = q ∗ ( ˆ V ∩ ˇ V ( ∞ ) ) ⊕ ˇ Z, ˇ W ( ∞ ) = q ∗ ( ˇ V ( ∞ ) ) ⊕ ˇ Z, (3.14b) ˆ W = q ∗ ( ˆ V ) ⊕ ˆ Z, ˇ W = q ∗ ( ˆ V ) ⊕ ˇ Z, and (3.14c) ˇ W ∩ ˆ W = q ∗ ( ˇ V ∩ ˆ V ) + ˆ Z = q ∗ ( ˇ V ∩ ˆ V ) + ˇ Z. (3.14d)If ˆ ζ and ˇ ζ are local basis of sections for ˆ Z and ˇ Z then, because ˆ ζ ∈ ˆ W ( ∞ ) and ˇ ζ ∈ ˇ W ( ∞ ) , d ˆ ζ ≡ ˆ W ( ∞ ) and d ˇ ζ ≡ ˇ W ( ∞ ) , or d ˆ ζ ≡ { q ∗ ( ˆ V ( ∞ ) ) , ˆ ζ } and d ˇ ζ ≡ { q ∗ ( ˇ V ( ∞ ) ) , ˇ ζ } . These equations, together with (3.14c), show that the Pfaffian systems ˆ W and ˇ W are algebraically generated as ˆ W = { q ∗ ( ˆ V ) , ˆ ζ } and ˇ W = { q ∗ ( ˇ V ) , ˇ ζ } . (3.15)To complete the proof of the theorem, we now make the key observation thatsince ˇ ζ ∈ ˆ W ∩ ˇ W , (3.14c) implies that we may express these forms as linearcombinations of the ˆ ζ and the forms in q ∗ ( ˆ V ∩ ˇ V ). Hence ˇ W is also algebraicallygenerated as ˇ W = { q ∗ ( ˇ V ) , ˇ ζ } . We may list an explicit set of local generatorsfor q ∗ ( ˆ V ) and q ∗ ( ˇ V ) using the 1-adapted coframe constructed for the Darbouxpair { ˆ V , ˇ V } in Theorem 2.9. The arguments given in the proof of Theorem2.10 can then be repeated to show that ˆ W ∩ + ˇ W = { q ∗ ( ˆ V ∩ + ˇ V ) , ˆ ζ } and hence( ˆ W ∩ + ˇ W ) sb = q ∗ ( ˆ V ∩ + ˇ V ), as required. .2 Symmetry Reduction of Darboux Pairs Corollary 3.4.
Let W and W be Pfaffian systems on manifolds M and M with W ( ∞ )1 = W ( ∞ )2 = 0 . Suppose that [i] a Lie group G acts freely on M and M and as symmetry groups of both W and W ; [ii] G is transverse to both W and W ; and [iii] the diagonal action of G on M × M is regular.Then the quotient differential system J = ( W + W ) /G (3.16) is a decomposable, Darboux integrable, differential system with respect to theDarboux pair ˆ U = ( W ⊕ Λ ( M )) /G and ˇ U = (Λ ( M ) ⊕ W ) /G. (3.17) Proof.
As in Theorem 3.1, define the Darboux pair ˆ W = W ⊕ Λ ( M ) and ˇ W = Λ ( M ) ⊕ W . To conclude from Theorem 3.2 that ˆ U = ˆ W /G , ˇ U = ˇ W /G are also a Darboux pair, we observe that [i] and [ii] imply that the diagonalaction of G on M × M is transverse to the Pfaffian systems W and W pulledback to M × M (a fact which is not true without [i] . We also note that, by(3.3), ˆ W ∩ ˇ W ( ∞ ) = W and ˆ W ( ∞ ) ∩ ˇ W = W . (3.18)We now use (3.2) and Theorem 3.3 to deduce that J = ( W + W ) /G = ( ˆ W ∩ + ˇ W ) /G = ( ˆ W /G ) ∩ + ( ˇ W /G ) = ˆ U ∩ + ˇ U . An application of Corollary 2.10 completes the proof. he coframe adaptations for a Darboux pair In this section we present a series of coframe adaptations for any Darbouxpair { ˆ V , ˇ V } . These coframe adaptations lead to the proof of the followingtheorem. Theorem 4.1.
Let { ˆ V , ˇ V } be a Darboux pair of Pfaffian differential systemson a manifold M . Then around each point of M there are 2 coframes { ˆ θ i , ˆ π a , ˇ π α } and { ˇ θ i , ˆ π a , ˇ π α } (4.1) such that ˆ V = { ˆ θ i , ˆ π a } , ˆ V ∞ = { ˆ π a } , ˇ V = { ˇ θ i , ˇ π a } , ˇ V ∞ = { ˇ π a } (4.2) and with structure equations d ˆ θ i = 12 G iαβ ˇ π α ∧ ˇ π β + 12 C ijk ˆ θ j ∧ ˆ θ k and d ˇ θ i = 12 H iab ˆ π a ∧ ˆ π b − C ijk ˇ θ j ∧ ˇ θ k . (4.3) The coefficients C ijk are the structure constants of a Lie algebra whose isomor-phism class is an invariant of the Darboux pair { ˆ V , ˇ V } . We have already constructed the first coframe adaptation in Section 2. Thesecond coframe (Section 4.1) adaptation provides us with a stronger form forthe decomposition of the structure equations than that initially given by (2.5).The third and fourth coframe adaptations (Section 4.2 and 4.3) lead to thedefinition of the Vessiot Lie algebra vess ( ˆ V , ˇ V ), with structure constants C ijk ,as a fundamental invariant for any Darboux integrable differential system. Theimportance of this algebra was first observed by Vassiliou [26] who introducedthe notion of the tangential symmetry algebra for certain special classes ofDarboux integrable systems. The tangential symmetry algebra is a Lie algebraof vector fields Γ on M which is isomorphic (as an abstract Lie algebra) to theVessiot Lie algebra. The tangential symmetry algebra also plays a significantrole in Eendebak’s projection method [10] for Darboux integrable equations.Of course, the tangential symmetry algebra Γ determines a local transforma-tion group on M but this transformation group will generally not be the correctone required for constructing the superposition formula (See Example 6.1). One .1 The second adapted coframe for a Darboux pair. d ( α ) = β , where β is a closed 1 or 2-form (see (4.64) and (4.73)). Let { ˆ V , ˇ V } be a Darboux pair and let { θ , ˆ σ , ˆ η , ˇ σ , ˇ η } be a 1-adaptedcoframe with dual frame { ∂ θ , ∂ ˆ σ , ∂ ˆ η , ∂ ˇ σ , ∂ ˇ η } . From this point forward, weare solely interested in adjustments to the forms θ that will simplify the 1-adapted structure equations d θ = A ˆ σ ∧ ˆ σ + B ˇ σ ∧ ˇ σ mod { θ , ˆ η , ˇ η } . (4.4)Of the “mixed” wedge products ˆ σ ∧ ˇ σ , ˆ σ ∧ ˇ η , ˆ η ∧ ˇ σ , and ˆ η ∧ ˇ η (4.5)the products ˆ σ ∧ ˇ σ are the only ones definitely not present in (4.4). We shallnow show that it is possible to make an adjustment to the 1-forms θ of the type θ ′ = θ + P ˆ η + Q ˇ η (4.6)so that the structure equations for the modified forms θ ′ are free of all the wedgeproducts (4.5). We begin with the following simple observation. Lemma 4.2. If { ∂ θ , ∂ ˆ σ , ∂ ˆ η , ∂ ˇ σ , ∂ ˇ η } is the dual frame to a 1-adapted coframefor the Darboux pair { ˆ V , ˇ V } , then [ ∂ ˆ σ , ∂ ˇ σ ] = 0 . (4.7) Proof.
It suffices to note that none of the structure equations for a 1-adaptedcoframe contain any of the wedge products ˆ σ ∧ ˇ σ .The construction of the second adapted coframe is completely algebraic inthat only differentiations and linear algebraic operations are involved. Let ˆ S a = .1 The second adapted coframe for a Darboux pair. ∂ ˆ σ a and ˆ S α = ∂ ˇ σ α and define two sequences of vector fields inductively by ˆ S a a ··· a ℓ = [ ˆ S a a ··· a ℓ − , ˆ S a ℓ , ] and ˇ S α α ··· α ℓ = [ ˇ S α α ··· α ℓ − , ˇ S α ℓ ] . (4.8)Because the vector fields ˆ S a ℓ ∈ ˆ H , the vector fields S a a ··· a ℓ belong to ˆ H ( ∞ ) andare therefore linear combinations of the vector fields { ∂ θ , ∂ ˆ σ , ∂ ˆ η } . However,because d ˆ σ = , one can deduce by a straightforward induction that in fact S a a ··· a ℓ ∈ span { ∂ θ , ∂ ˆ η } and, likewise, ˇ S α α ··· α ℓ ∈ span { ∂ θ , ∂ ˇ η } . (4.9)The Jacobi identity also shows that ˆ H ( ∞ ) = span { ˆ S a a ··· a ℓ } ℓ ≥ and ˇ H ( ∞ ) = span { ˇ S α α ··· α ℓ } ℓ ≥ . (4.10)From (4.7) it also follows that[ ˆ S a a ··· a k , ˇ S α α ··· α ℓ ] = 0 . (4.11)for k, l ≥
1. By virtue of (4.10) we can choose a specific collection of iter-ated Lie brackets ˆ S a a ··· a ℓ and ˇ S α α ··· α ℓ , ℓ ≥
2, which complete { ∂ θ , ∂ ˆ σ } and { ∂ θ , ∂ ˇ σ } to local bases for ˆ H ( ∞ ) and ˇ H ( ∞ ) respectively. Denote theiterated brackets so chosen by { ˆ s ′ } and { ˇ s ′ } . Since span { ∂ θ , ∂ ˆ σ , ∂ ˆ η } =span { ∂ θ , ∂ ˆ σ , ˆ s ′ } and span { ∂ θ , ∂ ˇ σ , ∂ ˇ η } = span { ∂ θ , ∂ ˇ σ , ˇ s ′ } , this gives us apreferred 0-adapted frame { ∂ θ , ∂ ˆ σ , ˆ s ′ , ∂ ˇ σ , ˇ s ′ } where, on account of (4.11),[ ∂ ˆ σ , ∂ ˇ σ ] = 0 , [ ∂ ˆ σ , ˇ s ′ ] = 0 [ ˆ s ′ , ∂ ˇ σ ] = 0 , [ ˆ s ′ , ˇ s ′ ] = 0 . (4.12)It follows that the coframe { θ ′ , ˆ σ ′ , ˆ η ′ , ˇ σ ′ , ˇ η ′ } dual to { ∂ θ , ∂ ˆ σ , ˆ s ′ , ∂ ˇ σ , ˇ s ′ } isa 0-adapted coframe for the Darboux pair { ˆ V , ˇ V } . Equations (4.9) imply (4.6).On account of (4.12), the structure equations for the forms { θ ′ } are free ofall the wedge products ˆ σ ′ ∧ ˇ σ ′ , ˆ σ ′ ∧ ˇ η ′ , ˆ η ′ ∧ ˇ σ ′ , and ˆ η ′ ∧ ˇ η ′ . We can expressthis result by writing d θ ′ ∈ Ω ( ˆ V ( ∞ ) ) + Ω ( ˇ V ( ∞ ) ) mod { θ ′ } . The coframe { θ ′ , ˆ σ , ˆ η , ˇ σ , ˇ η } therefore satisfies the criteria of the followingtheorem. Theorem 4.3.
Let { ˆ V , ˇ V } be a Darboux pair on a manifold M . Then abouteach point of M there exists a 1-adapted coframe { θ , ˆ σ , ˆ η , ˇ σ , ˇ η } with structureequations (2.13) and d θ = A ˆ π ∧ ˆ π + B ˇ π ∧ ˇ π mod { θ } , (4.13) where ˆ π and ˇ π denote the tuples of forms ( ˆ σ , ˆ η ) and ( ˇ σ , ˇ η ) . .1 The second adapted coframe for a Darboux pair. . .2 The third adapted coframes for a Darboux pair Written out in full, the structure equations (4.13) are d θ = A ˆ π ∧ ˆ π + B ˇ π ∧ ˇ π + C θ ∧ θ + M ˆ π ∧ θ + N ˇ π ∧ θ . (4.14)We obtain the third coframe reduction for a Darboux pair by showing that achange of coframe θ ′ = P θ can be made so as to eliminate either all the wedgeproducts ˆ π ∧ θ or all the wedge products ˇ π ∧ θ in (4.14). The construction ofthis coframe uses another set of iterated Lie brackets.We start with a 2-adapted coframe { θ , ˆ σ , ˆ η , ˇ σ , ˇ η , } , where ˆ π = ( ˆ σ , ˆ η ) and ˇ π = ( ˇ σ , ˇ η ), and introduce the provisional frame { θ , ˆ ι , ˇ ι } , where ˆ ι = d ˆ I and ˇ ι = d ˇ I . This is not a 0-adapted coframe although it is the case thatspan { ˆ ι } = span { ˆ π } and span { ˇ ι } = span { ˇ π } . (4.15)From (4.14) and (4.15) we find that the structure equations for this coframe are d ˆ ι = 0 , d ˇ ι = 0 , and d θ = A ˆ ι ∧ ˆ ι + B ˇ ι ∧ ˇ ι + C θ ∧ θ + M ˆ ι ∧ θ + N ˇ ι ∧ θ . (4.16)Denote the dual to this provisional coframe by { ∂ θ , ˆ U , ˇ U } . Sincespan { ∂ ˆ σ } = ann ˇ V = ann { θ , ˆ ι , ˇ σ } ⊂ ann { θ , ˆ ι } = span { ˇ U } , we have that ∂ ˆ σ ⊂ span { ˇ U } and ∂ ˇ σ ⊂ span { ˆ U } . (4.17)From (4.16) it follows that the structure equations for the vectors fields ˆ U and ˆ U are [ ˆ U , ˆ U ] = − A∂ θ , [ ˆ U , ∂ θ ] = − M ∂ θ , [ ˇ U , ˇ U ] = − B∂ θ , [ ˇ U , ∂ θ ] = − N ∂ θ , and [ ˇ U , ˆ U ] = 0 . (4.18)As in Section 4.2, define two sequences of vector fields inductively by ˆ U a a ··· a ℓ = [ ˆ U a a ··· a ℓ − , ˆ U a ℓ ] and ˇ U α α ··· α ℓ = [ ˆ U α α ··· α ℓ − ˆ U α ℓ ] . (4.19)A simple induction argument, based upon the last of (4.18) and the Jacobiidentity, shows that [ ˆ U a a ··· a k , ˇ U α α ··· α ℓ ] = 0 . (4.20) .2 The third adapted coframes for a Darboux pair k, l ≥
1. On account of (2.7), (2.9) and (4.17) the iterated brackets ˆ U a a ··· a ℓ will span all of ˇ H and therefore, by the first two equations in (2.9), we can choosea basis X for K = ˆ H ( ∞ ) ∩ ˇ H ( ∞ ) (see (2.7)) consisting of a linear independentset of the vector fields ˆ U a a ··· a ℓ . Alternatively, we can choose a basis Y for K consisting of a linear independent set of the vector fields ˇ U a a ··· a ℓ . Due to(4.20) these two bases for K satisfy[ X , ˇ U ] = 0 , [ X , Y ] = 0 , [ Y , ˆ U ] = 0 . (4.21)Denote the dual of the coframe { X , ˆ U , ˇ U } by { θ X , ˆ ι , ˇ ι } and the dual of thecoframe { Y , ˆ U , ˇ U } by { θ Y , ˆ ι , ˇ ι } . Theorem 4.4.
Let { ˆ V , ˇ V } be a Darboux pair on a manifold M . Then abouteach point of M there are two 2-adapted coframes { θ X , ˆ σ , ˆ η , ˇ σ , ˇ η } and { θ Y , ˆ σ , ˆ η , ˇ σ , ˇ η } with structure equations (2.13) , d θ X = A ˆ π ∧ ˆ π + B ˇ π ∧ ˇ π + C θ X ∧ θ X + M ˆ π ∧ θ X , (4.22) and d θ Y = E ˆ π ∧ ˆ π + F ˇ π ∧ ˇ π + K θ Y ∧ θ Y + N ˇ π ∧ θ Y . (4.23) Moreover, span { θ X } = span { θ Y } and the vector fields X , Y belonging to thedual frames { X , ∂ ˆ σ , ∂ ˆ η , ∂ ˇ σ , ∂ ˇ η } and { Y , ∂ ˆ σ , ∂ ˆ η , ∂ ˇ σ , ∂ ˇ η } satisfy [ X , Y ] = 0 . (4.24)Coframes which satisfy the conditions of Theorem 4.4 are called . Remark 4.5.
In the special case where the pair { ˆ V , ˇ V } admits an involution(see Remark 2.13) the forms θ Y may be defined by θ Y = Φ ∗ ( θ X ). Then thestructure equations (4.23) can be immediately inferred from (4.22). .3 The fourth adapted coframe and the Vessiot algebra We now show that any 3-adapted coframe may be adjusted so that thestructure functions C ijk and K ijk (see (4.22)–(4.24)) are constants and satisfy C ijk = − K ijk . Theorem 4.6.
Let { ˆ V , ˇ V } be a Darboux pair on M . [i] Then, about each point of M , there exist local coframes { θ iX , ˆ π a , ˇ π α } and { θ iY , ˆ π a , ˇ π α } which are 3-adapted and with structure equations d θ iX = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β + 12 C ijk θ jX ∧ θ kX + M iaj ˆ π a ∧ θ jX (4.25) and d θ iY = 12 E iab ˆ π a ∧ ˆ π b + 12 F iαβ ˇ π α ∧ ˇ π β − C ijk θ jY ∧ θ kY + N iαj ˇ π α ∧ θ jY . (4.26) The structure functions C ijk = − C ikj are constant on M and are the structureconstants of a real Lie algebra. [ii] The isomorphism class of the Lie algebra defined by the structure constants C ijk is an invariant of the Darboux pair { ˆ V , ˇ V } . Coframes which satisfy the conditions of Theorem 4.6 are called . As with the 2 and 3-adapted coframes defined previously, ˆ π =( ˆ σ , ˆ η ) and ˇ π = ( ˇ σ , ˇ η ), with structure equations (2.13). In passing from the3-adapted coframes to the 4-adapted coframes only the forms θ iX and θ iY aremodified. The (abstract) Lie algebra defined by the structure constants C ijk is called the Vessiot algebra for the Darboux pair { ˆ V , ˇ V } and is denotedby vess ( ˆ V , ˇ V ). The vectors fields X i in the dual basis { X i , ˆ U a , ˇ U α } to the 4-adapted coframe { θ iX , ˆ π a , ˇ π α } define a realization for vess ( ˆ V , ˇ V ), with structureequations [ X j , X k ] = − C ijk X i . Proof of Theorem 4.6 .
Theorem 4.4 provides us with two locally defined 3-adapted frames { X i , ˆ U a , ˇ U α } and { Y i , ˆ U a , ˇ U α } and dual coframes { θ iX , ˆ π a , ˇ π α } and { θ iY , ˆ π a , ˇ π α } . The structure equations are (4.22)–(4.24). We begin theproof of part [i] by showing that the C ijk are functions only of the first integrals ˆ I a while the K ijk are functions only of the first integrals ˇ I α .Since span { θ X } = span { θ Y } , there is an invertible matrix Q such that θ iX = Q ij θ jY . (4.27) .3 The fourth adapted coframe and the Vessiot algebra X i and Y j commute, the identity d θ iX ( X j , Y k ) = X j ( θ iX ( Y k )) − Y k ( θ iX ( X j )) − θ iX ([ X j , Y k ])leads to X j ( Q ik ) = C ijℓ Q ℓk . (4.28)We shall use this result repeatedly in what follows – it is equivalent to the factthat the vector fields X i and Y j commute, a property of the two 3-adaptedcoframes which is not encoded in the structure equations (4.22) and (4.23).We next substitute (4.27) into (4.22) and equate the coefficients of θ jY ∧ θ kY to deduce that X ℓ ( Q ik ) Q ℓj − X l ( Q ij ) Q lk + Q il K ljk = C iℓm Q ℓj Q mk . By virtue of (4.28), this equation simplifies to Q il K ljk = − C ilm Q lj Q mk . (4.29)Also, by equating to zero the coefficients of ˇ π α ∧ θ iX ∧ θ jX and ˆ π a ∧ θ iY ∧ θ jY inthe expansions of the identities d θ iX = 0 and d θ iY = 0, we find that ˇ U a ( C ijk ) = 0 and ˆ U α ( K ijk ) = 0 . By Lemma 2.8, these equations imply that C ijk ∈ Int( ˆ V ) and K ijk ∈ Int( ˇ V ) . (4.30)Our next goal is to show that the coframes θ iX and θ iY may be adjusted so that K ijk = − C ijk while still preserving (4.22)–(4.24). To this end we introduce localcoordinates ( ˆ I a , ˇ I α , z m ) on M satisfying (2.10) and (2.11). Then, on account of(4.30), equation (4.29) becomes Q ℓk ( ˆ I a , ˇ I α , z m ) K kij ( ˇ I α ) = − C ℓhk ( ˆ I a ) Q hi ( ˆ I a , ˇ I α , z m ) Q kj ( ˆ I a , ˇ I α , z m ) . (4.31)Evaluate this equation, first at a fixed point ( ˆ I a , ˇ I α , z m ) and then at the point( ˆ I a , ˇ I α , z m ). With o K kij = K kij ( ˇ I α ) , o C kij = C kij ( ˆ I a ) , o Q ji = Q ji ( ˆ I a , ˇ I α , z m ) , and Q ji ( ˆ I a ) = Q ji ( ˆ I a , ˇ I α , z m ) .3 The fourth adapted coframe and the Vessiot algebra o Q ℓk o K kij = − o C ℓhk o Q hi o Q kj and Q ℓk ( ˆ I a ) o K kij = − C ℓhk ( ˆ I a ) Q hi ( ˆ I a ) Q kj ( ˆ I a ) . (4.32)It then readily follows that the matrix P ij ( ˆ I a ) = Q iℓ ( ˆ I a )( o Q − ) ℓj . (4.33)satisfies P ℓk ( ˆ I a ) o C kij = C ℓhk ( ˆ I a ) P hi ( ˆ I a ) P kj ( ˆ I a ) . (4.34)The 1-forms o θ iX defined by o θ jX P ij = θ iX (4.35)then satisfy structure equations of the required form (4.25), where the structurefunctions C ijk coincide with the constants o C ijk . Finally, by evaluating (4.31) at( ˆ I a , ˇ I α , z m ), it follows that R ij ( ˇ I α ) = ( Q − ( ˆ I a , ˇ I α , z m )) ij (4.36)satisfies − R ℓk ( ˇ I α ) o C kij = K ℓhk ( ˇ I α ) R hi ( ˇ I α ) R kj ( ˇ I α ) (4.37)and the 1-forms o θ iY defined by o θ jY R ij = θ iY (4.38)satisfy the required structure equations (4.26), again with C ijk = o C ijk .The next step in the proof of [i] requires us to check that the coframes wehave just constructed are still 3-adapted. We must therefore show that the dualvector fields o X i and o Y j commute and for this it suffices to show, because of ourremarks at the beginning of this proof, that o X j ( o Q ik ) = o C ijl o Q lk where o θ iX = o Q ij o θ jY , (4.39)The proof of this formula follows from three simple observations. We firstnote that (4.35) and (4.38) imply o Q ik = ( P − ) iℓ Q ℓm R mk . Secondly, because { o X i , ˆ π a , ˇ π α } is the dual frame to { o θ iX , ˆ π a , ˇ π α } , we have o X i = P ji X j . Finally,because P ij = P ij ( ˆ I a ) and R ij = R ij ( ˇ I α ), it follows that o X j ( P ik ) = o X j ( R ik ) = 0. Astraightforward calculation based upon these three observations and equations(4.28) and (4.34) leads to (4.39), as required. .3 The fourth adapted coframe and the Vessiot algebra θ iX ∧ θ jX ∧ θ kX in the expansionof the equations d θ iX = 0 one finds that the C ijk satisfy the Jacobi identity andare therefore the structure constants of a real r -dimensional Lie algebra.To prove [ii] , let { θ ′ iX , ˆ π ′ a , ˇ π ′ α } be another coframe adapted to the Darbouxpair { ˆ V , ˇ V } with structure equations d θ ′ iX = 12 A ′ iab ˆ π ′ a ∧ ˆ π ′ b + 12 B ′ iαβ ˇ π ′ α ∧ ˇ π ′ β + 12 C ′ ijk θ ′ j ∧ θ ′ k + M ′ iaj ˆ π ′ a ∧ θ ′ j . (4.40)From the definition of a 0-adapted coframe (see (2.6b)) we have θ ′ iX = R ij θ j + ˆ S ip ˆ η p + ˇ S iq ˇ η q , where the matrix R ij is invertible. Substitute this equation into (4.40) and thensubstitute from (2.13) and (4.25). From the coefficients of ˆ σ ∧ θ we deduce that ∂ ˆ σ R ij = 0. By Lemma 2.8 this implies that R ij ∈ Inv( ˆ V ) in which case one findsfrom the coefficients of θ j ∧ θ k that C ′ ilm R lj R mk = R il C ljk . Hence the structure constants C ijk and C ′ ijk define the same abstract Lie algebraand the proof of part [ii] is complete. Corollary 4.7.
Let { ˆ V , ˇ V } and { ˆ W , ˇ W } be Darboux pairs on manifolds M and N respectively and suppose that φ : M → N is smooth map satisfying φ ∗ ( ˆ W ) ⊂ ˆ V and φ ∗ ( ˇ W ) ⊂ ˇ V . (4.41)
Then φ induces a Lie algebra homomorphism ˜ φ : vess ( ˆ V , ˇ V ) → vess ( ˆ W , ˇ W ) . (4.42) Proof.
The proof of this corollary is similar to that of part [ii] of Theorem 4.6.Let { θ ′ sX , ˆ π ′ c , ˇ π ′ γ } be a 4-adapted coframe on N for the Darboux pair { ˆ W , ˇ W } ,with structure equations d θ ′ sX = 12 A ′ scd ˆ π ′ c ∧ ˆ π ′ d + 12 B ′ sγδ ˇ π ′ γ ∧ ˇ π ′ δ + 12 K srt θ ′ r ∧ θ ′ t + M ′ sat ˆ π ′ a ∧ θ ′ t . (4.43)The constants K srt are the structure constants for the Lie algebra vess ( ˆ W , ˇ W ).Denote the dual frame by { X ′ r , U ′ c , U ′ γ } .Let { θ iX , ˆ π a , ˇ π α } be a 4-adapted coframe on M for the Darboux pair { ˆ V , ˇ V } with the structure equations (4.25). The inclusions (4.41) imply that φ ∗ ( ˆ W ∩ ˇ W ) ⊂ ˆ V ∩ ˇ V , φ ∗ ( ˆ W ( ∞ ) ) ⊂ ˆ V ( ∞ ) and φ ∗ ( ˇ W ( ∞ ) ) ⊂ ˇ V ( ∞ ) .3 The fourth adapted coframe and the Vessiot algebra R si , ˆ S sp , ˇ S sq ˆ T ca and ˇ T γα on M such that φ ∗ ( θ ′ sX ) = R si θ iX + ˆ S sp ˆ η p + ˇ S sq ˇ η q , φ ∗ ( ˆ π ′ c ) = ˆ T ca ˆ π a and φ ∗ ( ˇ π ′ γ ) = ˇ T γα ˇ π α . We pullback (4.43) using these equations and substitute from (4.25). The samearguments as in the proof of [ii] of Theorem 4.6 now show that R si ∈ Inv( ˆ V )and R ri C ijk = K rst R sj R tk . This proves that the Jacobian mapping φ ∗ ( X i ) = R ti X ′ t induces a Lie algebra homomorphism of Vessiot algebras. Remark 4.8.
Let { θ iX , ˆ π a , ˇ π α } and { θ iY , ˆ π a , ˇ π α } be two coframes which are4-adapted to the Darboux pair { ˆ V , ˇ V } and which satisfy the structure equations(4.25) and (4.26). Then it is not difficult to check that the commutativity ofthe dual vector fields X i and Y j is equivalent to the supposition that the changeof frame (4.27) defines an automorphism of the Vessiot algebra, that is, Q iℓ C ℓjk = C ilm Q ℓj Q mk . (4.44)In the next section we shall need all the derivatives of the matrix Q ij . The ˆ π a and ˇ π α components of d Q ij are easily determined by substituting (4.27) into(4.25) and comparing the result with (4.26). If we then take into account (4.28)and (4.44) we find that d Q ij = Q ℓj M iℓ − Q iℓ N ℓj + C ijℓ Q ℓk θ jX , (4.45)where M ij = M iaj ˆ π a and N ij = N iαj ˇ π α . We note, also for future use, that A iab = Q ij E jab and B iαβ = Q ij F jαβ . (4.46)where E jab and F jαβ are defined by (2.13). .4 The proof of Theorem 4.1 Let { ˆ V , ˇ V } be a Darboux pair on M and let { θ iX , ˆ π a , ˇ π α } and { θ iY , ˆ π a , ˇ π α } be local coframes on M which are 4-adapted to the Darboux pair { ˆ V , ˇ V } andwhich therefore satisfy the structure equations (4.25) and (4.26). In this sectionwe shall prove that it is possible to define forms ˆ θ i = ˆ R ij θ jX + φ ia ˆ π a and ˇ θ i = ˇ R ij θ jY + ψ ia ˇ π a , (4.47)where ˆ R ij , φ ia ∈ Int( ˆ V ) and ˇ R ij , ψ iα ∈ Int( ˇ V ) , which satisfy structure equations d ˆ θ i = 12 G iαβ ˇ π α ∧ ˇ π β + 12 C ijk ˆ θ j ∧ ˆ θ k (4.48)and d ˇ θ i = 12 H iab ˆ π a ∧ ˆ π b − C ijk ˇ θ j ∧ ˇ θ k . (4.49)These are the structure equations for the Darboux pair { ˆ V , ˇ V } announced inTheorem 4.1.We shall focus on (4.48) and simply note that the proof of (4.49) is similar.Our starting point are the equations (4.25) and (4.26), in particular, d θ iX = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β + 12 C ijk θ jX ∧ θ kX + M iaj ˆ π a ∧ θ jX . (4.50)In what follows it will be useful to introduce the 1-forms and 2-forms M ℓj = M ℓaj ˆ π a , N ℓj = N ℓαj ˇ π α , A ℓ = A ℓab ˆ π a ∧ ˆ π b F ℓ = F ℓαβ ˇ π α ∧ ˇ π β . (4.51)By setting to zero the coefficients of ˆ π a ∧ ˇ π α ∧ θ iX and ˇ π α ∧ ˆ π a ∧ ˆ π b inthe equations d θ iX = 0, we find that ˇ U α ( M iaj ) = 0 and ˇ U α ( A iab ) = 0. Theseequations imply that M iaj ∈ Int( ˆ V ) and A iab ∈ Int( ˆ V ) and therefore (see Lemma2.8) d M iaj = ˆ U b ( M iaj ) ˆ π b and d A iab = ˆ U c ( A iab ) ˆ π c . (4.52)Bearing these two results in mind, we then respectively deduce from the coeffi-cients of ˆ π a ∧ θ jX ∧ θ kX , ˆ π a ∧ ˆ π b ∧ θ jX and ˆ π c ∧ ˆ π a ∧ ˆ π b in d θ iX = 0 that M ℓaj C iℓk + M ℓak C ijℓ − M iaℓ C ℓjk = 0 , (4.53) d M ij − M iℓ ∧ M ℓj + C iℓj A ℓ = 0 , and (4.54) d A i − A ℓ M iℓ = 0 . (4.55) .4 The proof of Theorem 4.1 a , the linear transformation M a : vess → vess defined by M a ( X j ) = M iaj X i is a derivation or infinitesimal automorphism of the Vessiot Lie algebra vess = vess ( ˆ V , ˇ V ). Consequently, to analyze this equation we shall invoke some basicstructure theory for Lie algebras. Specifically, we shall consider separately thecases where vess is semi-simple, where vess is abelian, and where vess is solvable.Then we shall make use of the Levi decomposition of vess to solve the generalcase. Case I.
We first consider the case where the Lie algebra vess is semi-simple.Here the proof of (4.48) is relatively straightforward and is based upon the factthat every derivation of a semi-simple Lie algebra is an inner derivation (see, forexample, Varadarajan[25], page 215). In fact, because the proof of this result isconstructive, we can deduce from (4.53) that there are uniquely defined smoothfunctions φ ℓa ∈ Int( ˆ V ) such that M iaj = φ ℓa C iℓj . (4.56)The forms ˆ θ i defined by ˆ θ i = θ iX + φ ia ˆ π a (4.57)then satisfy structure equations of the form d ˆ θ i = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β + 12 C ijk ˆ θ j ∧ ˆ θ k . (4.58)For these structure equations, equations (4.54) reduce to C iℓj A ℓab = 0. Since weare assuming that vess is semi-simple, the center of vess is trivial and thereforethis equation implies that A ℓab = 0. The structure equations (4.58) then reduceto the form (4.50), as desired. Case II.
Now we consider the other extreme case, namely, the case where vess is abelian. Strictly speaking, we need not treat this as a separate case butthe analysis here will simplify our subsequent discussion of the case where vess is solvable. When vess is abelian the structure equations (4.50) are d θ iX = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β + M iaj ˆ π a ∧ θ jX , (4.59) .4 The proof of Theorem 4.1 d M ij − M iℓ ∧ M ℓj = 0 , where M ij = M iaj ˆ π a . (4.60)By the Frobenius theorem we conclude that there are locally defined functions R ij ∈ Int( ˆ V ) such that d ( R ij ) + R iℓ M ℓj = 0 and det( R ij ) = 0 . (4.61)(For this precise application of the Frobenius theorem see, for example, Flanders[11], page 102.) We remark that this is the first (and only) step in all our coframeadaptations that require the solution to systems of linear ordinary differentialequationsThe forms ˆ θ i = R ij θ jX are then easily seen to satisfy d ˆ θ i = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β . (4.62)Since the structure functions A iab = R ij A jab ∈ Int( ˆ V ) it follows, either directlyfrom (4.62) or from (4.55) (with M iaℓ = 0) that the 2-forms χ i = 12 A iab ˆ π a ∧ ˆ π b (4.63)are all d -closed. If we pick 1-forms φ i = φ ia ˆ π a , with φ ia ∈ Int( ˆ V ), such that d φ i = − χ i , then the forms ˆ θ i = ˆ θ i + φ ia ˆ π a = R ij θ jX + φ ia ˆ π a (4.64)will satisfy the required structure equations (4.48), with C ijk = 0. By applyingthe usual homotopy formula for the de Rham complex, we see that this lastcoframe adaptation can be implemented by quadratures. Case III.
Now we suppose that vess is solvable. Recall that a Lie algebra g is said to be p -step solvable if the derived algebras g ( i ) = [ g ( i − , g ( i − ] satisfy g = g (0) ⊃ g (1) ⊃ g (2) · · · ⊃ g ( p − ⊃ g ( p ) = { } . The annihilating subspaces Λ ( i ) = ann( g ( i ) ) therefore satisfy { } = Λ (0) ⊂ Λ (1) ⊂ Λ (2) · · · ⊂ Λ ( p − ⊂ Λ ( p ) = g ∗ (4.65)and, because the g ( i ) are all ideals, d Λ ( i ) ⊂ Λ ( i − ⊗ Λ ( i ) . (4.66) .4 The proof of Theorem 4.1 s = dim( g ) and s i = dim Λ ( i ) . If M : g → g is a derivation, then a simpleinduction shows that M : g ( i ) → g ( i ) and therefore M ∗ : Λ ( i ) → Λ ( i ) . (4.67)Thus the matrix representing M ∗ , with respect to any basis for g ∗ adapted tothe flag (4.65), is block triangular.We apply these observations to the forms { θ iX } and the structure equations(4.50). The forms { θ iX } , restricted to the vectors X i , define a basis for vess ∗ and consequently, by a constant coefficient change of basis, we can supposethat the basis { θ iX } is adapted to the derived flag (4.65). Since the notationfor this adaptation becomes rather cumbersome in the general case of a p -stepLie algebra, we shall consider just the cases where vess is a 2-step or a 3-stepsolvable Lie algebra. The construction of the coframe with structure equations(4.48) in these two special cases will make the nature of the general constructionapparent.In the case where vess is a 2-step solvable Lie algebra, we begin with a4-adapted coframe { θ , . . . , θ s , θ s +12 , . . . , θ s , ˆ π a , ˇ π α } , where span { θ X , . . . , θ sX } = span { θ , . . . , θ s , θ s +12 , . . . , θ s } over R , and where { θ , . . . , θ s } is a basis for Λ (1) ( vess ). In this basis the structureequations (4.50) become d θ r = 12 A rab ˆ π a ∧ ˆ π b + 12 B rαβ ˇ π α ∧ ˇ π β + M ras ˆ π a ∧ θ s , and (4.68a) d θ i = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β + 12 C irs θ r ∧ θ s + C irj θ r ∧ θ j (4.68b)+ M iar ˆ π a ∧ θ r + M iaj ˆ π a ∧ θ j . In these equations the indices r, s range from 1 to s and the indices i, j rangefrom s + 1 to s . On account of (4.66), there are no quadratic θ terms in(4.68a) because θ r ∈ Λ (1) ( vess ) and there are no θ i ∧ θ j terms in (4.68b) because θ ∈ Λ (2) ( vess ). There are no ˆ π a ∧ θ i terms in (4.68a) by virtue of (4.67).Since the structure equations (4.68a) are identical in form to the structureequations (4.59) for the abelian case, we can invoke the arguments there todefine new forms ˆ θ r = R rs θ s + φ ra ˆ π a (4.69) .4 The proof of Theorem 4.1 d ˆ θ r = 12 B rαβ ˇ π α ∧ ˇ π β , and (4.70a) d θ i = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β + 12 C irs ˆ θ r ∧ ˆ θ s + C irj ˆ θ r ∧ θ j (4.70b)+ M iar ˆ π a ∧ ˆ θ r + M iaj ˆ π a ∧ θ j . It is important to track the coordinate dependencies of the coefficients in thesestructure equations as we perform these frame changes. Since the coefficients A iab , M iar and M iaj in (4.68b) and the coefficients R rs and φ ra in (4.69) are all ˆ V first integrals, it is easily checked that the same is true of the coefficients A iab , C irs , C irj , M iar and M iaj in (4.70b).The coefficients of ˆ π a ∧ ˆ π b ∧ θ i and ˆ π a ∧ ˆ π b ∧ ˆ π c in d θ i = 0 give the sameequations as we had in the abelian case and consequently we can define ˆ θ i = R ij θ j + φ ia ˆ π a (4.71)so as to eliminate the ˆ π a ∧ ˆ π b and ˆ π a ∧ θ j terms in (4.70b). The structureequations are now d ˆ θ r = 12 B rαβ ˇ π α ∧ ˇ π β , and (4.72a) d ˆ θ i = 12 B iαβ ˇ π α ∧ ˇ π β + 12 C irs ˆ θ r ∧ ˆ θ s + C irj ˆ θ r ∧ ˆ θ j + M iar ˆ π a ∧ ˆ θ r . (4.72b)Finally, from the coefficients of ˆ π a ∧ ˆ π b ∧ ˆ θ r in d ˆ θ i = 0 we find that d M ib = 0 , where M ib = M ibr ˆ π r , (4.73)and therefore, again by the de Rham homotopy formula, there are functions R ir ∈ Int( ˆ V ) such that M iar = ˆ U a ( R ir ). The change of frame ˆ θ i = ˆˆ θ i + R ir ˆ θ r (4.74)then leads to the desired structure equations d ˆ θ r = 12 B rαβ ˇ π α ∧ ˇ π β , and (4.75a) d ˆˆ θ i = 12 B iαβ ˇ π α ∧ ˇ π β + 12 C irs ˆ θ r ∧ ˆ θ s + C irj ˆ θ r ∧ ˆˆ θ j . (4.75b)At this point it is a simple matter to check that the equations d ˆˆ θ i = 0 force thecoefficients C irs and C irj (which belong to Int( ˆ V )) to be constant. Equations(4.75) establish (4.48) for the case of a 2-step solvable Vessiot algebra. .4 The proof of Theorem 4.1 vess is a 3-step solvable Lie algebra we assume that the 4-adaptedcoframe { θ r , θ u , θ i , ˆ π a , ˇ π α } is adapted to the flag (4.65) in the sense thatΛ (1) = span { θ r } , Λ (2) = span { θ r , θ u } and Λ (3) = span { θ r , θ u , θ i } . The structure equations (4.76b) are then of the form d θ r = 12 A rab ˆ π a ∧ ˆ π b + 12 B rαβ ˇ π α ∧ ˇ π β + M ras ˆ π a ∧ θ s , (4.76a) d θ u = 12 A uab ˆ π a ∧ ˆ π b + 12 B uαβ ˇ π α ∧ ˇ π β + 12 C urs θ r ∧ θ s + C urv θ r ∧ θ v (4.76b)+ M uar ˆ π a ∧ θ r + M uav ˆ π a ∧ θ v , and d θ i = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β (4.76c)+ 12 C irs θ r ∧ θ s + C iru θ r ∧ θ u + C irj θ r ∧ θ j + 12 C iuv θ u ∧ θ v + C iuj θ u ∧ θ j + M iar ˆ π a ∧ θ r + M iau ˆ π a ∧ θ u + M iaj ˆ π a ∧ θ j . In these equations r, s range from 1 to s , u, v from s + 1 to s , and i, j from s + 1 to s . Note that the form of the structure equations (4.76a)–(4.76c) ispreserved by changes of coframe of the type θ → R θ + φ ˆ π , θ → R θ + R θ + φ ˆ π , and θ → R θ + R θ + R θ + φ ˆ π , where the coefficients R ij ∈ Int( ˆ V ). Exactly as in the previous case of a 2-step solvable algebra, we use such a change of coframe for θ , θ to reduce thestructure equations (4.76) to d ˆ θ r = 12 B rαβ ˇ π α ∧ ˇ π β , (4.77a) d ˆ θ u = 12 B uαβ ˇ π α ∧ ˇ π β + 12 C urs ˆ θ r ∧ ˆ θ s + C urv ˆ θ r ∧ ˆ θ v , and (4.77b) d θ i = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β (4.77c)+ 12 C irs ˆ θ r ∧ ˆ θ s + C iru ˆ θ r ∧ ˆ θ u + C irj ˆ θ r ∧ θ j + 12 C iuv ˆ θ u ∧ ˆ θ v + C iuj ˆ θ u ∧ θ j + M iar ˆ π a ∧ ˆ θ r + M iau ˆ π a ∧ ˆ θ u + M iaj ˆ π a ∧ θ j . Again, as in Case II , a change of frame ˆ θ = R θ + φ ˆ π leads to the simpli- .4 The proof of Theorem 4.1 d ˆ θ i = 12 B iαβ ˇ π α ∧ ˇ π β + M iar ˆ π a ∧ ˆ θ r + M iau ˆ π a ∧ ˆ θ u (4.78)+ 12 C irs ˆ θ r ∧ ˆ θ s + C iru ˆ θ r ∧ ˆ θ u + C irj ˆ θ r ∧ ˆ θ j + 12 C iuv ˆ θ u ∧ ˆ θ v + C iuj ˆ θ u ∧ ˆ θ j . Finally, just as in the reduction from (4.72) to (4.75), we use a change of frame ˆ θ → ˆ θ + R ˆ θ + R ˆ θ to transform (4.78) to d ˆ θ i = 12 B iαβ ˇ π α ∧ ˇ π β (4.79)+ 12 C irs ˆ θ r ∧ ˆ θ s + C iru ˆ θ r ∧ ˆ θ u + C irj ˆ θ r ∧ ˆ θ j + 12 C iuv ˆ θ u ∧ ˆ θ v + C iuj ˆ θ u ∧ ˆ θ j . Equations (4.77a), (4.77b) and (4.79) give the desired result. We note oncemore that the structure functions C ∗∗∗ in (4.77c) and(4.78) are not necessarilyconstant but they are constant in (4.79).The reduction of the structure equations for a general p -step solvable algebrafollows this construction and can be formally defined by induction on p . Case IV:
Finally, we consider the case where vess is a generic Lie algebra.We use the fact that every real Lie algebra g admits a Levi decomposition intoa semi-direct sum g = s ⊕ r , where r is the radical of g and s is a semi-simplesubalgebra of g . The radical r is the unique maximal solvable ideal in g – thesemi-simple component s in the Levi decomposition is not unique. The dualspace g ∗ then decomposes according to g ∗ = ann( r ) ⊕ ann( s ) . (4.80)The fact that r is an ideal implies that d ann( r ) ⊂ Λ (ann( r )) and d ann( s ) ⊂ g ∗ ⊗ ann( s ) . (4.81)If M : g → g is a derivation, then M preserves r and hence M ∗ : ann( r ) → ann( r ).Now choose, by a constant coefficient change of basis, 1-forms { θ r , θ i } adapted to the decomposition (4.80) with ann( r ) = span { θ r } and ann( s ) =span { θ i } . In view of (4.81), the structure equations (4.50) become d θ r = 12 A rab ˆ π a ∧ ˆ π b + 12 B rαβ ˇ π α ∧ ˇ π β + 12 C rst θ s ∧ θ t + M ras ˆ π a ∧ θ s , (4.82a) d θ i = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β + 12 C ijk θ j ∧ θ k + C irj θ r ∧ θ j (4.82b)+ M iar ˆ π a ∧ θ r + M iaj ˆ π a ∧ θ j . .4 The proof of Theorem 4.1 r , s , t range from 1 to s = dim(ann( r )) and i , j , k range from s + 1 to s = dim( g ).Since the structure constants C rst are those for the semi-simple Lie algebra s ,we can return to the analysis presented in Case I to prove that there is a changeof coframe ˆ θ r = θ r + φ ra ˆ π a (see (4.57)) which reduces the structure equations(4.82) to d ˆ θ r = 12 B rαβ ˇ π α ∧ ˇ π β + 12 C rst ˆ θ s ∧ ˆ θ t , and (4.83a) d θ i = 12 A iab ˆ π a ∧ ˆ π b + 12 B iαβ ˇ π α ∧ ˇ π β + 12 C ijk θ j ∧ θ k + C irj ˆ θ r ∧ θ j (4.83b)+ M iar ˆ π a ∧ ˆ θ r + M iaj ˆ π a ∧ θ j . The structure functions C rst , C ijk and C irj in (4.83) are identical to the corre-sponding structure constants in (4.82). One now checks that the ˆ θ r terms in(4.83b) do not effect the arguments made in Cases II and III . Thus, by a changeof frame ˆ θ = Rθ + φ ˆ π , one can reduce (4.83) to d ˆ θ r = 12 B rαβ ˇ π α ∧ ˇ π β + 12 C rst ˆ θ s ∧ ˆ θ t , and (4.84a) d ˆ θ i = 12 B iαβ ˇ π α ∧ ˇ π β + 12 C ijk ˆ θ j ∧ ˆ θ k + C irj ˆ θ r ∧ θ j + M iar ˆ π a ∧ ˆ θ r . (4.84b)Finally, by a now familiar computation, one sees that a change of frame ˆ θ → ˆ θ + R ˆ θ allows one to eliminate the ˆ π a ∧ ˆ θ r terms in (4.84b).This completes our derivation of the structure equations (4.48) and the proofof Theorem 4.1 is, at last, finished. Remark 4.9.
We collect together a few additional formulas which will beneeded in the next section for the proof of Lemma 5.8 and the constructionof the superposition formula. If the forms ˆ θ i and ˇ θ i in (4.47) satisfy (4.48) and(4.49), then it easy to check that coefficients ˆ R ij , ˇ R ij , φ i = φ ia ˆ π a and ψ i = ψ iα ˇ π α satisfy (see (4.51)) d ˆ R ij = − C iℓk ˆ R ℓj φ k − ˆ R iℓ M ℓj , d ˇ R ij = C iℓk ˇ R ℓj ψ k − ˇ R iℓ N ℓj , (4.85a) ˆ R iℓ B ℓαβ = G iαβ , ˇ R iℓ E ℓab = H iab , (4.85b) d φ i = 12 C ijk φ j ∧ φ k − ˆ R iℓ A ℓ , d ψ i = − C ijk ψ j ∧ ψ k − ˇ R iℓ F ℓ . (4.85c) .4 The proof of Theorem 4.1 ˆ R ij and ˇ R ij define automorphisms of the Vessiot algebra, that is ˆ R iℓ C ℓjk = C iℓm ˆ R ℓj ˆ R mk , and ˇ R iℓ C ℓjk = C iℓm ˇ R ℓj ˇ R mk . (4.86)Finally, a series of straightforward computations, based on (4.44), (4.45), (4.85a)and (4.86), shows that the matrix λ = ˆ RQ ˇ R − satisfies (4.87) λ iℓ C ℓjk = C iℓm λ ℓj λ mk and d λ ij = C iℓm λ mj ˆ θ ℓ + λ ih C hmj ψ m . (4.88)We conclude this section by determining the residual freedom in the de-termination of the 1-forms ˆ θ i . Specifically, we compute the group of coframetransformations which fix the forms ˆ π a and ˇ π α and which transform the ˆ θ i by˜ θ i = Λ ij ˆ θ i + σ i , where σ i = S ia ˇ π a (4.89)in such a manner as to preserve the form of the structure equations (4.48).If we take the exterior derivative of (4.89) and substitute into the structureequations for ˜ θ i we find first, from the ˇ π α ∧ θ i and the ˆ π a ∧ ˇ π α terms, thatΛ ij , S ia ∈ Int( ˆ V ) and then that d Λ ij = C iℓk Λ ℓj σ k , Λ iℓ C ℓjk = C iℓm Λ ℓj Λ mk , d σ i = 12 C ijk σ j ∧ σ k . (4.90)To integrate these equations, let G be a local Lie group whose Lie algebra isthe Vessiot algebra vess with structure constants C ijk . On G construct a coframe η i of invariant 1-forms with structure equations d η i = C ijk η j ∧ η k . Then thereexists ([17], Proposition 1.3) a map σ : Int( ˆ V ) → G such that σ i = σ ∗ ( η i ).Define S ( ˆ I ) = Ad( σ ( ˆ I ). Then d S ij = C iℓk σ ℓ S kj and d (Λ iℓ ( S ( − ) ℓj ) = 0 . and hence the general solution to (4.90) is σ i = σ ∗ ( η i ) , Λ ij = Λ iℓ S ℓj , S = Ad( σ ( ˆ I )) , (4.91)where the matrix Λ iℓ is a constant automorphism of the Vessiot algebra. When σ is a constant map, S is constant inner automorphism so that as far as the generalsolution (4.91) is concerned, one can restrict the Λ iℓ to a set of representativesof the group of outer automorphisms of the Vessiot algebra. .4 The proof of Theorem 4.1 Remark 4.10.
The bases for the infinitesimal Vessiot transformation groups ˆ Γ = { ˆ X i } and ˇ Γ = { ˇ X i } are related by ˇ X j = λ ij ˆ X i , where λ is the matrix(4.87). Let ˆ X r , r = 1 . . . m , be a basis for the center of ˆ Γ and pick a comple-mentary set of vectors ˆ X u , u = m + 1 . . . s , which complete the ˆ X r to a basisfor ˆ Γ. Choose a similar basis for ˇ Γ. Then, because λ defines a Lie algebraautomorphism, we have that ˇ X r = λ tr ˆ X t and ˇ X u = λ vu ˆ X v + λ tu ˆ X t , where t = 1 . . . m and v = m +1 . . . s . The second equation in (4.88) now impliesthat d λ rs = 0. Consequently, we may always pick bases for the infinitesimalVessiot transformation groups ˆ Γ and ˇ Γ so that the infinitesimals generators forthe center coincide. In turn, this implies that any vector in the center of eitherinfinitesimal Vessiot transformation group is a infinitesimal symmetry of both ˆ V and ˇ V . he Superposition Formula for Darboux Pairs The proof of the superposition formula for a Darboux pair will dependupon the following technical result concerning group actions and Maurer-Cartanforms.
Theorem 5.1.
Let M be a manifold and G a connected Lie group. Let ω iL and ω iR be the left and right invariant Maurer-Cartan forms on G , with structureequations d ω iL = 12 C ijk ω jL ∧ ω kL and d ω iR = − C ijk ω jR ∧ ω kR . (5.1) Let X Li and X Ri be the dual basis of left and right invariant vector fields on G . [i] Suppose that there are right and left commuting, regular, free group actionsof G on M , ˆ µ : G × M → M and ˇ µ : G × M → M, (5.2) with common orbits. Denote the infinitesimal generators of these actions by ( ˆ µ x ) ∗ ( X Li ) = ˆ X i and ( ˇ µ x ) ∗ ( X Ri ) = ˇ X i . [ii] Suppose there are 1-forms ˆ ω i and ˇ ω i on M such that [a] ˆ ω i ( ˆ X j ) = δ ij and ˇ ω i ( ˇ X j ) = δ ij ; [b] span { ˆ ω i } = span { ˇ ω i } with ˆ ω i ( x ) = ˇ ω i ( x ) at some fixedpoint of x ∈ M and; [c] d ˆ ω i = 12 C ijk ˆ ω j ∧ ˆ ω k and d ˇ ω i = − C ijk ˇ ω j ∧ ˇ ω k . (5.3) Then about each point x of M there is an open G bi-invariant neighborhood U of M and a mapping ρ : U → G such that ρ ∗ ( ω iL ) = ˆ ω i , ρ ∗ ( ω iR ) = ˇ ω i , ρ ( ˆ µ ( x, g )) = ρ ( x ) · g, ρ ( ˇ µ ( g, x )) = g · ρ ( x ) . (5.4) If M/G is simply connected, then one can take U = M . Remark 5.2.
This theorem can be viewed as a simple extension (or refinement)of three different, well-known results. First, suppose that [i] holds. Then,because the actions ˆ µ and ˇ µ are free and regular, we may construct ˆ µ and ˇ µ invariant sets ˆ U and ˇ U and maps ˆ ρ : ˆ U → G and ˇ ρ : ˇ U → G (5.5) .1 A preliminary result G equivariant (see, for example, [12]) with respect to the actions ˆ µ and ˇ µ , respectively. Theorem 5.1 states that these maps can be chosen such thatthey are equal on a common bi-invariant domain, ˆ ρ ∗ ( ω iL ) = ˆ ω i and ˇ ρ ∗ ( ω iR ) = ˇ ω i .Secondly, assume [ii] . Then a fundamental theorem in Lie theory (see Griffiths[17] or Sternburg [23], page 220) asserts that there always exists maps ˆ ϕ : M → G and ˇ ϕ : M → G such that ˆ ϕ ∗ ( ω iL ) = ˆ ω i and ˇ ϕ ∗ ( ω iR ) = ˇ ω i . (5.6)From this viewpoint, Theorem 5.1 states that these maps can be taken to beequal and equivariant with respect to the actions ˆ µ and ˇ µ . Thirdly, supposethat just one of the actions in [i] is given, that G acts regularly on M and thecorresponding forms (say ˆ ω i ) in [ii] are given. Then π : M → M/G may beviewed as a principal G bundle with (by [ii] ) flat connection 1-forms ˆ ω i . ThenKobayashi and Nomizu [18](Theorem 9.1, Volume I, page 92) state that thereis an open set U in M/G and a diffeomorphism ψ : π − ( U ) → U × G suchthat ψ ∗ ( ω iL ) = ˆ ω i (where the Maurer-Cartan forms ω iL define the canonical flatconnection on U × G ). Moreover when M/G is simply connected, then M isglobally the trivial G bundle with canonical flat connection. Compare this resultwith Corollary 5.6 to Theorem 5.1.We establish Theorem 5.1 with the following sequence of lemmas. Lemma 5.3.
Under the hypothesis of Theorem 5.1 there is a bi-invariant, con-nected neigborhood U around x and a mapping ρ : U → G such that ρ ∗ ( ω iL ) = ˆ ω i and ρ ∗ ( ω iR ) = ˇ ω i . The map ρ is uniquely defined for a given domain U .Proof. As is customary, we suppose that ω iL ( e ) = ω iR ( e ), where e is the identityof G . In accordance with the aforementioned Theorem 9.1 in [18] there is a ˆ µ invariant open set U in M and a unique smooth map ρ : U → G such that ρ ∗ ( ω iL ) = ˆ ω i and ρ ( x ) = e. (5.7)Since the ˆ µ and ˇ µ orbits coincide, the set U is actually bi-invariant. Because G is connected, we may assume that U is connected. The uniqueness of ρ isestablished in Theorem 2.3 (page 220) of [23].In order to prove that ρ satisfies ρ ∗ ( ω iR ) = ˇ ω i , let Λ ij ( g ) be the matrix forthe linear transformationAd ∗ ( g ) = R ∗ g ◦ L ∗ g − : T ∗ e G → T ∗ e G .1 A preliminary result ˆ ω i ( e ). Then an easy computation shows that ω iL ( g ) = Λ ij ( g ) ω jR ( g ) and Λ ik ( g g ) = Λ ij ( g )Λ jk ( g ) . (5.8)The Jacobian of the multiplication map m : G × G → G may be computed interms of Λ ij as m ∗ ( ω iL )( g g ) = Λ ij ( g ) ω jL ( g ) + ω iL ( g ) . (5.9)We take the Lie derivative of the first equation in (5.8) with respect to X Lk tofind that L X Lk Λ ij = C ikℓ Λ ℓj and hence d Λ ij = C ikℓ Λ ℓj ω kL . (5.10)Similarly, by [ii][b] , we may write ˆ ω i = λ ij ˇ ω j and calculate from (5.3) d λ ij = C ikℓ λ ℓj ˆ ω k . (5.11)The combination of equations (5.7), (5.10) and (5.11) (and the fact that Λ ij ( e ) = λ ij ( x )) now shows that d (cid:0) ρ ∗ (Λ ij )( λ − ) jk (cid:1) = 0 and hence ρ ∗ (Λ ij ) = λ ij . (5.12)(Here we use the connectivity of U ). The equations ρ ∗ ( ω iR ) = ˇ ω i now followsimmediately from (5.7), (5.8), and (5.12).We note, for future use, that (5.11) also implies L ˆ X i ˇ ω j = L ˆ X i ( λ − ) jk ˆ ω k + ( λ − ) jk C kiℓ ˆ ω ℓ = 0 . (5.13)This shows, by virtue of the connectivity of G , that( ˆ µ g ) ∗ ( ˇ ω j ) = ˇ ω j . (5.14) Lemma 5.4.
With ρ : U → G as in Lemma 5.3, the maps ˆ ϕ = ρ ◦ ˆ µ x : G → G and ˇ ϕ = ρ ◦ ˇ µ x : G → G satisfy ˆ ϕ ∗ ( ω iL ) = ω iL and ˇ ϕ ∗ ( ω iR ) = ω iR (5.15) and hence ˆ ϕ = ˇ ϕ = id G .Proof. The definitions ( ˆ µ x ) ∗ ( X Li ) = ˆ X i and ( ˇ µ x ) ∗ ( X Ri ) = ˇ X i , together with [ii](a) , imply that ˆ µ ∗ x ( ˆ ω i ) = ω iL and ˇ µ ∗ x ( ˇ ω i ) = ω iR . Equations (5.15) thenfollow directly from Lemma 5.3. These equations imply that ˆ ϕ and ˇ ϕ are trans-lations in G and therefore, since G is connected and ˆ ϕ ( e ) = ˇ ϕ ( e ) = e , we havethat ˆ ϕ = ˇ ϕ = id G . .1 A preliminary result Lemma 5.5.
The maps ρ : U → G satisfies ρ ( ˆ µ ( x, g )) = ρ ( x ) · g and ρ ( ˇ µ ( x, g )) = g · ρ ( x ) for all x ∈ U and g ∈ G .Proof. To prove these equivariance properties of ρ , define maps ˆ ρ = R g − ◦ ρ ◦ ˆ µ g : M → G and ˇ ρ = L g − ◦ ρ ◦ ˇ µ g : M → G, where ˆ µ g ( x ) = ˆ µ ( g, x ) and ˇ µ g ( x ) = ˇ µ ( g, x ). We wish to prove that ˆ ρ = ˇ ˇ ρ = ρ .Lemma 5.4 shows that ˆ ρ ( x ) = ˇ ρ ( x ) = e so that it suffices to show that ˆ ρ ∗ ( ω iR ) = ˇ ω i and ˇ ρ ∗ ( ω iL ) = ˇ ω i (5.16)and then to appeal to the uniqueness statement in Lemma 5.3. The first equa-tion (5.16) follows from the fact that the forms ω iR are right invariant, thesecond equation in (5.4) and (5.14). The proof of the second equation in (5.16)is similar. Corollary 5.6 (Corollary to Theorem 5.1) . Let S be the submanifold of M defined by S = ρ − ( e ) , where ρ is the map (5.4) . Then the map Ψ :
U → S × G defined by Ψ( x ) = ( ˆ µ ( ρ ( x ) − , x ) , ρ ( x )) is a G bi-equivariant diffeomorphism to an open subset of S × G and satisfies Ψ ∗ ( ˆ X i ) = X Ri , Ψ ∗ ( ˇ X i ) = X Li , Ψ ∗ ( ω iL ) = ˆ ω i , Ψ ∗ ( ω iR ) = ˇ ω i . (5.17) If M/G is simply-connected, then
Ψ : M → S × G is a global diffeomorphismProof. It is easy to check that Ψ sends the intersection of U with the domainof ˆ µ in M into S × G . To check the equivariance of Ψ, it is convenient to write ˆ µ ( g, x ) = x ˆ ∗ g and ˇ µ ( g, x ) = g ˇ ∗ x . Because the orbits for the two actionscoincide, we know that for each g ∈ G there is a g ′ such that g ˇ ∗ x = x ˆ ∗ g ′ .We then use (5.4) to calculateΨ( x ˆ ∗ g ) = (cid:0) ( x ˆ ∗ g ) ˆ ∗ ρ ( x ˆ ∗ g ) ( − , ρ ( x ˆ ∗ g ) (cid:1) = (cid:0) x ˆ ∗ ρ ( x ) ( − , ρ ( x ) ˆ ∗ g (cid:1) = Ψ( x ) ∗ g, andΨ( g ˇ ∗ x ) = (cid:0) ( g ˇ ∗ x ) ˆ ∗ ρ ( g ˇ ∗ x ) ( − , ρ ( g ˇ ∗ x ) (cid:1) = (cid:0) x ˆ ∗ g ′ ) ˆ ∗ ρ ( x ˆ ∗ g ′ ) ( − , g ˇ ∗ ρ ( x ) (cid:1) = (cid:0) ( x ˆ ∗ ρ ( x ) ( − , g ˇ ∗ ρ ( x ) (cid:1) = g ∗ Ψ( x ) , as required. .2 The Superposition Formula for Darboux Pairs We are now ready to establish, using the 5-adapted coframe of Section 4.4and the technical results of Section 5.1, the superposition formula for any Dar-boux pair.By way of summary, let us recall that our starting point is the local (0-adapted) coframe { θ , ˆ η , ˆ σ , ˇ η , ˇ σ } which is adapted to a given Darboux pair { ˆ V , ˇ V } in the sense that I = span { θ , ˆ η , ˇ η } , ˆ V = span { θ , ˆ σ , ˆ η , ˇ η } , ˇ V = span { θ , ˆ η , ˇ σ , ˇ η } . (5.18)In our first coframe adaption we adjusted the forms ˆ η and ˇ η to the form ˆ η = d ˆ I + ˆR ˆ σ and ˇ η = d ˇ I + ˇ R ˇ σ . (5.19)The fourth coframe adaptations { θ X , ˆ σ , ˆ η , ˇ σ , ˇ η } and { θ Y , ˆ σ , ˆ η , ˇ σ , ˇ η } al-lowed us to associate to the Darboux pair { ˆ V , ˇ V } an abstract Lie algebra vess ( ˆ V , ˇ V ). We have θ iX = Q ij θ jY and we can write ˆ V ∩ ˇ V = span { θ X , ˆ η , ˇ η } , (5.20)From the 4-adapted coframes we then constructed the final coframes ˆ θ i = ˆ R ij θ jX + φ ia ˆ π a , ˇ θ i = ˇ R ij θ jY + ψ iα ˇ π α , (5.21)where ˆ π = [ ˆ σ , ˆ η ] , and ˇ π = [ ˇ σ , ˇ η ], with structure equations d ˆ θ i = 12 G iαβ ˇ π α ∧ ˇ π β + 12 C ijk ˆ θ j ∧ ˆ θ k , d ˇ θ i = 12 H iab ˆ π a ∧ ˆ π b − C ijk ˇ θ j ∧ ˇ θ k . (5.22)The coefficients G iαβ are functions of the first integrals ˇ I α while the H iab arefunctions of the ˆ I a . The two 5-adapted coframes { ˆ θ , ˆ σ , ˆ η , ˇ σ , ˇ η } and { ˇ θ , ˆ σ , ˆ η , ˇ σ , ˇ η } (5.23)are not adapted to the Darboux pair { ˆ V , ˇ V } (in the sense of (2.6)) althoughwe do have ˆ V = span { ˆ θ , ˆ σ , ˆ η , ˇ η } and ˇ V = span { ˇ θ , ˆ η , ˇ σ , ˇ η } . (5.24) Definition 5.7.
Let { ˆ V , ˇ V , } be a Darboux pair with 5-adapted coframes (5.23) .Let { ˆ X i , ˆ V a , ˇ V α } and { ˇ X i , ˇ W a , ˇ W α } be the dual frames corresponding to the5 adapted coframes so that, in particular, ˆ θ i ( ˆ X j ) = δ ij , ˆ π a ( ˆ X j ) = 0 , ˇ θ i ( ˇ X j ) = δ ij , ˇ π a ( ˇ X j ) = 0 . (5.25) .2 The Superposition Formula for Darboux Pairs The infinitesimal Vessiot group actions for the Darboux pair { ˆ V , ˇ V } aredefined by the Lie algebras of vector fields ˆ X i and ˇ X i . These define (local)groups actions ˆ µ : G × M → M and ˇ µ : G × M → M. (5.26) We take ˆ µ to be a right action on M and ˇ µ to be a left action. The vectors fields ˆ X i and ˇ X i are related by (see (4.87) and (4.88)) ˆ X i = λ ji ˇ X j and satisfy [ ˆ X i , ˇ X j ] = 0 . (5.27)From the structure equations (5.22) we find that L ˆ X j ˆ θ i = C ijk ˆ θ k and L ˇ X j ˇ θ i = C ijk ˇ θ k . (5.28) L ˆ X j ˆ π a = L ˆ X j ˇ π α = L ˇ X j ˆ π a = L ˇ X j ˇ π α = 0 . (5.29)The actions ˆ µ and ˇ µ commute because [ ˆ X i , ˇ X j ] = 0. In what follows we shallassume that these actions are regular.Now let G be a Lie group with Lie algebra vess ( ˆ V , ˇ V ). Let ω iL and ω iR bethe left and right invariant Maurer-Cartan forms on G , with structure equations(5.1). We shall assume that these two coframes on G coincide at the identity e .Let X Li and X Ri be the dual basis of left and right invariant vector fields.To apply Theorem 5.1 we need to identify the forms ˆ ω i and ˇ ω i . Lemma 5.8.
The forms ˆ ω i and ˇ ω i defined in terms of the 5-adapted coframes( (4.47) or (5.21) ) by ˆ ω i = ˆ θ i + λ ij ψ jα ˇ π α and ˇ ω i = ˇ θ i + µ ij φ ja ˆ π a , (5.30) where λ = ˆ RQ ˇ R − and µ = λ − (see Remark 4.9), have the same span andsatisfy the structure equations d ˆ ω i = 12 C ijk ˆ ω j ∧ ˆ ω k and d ˇ ω i = − C ijk ˇ ω j ∧ ˇ ω k . (5.31) Proof.
To check that the forms ˆ ω i and ˇ ω i have the same span we simply use thevarious definitions given here to calculate ˆ ω i = ˆ θ i + λ ij ψ jα ˇ π α = ˆ R ij θ jX + φ ia ˆ π a + λ ij ψ jα ˇ π α = ˆ R ij Q jk θ kY + φ ia ˆ π a + λ ij ψ jα ˇ π α = ( ˆ R Q ˇ R − ) ij ˇ R jk θ kY + φ ia ˆ π a + λ ij ψ jα ˇ π α = λ ij ( ˇ R jk θ kY + ψ ja ˇ π a + ( λ − ) jk φ kα ˆ π α ) = λ ij ( ˇ θ j + µ jk φ kα ˆ π α ) = λ ij ˇ ω j . (5.32) .2 The Superposition Formula for Darboux Pairs d ˆ ω i using (4.48), (4.85), (4.86) and (4.88) to find that d ˆ ω i = d ˆ θ i + d λ ij ∧ ψ j + λ ij ∧ d ψ j = 12 G iαβ ˇ π α ∧ ˇ π β + 12 C ijk ˆ θ j ∧ ˆ θ k + ( C iℓm λ mj ˆ θ ℓ + λ ih C hmj ψ m ) ∧ ψ j + λ ij ( − C jhk ψ h ∧ ψ k − ˇ R jℓ F ℓαβ ˇ π α ∧ ˇ π β )= 12 C ijk ˆ θ j ∧ ˆ θ k + C iℓm λ mj ˆ θ ℓ ∧ ψ j + 12 λ iℓ C ℓhk ψ h ∧ ψ k = 12 C ijk ˆ ω j ∧ ˆ ω k . The derivation of the structure equations for ˇ ω i is similar.Let x ∈ M . By a change of frame at x we may suppose that ˆ ω i ( x ) = ˇ ω i ( x ). At this point all the hypothesis for Theorem 5.1 are satisfied and, inaccordance with this theorem and Corollary 5.6, we can construct maps ρ : U → G and Ψ : U → S × G (5.33)which satisfy (5.4), (5.17) and ρ ( x ) = e .Define S and S to be the integral manifolds through x for the restrictionof ˆ V ( ∞ ) and ˇ V ( ∞ ) to S . Because T ∗ S x = ˆ V ( ∞ ) x ⊕ ˇ V ( ∞ ) x for all x ∈ S , (5.34)we may choose S and S small enough so as to be assured of the existence ofa local diffeomorphism χ : S × S → S , (5.35)where S is an open set in S . The map χ satisfies (with s = χ ( s , s ) and s i ∈ S i ) χ ∗ ( ˆ V ( ∞ ) s ) = T ∗ s S and χ ∗ ( ˇ V ( ∞ ) s ) = T ∗ s S . (5.36)Set χ ( x , x ) = x . Hence, for any first integrals ˆ I and ˇ I of ˆ V and ˇ V , we have ˆ I ( χ ( s , s ) = ˆ I ( χ ( x , s )) and ˇ I ( χ ( s , s )) = ˇ I ( χ ( s , x )) . (5.37)Put U = Φ − ( S ).To define the superposition formula for the Darboux pair { ˆ V , ˇ V } , we take M to be the (maximal, connected) integral manifold of ˆ V ( ∞ ) through x in U .2 The Superposition Formula for Darboux Pairs M to be the (maximal, connected) integral manifold of ˇ V ( ∞ ) through x in U . The group actions ˆ µ and ˇ µ and the maps ρ and Ψ all restrict to maps ˆ µ i : G × M i → M i , ˇ µ i : G × M i → M i ,ρ i : M i → G, and Ψ i : M i → S i × G (5.38)for i = 1 , χ defined by (5.4) and (5.35) and with these choicesfor M and M , we defineΣ : M × M → M by Σ( x , x ) = ˆ µ (cid:0) g · g , χ ( s , s ) (cid:1) , (5.39)where x i ∈ M i , and Ψ( x i ) = ( s i , g i ).To prove that Σ defines a superposition formula, we must calculate thepullback by Σ of an adapted coframe on M . To this end, we introduce thefollowing notation. For any differential form α on M , denote the restriction of α to M i by α i . If f is a function on M and x i ∈ M i , then f ( x i ) denotes thevalue of f at x i , viewed as a point of M . We also remark that the equivarianceof ρ with respect to the group action ˆ µ and ˇ µ implies that Σ satisfies( ρ ◦ Σ)( x , x ) = ρ ( ˆ µ ( g · g , χ ( s , s )) = g · g · ρ ( χ ( s , s ))= g · g = ρ ( x ) · ρ ( x ) = ( m ◦ ( ρ × ρ ))( x , x ) . (5.40) Lemma 5.9.
The pullback by Σ of the 4-adapted coframe (see Theorm 4.6) on M is given in terms of the restrictions of the 5-adapted coframes (5.23) on M and M by Σ ∗ ( ˆ σ ) = ˆ σ , Σ ∗ ( ˇ σ ) = ˇ σ , Σ ∗ ( ˆ η ) = ˆ η , Σ ∗ ( ˇ η ) = ˇ η , (5.41)Σ ∗ ( Rθ X ) = λ ( ˆ θ + ˇ θ ) . (5.42) Proof.
We begin with the observation that the first integrals ˆ I a and ˇ I α areinvariants of ˆ µ and therefore, by (5.37),Σ ∗ ( ˆ I a )( x , x ) = ˆ I a ( ˆ µ ( g g , χ ( s , s ))) = ˆ I a ( χ ( s , s ))= ˆ I a ( χ ( x , s )) = ˆ I a ( ˆ µ ( g , χ ( x , s )))= ˆ I a ( x ) , and similarly (5.43)Σ ∗ ( ˇ I α )( x , x ) = ˇ I α ( x ) . (5.44) .2 The Superposition Formula for Darboux Pairs ∗ ( G iαβ )( x , x ) = G iαβ ( x ) and Σ ∗ ( H iab )( x , x ) = G iab ( x ) . (5.45)From equations (5.43) and (5.44) it follows directly thatΣ ∗ ( ˆ σ ) = ˆ σ , Σ ∗ ( ˇ σ ) = ˇ σ , (5.46)Σ ∗ ( ˆ η ) = Σ ∗ ( d ˆ I + ˆR ˆ σ ) = ˆ η and Σ ∗ ( ˇ η ) = Σ ∗ ( d ˇ I + ˇR ˇ σ ) = ˇ η . (5.47)Equations (5.41) are therefore established.Equations (5.21) and (5.30) imply that ˆ R ij θ jX = ˆ ω i − φ ia ˆ π a − λ ik ψ ka ˇ π α . (5.48)We use this equation to calculate Σ ∗ ( ˆ R ij θ jX ).Since φ ia ∈ Int( ˆ V ) and ψ iα ∈ Int( ˇ V ), equations (5.43) and (5.44) imply thatΣ ∗ ( φ ia ˆ π a )( x , x ) = φ ia ( x ) ˆ π a and Σ ∗ ( ψ ia ˇ π α )( x , x ) = ψ iα ( x ) ˆ π α . (5.49)We deduce from equations (5.8), (5.12) and (5.40) thatΣ ∗ ( λ ik )( x , x ) = Σ ∗ ( ρ ∗ Λ ik )( x , x ) = Λ ik ( g · g ) == Λ ij ( g )Λ jk ( g ) = λ ij ( x ) λ jk ( x ) . (5.50)To calculate Σ ∗ ( ˆ ω i ) it is helpful to introduce the projection maps π i : G × G → G and to note that π i ( ρ × ρ )( x , x ) = g i , i = 1 ,
2. We can then re-write(5.9) as m ∗ ( ω iL ) = π ∗ ( λ ij ) π ∗ ( ω jL ) + π ∗ ( ω iL ) . This equation, together with (5.4) and (5.40) leads toΣ ∗ ( ˆ ω i )( x , x ) = Σ ∗ ( ρ ∗ ( ω iL ))( x , x ) = ( ρ ∗ × ρ ∗ )( m ∗ ( ω iL ))( x , x )= ( ρ ∗ × ρ ∗ )( π ∗ ( λ ij ) π ∗ ( ω jL ) + π ∗ ( ω iL ))( x , x ) == λ ij ( x ) ˆ ω j + ˆ ω i . (5.51)Finally, the combination of equations (5.32) and (5.48) – (5.51) allows us tocalculateΣ ∗ ( ˆ R ij θ jX ) = Σ ∗ (cid:0) ˆ ω i − φ ia ˆ π a − λ ik ψ kα ˇ π α (cid:1) = λ ij ( x ) ˆ ω j + ˆ ω i − φ ia ( x ) π a − λ ij ( x ) λ jk ( x ) ψ kα ( x ) ˇ π α = λ ij ( x ) (cid:0) ˆ ω j − λ jk ( x ) ψ kα ( x ) ˇ π α (cid:1) + λ ij ( x ) (cid:0) ˇ ω j − µ jk ( x ) φ ka ( x ) ˆ π a (cid:1) = λ ij ( x ) ˆ θ j + λ ij ( x ) ˇ θ j , (5.52) .2 The Superposition Formula for Darboux Pairs { ˆ V , ˇ V } , the EDS ˆ V ∩ + ˇ V is given by Defi-nition 2.5. Theorem 5.10 ( The Superposition Formula) . Let { ˆ V , ˇ V } define a Dar-boux pair on M. Define M and M as above and let W and W be the restric-tions of ˇ V and ˆ V to M and M , respectively. Then the map Σ : M × M → M ,defined by (5.39) , satisfies Σ ∗ ( ˆ V ∩ ˇ V ) ⊂ W + W and Σ ∗ ( ˆ V ∩ + ˇ V ) ⊂ W + W . (5.53) Thus Σ defines a superposition formula for ˆ V ∩ + ˇ V with respect to the Pfaffiansystems W and W .Proof. To prove this theorem we need only explicitly list the generators for W + W , W + W , ˆ V ∩ ˇ V and ˆ V ∩ + ˇ V and check that, using Lemma (5.17), thatthe latter pullback into the former by Σ. For this we use the definitions of ˆ V and ˇ V in terms of the 4-adapted and 5-adapted coframes by(5.20) and (5.24).From the definition of M and M as integral manifolds of ˆ V ( ∞ ) and ˇ V ( ∞ ) ,we have that ˆ η = 0 , ˆ σ = 0 on M and ˇ η = 0 , ˇ σ = 0 on M and hence the two 5-adapted coframes on M restrict to coframes { ˆ θ , ˇ η , ˇ σ } for M and { ˇ θ , ˆ η , ˆ σ } for M . (5.54)These naturally combine to give a coframe on M × M . The 1-form generatorsfor the Pfaffian systems W and W are then given by W = span { ˆ θ , ˇ η } and W = span { ˇ θ , ˆ η } . (5.55)The differential system W + W is algebraically generated by the 1-forms(5.55) and their exterior derivatives (see (2.13), (5.22)), that is, W + W = { ˆ θ , ˇ η , ˇ θ , ˆ η , d ˇ η , d ˆ η , d ˆ θ , d ˇ θ } . The restrictions of (5.22) to M and M lead to the structure equations d ˆ θ = 12 G ˇ π ∧ ˇ π mod { ˆ θ } = 12 G ˇ σ ∧ ˇ σ mod { ˆ θ , ˇ η } and d ˇ θ = 12 H ˆ π ∧ ˆ π mod { ˇ θ } = 12 H ˆ σ ∧ ˆ σ mod { ˆ θ , ˆ η } .2 The Superposition Formula for Darboux Pairs W + W is algebraically generated by W + W = { ˆ θ , ˇ η , ˇ θ , ˆ η , ˇ F ˇ σ ∧ ˇ σ , ˆ F ˆ σ ∧ ˆ σ , H ˆ σ ∧ ˆ σ } . (5.56)The Pfaffian system ˆ V ∩ ˇ V is generated by the forms { θ , ˆ η , ˇ η } or equivalently(see (5.20)) by { θ X , ˆ η , ˇ η } . The first inclusion in (5.53) follows immediately fromthe Lemma 5.9 and (5.55).By definition, the differential system ˆ V ∩ + ˇ V is algebraically generated, interms of the 1-adapted coframe (see Theorem 2.9), as ˆ V ∩ + ˇ V = { θ , ˆ η , ˇ η , d ˆ η , d ˇ η , A ˆ σ ∧ ˆ σ , B ˇ σ ∧ ˇ σ } or, equally as well, in terms of the 4-adapted coframe by ˆ V ∩ + ˇ V = { θ , ˆ η , ˇ η , ˇ F ˇ σ ∧ ˇ σ , ˆ F ˆ σ ∧ ˆ σ , A ˆ σ ∧ ˆ σ , B ˇ σ ∧ ˇ σ } . In this latter equation the coefficients A and B are those appearing in (4.25).By (4.46) and (4.85), we also have that ˆ V ∩ + ˇ V = { Rθ X , ˆ η , ˇ η , ˇ F ˇ σ ∧ ˇ σ , ˆ F ˆ σ ∧ ˆ σ , H ˆ σ ∧ ˆ σ , G ˇ σ ∧ ˇ σ } . (5.57)With the algebraic generators for ˆ V ∩ + ˇ V in this form, it is easy to calculate thepullback of ˆ V ∩ + ˇ V to M × M by Σ. By using (5.41), (5.42), (5.52) we find thatΣ ∗ ( ˆ V ∩ + ˇ V ) = { ˆ θ + ˇ θ , ˆ η , ˇ η , ˇ F ˇ σ ∧ ˇ σ , ˆ F ˆ σ ∧ ˆ σ , H ˆ σ ∧ ˆ σ , G ˇ σ ∧ ˇ σ } (5.58)and the second inclusion in (5.53) is firmly established.One may use the first integrals of ˆ V and ˇ V and the map ρ : U → G , andlocal coordinates z i on G to define local coordinates( ˆ I a = ˆ I a ( x ) , ˇ I α = ˇ I α ( x ) , z i = ρ i ( x )) (5.59)on M and induced coodinates ( ˇ I α , z i ) on M and ( ˆ I a , z i ) on M . In thesecoordinates the superposition formula becomesΣ (cid:0) ( ˇ I α , z j ) , ( ˆ I a , z k ) (cid:1) = (cid:0) ˆ I a = ˆ I a , ˇ I α = ˇ I α , z i = ( z j ) · ( z k ) (cid:1) . (5.60)We shall use this formula extensively in the examples in Section 6.To prove that the superposition map Σ is subjective, at the level of integralmanifolds, we use the concept of an integrable extension of a differential system .2 The Superposition Formula for Darboux Pairs I on M [5]. This is a differential system J on a manifold N together with asubmersion ϕ : N → M such that ϕ ∗ ( I ) ⊂ J and such that the quotient dif-ferential system J /ϕ ∗ I is a completely integrable Pfaffian system. This meansthat there are 1-forms ϑ ℓ ∈ J such that J is algebraically generated by theforms { ϑ ℓ } ∪ ϕ ∗ I and d ϑ ℓ ≡ { ϑ ℓ } ∪ ϕ ∗ I (5.61)or, equivalently, modulo the 1-forms in J and the 2-forms in ϕ ∗ I . Under theseconditions the map ϕ is guaranteed to be a local surjection from the integralmanifolds of J to the integral elements of I . Indeed, as argued in [5], let P ⊂ M be an integral manifold of I defined in a neighborhood of x ∈ M . Choose apoint y ∈ N such that ϕ ( y ) = x . Then ˜ P = ϕ − ( P ) is a submanifold on N containing y and the restriction of J to ˜ P is a completely integrable Pfaffiansystem ˜ J . By the Frobenius theorem (applied to ˜ J as a Pfaffian system on ˜ P )there is locally a unique integral manifold Q for ˜ J through y . The manifold Q is then an integral manifold of J which projects by ϕ to the original integralmanifold P on some open neighborhood of x . Corollary 5.11.
Let { ˆ V , ˇ V } be a Darboux pair on M . Then the EDS W + W on M × M is an integrable extension of ˆ V ∩ + ˇ V on M with respect to thesuperposition formula Σ : M × M → M .Proof. In view of (5.56) and (5.58) we may take the differential ideal generatedby F = { ˆ θ } as a complement to Σ ∗ ( ˆ V ∩ + ˇ V ) in W + W . The 1-forms ˆ θ areclosed modulo ˆ θ , ˆ η and G ˇ σ ∧ ˇ σ and therefore the generators for F areclosed, modulo F and modulo the 1-forms and 2-forms in Σ ∗ ( ˆ V ∩ + ˇ V ). Hence W + W is an integral extension of ˆ V ∩ + ˇ V . Corollary 5.12.
Let I be a decomposable, Darboux integrable Pfaffian system.Then the map Σ : M × M → M is a superposition formula for I with respectto the EDS W + W which is locally surjective on integral manifolds.Proof. If { ˆ V , ˇ V } is the Darboux pair for I , then I = ˆ V ∩ + ˇ V and the corollaryfollows from Theorem 5.10 and Corollary 5.11. .3 Superposition Formulas and Quotients of EDS In this section we shall use the superposition formula established in Theo-rem 5.10 to prove that if I is any decomposable differential system for whichthe singular Pfaffian systems { ˆ V , ˇ V } form a Darboux pair, then I can be real-ized as a quotient differential system, with respect to its Vessiot group, by theconstruction given in Corollary 3.4.To precisely formulate this result, it is useful to first summarize the essentialresults of Sections 4 and 5.1–5.2. We have constructed, through the coframeadaptation of Section 4, a local Lie group G and local right and left group actions ˆ µ, ˇ µ : G × M → M. For the sake of simplicity, let us suppose that G is a Liegroup and that these actions are globally defined. The infinitesimal generatorsfor these actions are the vector fields ˆ X i and ˇ X i , defined by the duals of the5-adapted coframes. [i] The actions ˆ µ and ˇ µ are free actions with the same orbits (the vector fields ˆ X i and ˇ X i are pointwise independent and related by ˆ X i = λ ji ˇ X j ). [ii] The actions ˆ µ and ˇ µ commute (see (5.27)). [iii] The actions ˆ µ and ˇ µ are symmetry groups of ˆ V and ˇ V , respectively (see(5.28)). [iv] Each integral manifold of ˆ V ∞ or ˇ V ∞ is fixed by both actions ˆ µ and ˇ µ (see(5.29)). [v] We have defined ι : M → M and ι : M → M to be fixed integralmanifolds of ˆ V ∞ and ˇ V ∞ and W and W to be the restrictions of ˆ V and ˇ V to these integral manifolds. [vi] Properties [i] – [iv] imply that the actions ˆ µ and ˇ µ restrict to actions on ˆ µ i and ˇ µ i on M i . These restricted actions are free, ˆ µ and ˇ µ are symmetriesof W and W respectively. The actions ˆ µ and ˇ µ are transverse to W and W (see (5.25)).The diagonal action δ : G × ( M × M ) → M × M is defined as the left action δ ( h, ( x , x )) = (cid:0) ˆ µ ( h − , x ) , ˇ µ ( h, x ) (cid:1) . (5.62) .3 Superposition Formulas and Quotients of EDS δ are Z i = − ˆ X i + ˇ X i . (5.63)Granted that the action δ is regular, we then have that all the hypothesis ofCorollary 3.4 are satisfied. We can therefore construct the quotient manifold q : M × M → ( M × M ) /G , the quotient differential system J = ( W + W ) /G and the Darboux pairs { ˆ U , ˇ U } (see (3.17)).We use the superposition formula (5.39) to identify the manifold M with( M × M ) /G and the original differential system I with the quotient system J . Theorem 5.13.
Let I be a decomposable differential system on M whose sin-gular Pfaffian systems { ˆ V , ˇ V } define a Darboux pair. Let G be the Vessiotgroup for { ˆ V , ˇ V } and define Pfaffian systems W on M and W on M asabove. Then the manifold M can be identified as the quotient of M × M bythe diagonal action δ of the Vessiot group G , the superposition formula Σ is thequotient map, and I = ( W + W ) /G . All the manifolds, actions, and differential systems appearing in Theorem5.13 are presented in the following diagram:
Proof of Theorem 5.13.
The following elementary facts are needed. [i]
The superposition map Σ : M × M → M is invariant with respect to thediagonal action δ of the Vessiot group G on M × M . [ii] For each point x = ( x , x ) ∈ M × M , ker Σ ∗ ( x ) = ker q ∗ ( x ). [iii] Σ ∗ ( ˆ V ) = [ W ⊕ Λ ( M )] sb and Σ ∗ ( ˇ V ) = [Λ ( M ) ⊕ W ] sb .Facts [i] and [ii] show that Σ : M × M → M can be identified with q : M × M → ( M × M ) /G . Fact [ iii ] shows that ˆ V = ˆ U and ˇ V = ˇ U (in the notationof Corollary 3.4) so that I = ˆ V ∩ + ˇ V = ˆ U ∩ + ˇ U = J .To prove [i] , let x ∈ M , x ∈ M and h ∈ G . If Ψ( x ) = ( s , g ) andΨ( x ) = ( s , g ) then it is a simple matter to check, using the G bi-equivarianceof ρ , that ˆ µ ( h − , x ) = ( s , g · h − ) and ˇ µ ( h, x ) = ( s , h · g ). The invarianceof Σ then follows immediately from its definition (5.39).Lemma 5.9 and (5.63) show thatker Σ ∗ = span { ∂ ˆ θ − ∂ ˇ θ } = span {− ˆ X i + ˇ X i } = span { Z i } (5.64) .3 Superposition Formulas and Quotients of EDS [ii] .Lemma 5.9 and equations (5.18) show thatΣ ∗ ( ˆ V ) = Σ ∗ ( { ˆ θ , ˆ σ , ˇ η , ˆ η } = { ˆ θ + ˇ θ , ˆ σ , ˆ η , ˇ η } (5.65)while (5.54) and (5.55) give W ⊕ Λ ( M ) = { ˆ θ , ˇ η , ˇ θ , ˆ η , ˆ σ } (5.66)in which case [iii] follows immediately from (5.64).To make precise the identification of M with the quotient manifold M =( M × M ) /G , define a map Υ : M → M as follows. For each point ¯ x ∈ M , picka point ( x , x ) ∈ M × M such that q ( x , x ) = ¯ x and let Υ(¯ x ) = Σ( x , x ). .3 Superposition Formulas and Quotients of EDS ✎✍ ☞✌ M × M ✍✌✎☞ M ✍✌✎☞ M Error: Incorrect label specification Error: Incorrect label specification ............................................................................................ q Error: Incorrect label specification Error: Incorrect label specification ............................................................................................
ΣError: Incorrect label specification Error: Incorrect label specification ............................................................................................
Υcommutes and, on the domain of any (local) cross-section ζ of q , one has Υ =Σ ◦ ζ . This observation and facts [i] and [ii] then suffice to show that Υ is awell-defined, smooth diffeomorphism. Moreover, the same computations usedto establish fact [i] show that Σ, and hence Υ, is G equivariant with respect tothe action ˇ µ on M × M and ˇ µ on M and also G equivariant with respect tothe action ˆ µ on M × M and ˆ µ on M .Finally, we recall that ˆ U and ˇ U may be calculated from the cross-section ζ .3 Superposition Formulas and Quotients of EDS δ semi-basis forms by ˆ U = ζ ∗ ([ W ⊕ T ∗ ( M )] sb ) and ˇ U = ζ ∗ ( T ∗ ( M ) ⊕ W ] sb ) (5.67)in which case [iii] implies that Υ ∗ ( ˆ V ) = ˆ U and Υ ∗ ( ˇ V ) = ˇ U . Remark 5.14.
Finally we remark that all the results of this section remainvalid in so long as the singular systems for I have algebraic generators ˆ V = { θ , ˆ η , ˇ η , ˆ σ , P ˇ σ ∧ ˇ σ } and ˇ V = { θ , ˆ η , ˇ η , ˇ σ , Q ˆ σ ∧ ˆ σ , } . (5.68)where P ∈ Int( ˇ V ) and Q ∈ Int( ˆ V ). Such systems need not be Pfaffian. xamples In this section we illustrate our general theory with a variety of examples.Examples 6.1 and 6.2 are taken from the classical literature and are simpleenough that most of the computations can be explicitly given. We consider PDEwhere the unknown functions take values in a group or in a non-commutativealgebra in Examples 6.3 and 6.4. In Example 6.5 we present some novel exam-ples of Darboux integrable systems constructed by the coupling of a nonlinear,Darboux integrable scalar equation to a linear or Moutard-type equation. AToda lattice system and a wave map system are explicitly integrated in Exam-ples 6.6 and 6.7. In Example 6.8, we solve some non-linear, over-determinedsystems in 3 independent variables.
Example 6.1.
As our first example we shall find the closed-form, general so-lution to u xy = u x u y u − x . (6.1)This example is taken from Goursat’s well-known classification (Equation VI)of Darboux integrable equations [16] and is simple enough that all the stepsleading to the solution can be explicitly given. See also Vessiot [28] (pages9–22) or Stomark [24] (pages 350–356).The canonical Pfaffian system for (6.1) is I = { α , α , α } , where α = du − p dx − q dy, α = dp − r dx − vpq dy, α = dq − vpq dx − t dy and v = 1 / ( u − x ). The associated singular Pfaffian systems are ˆ V = { α i , dx, dr − q ( vr + v p ) dy } and ˇ V = { α i , dy, dt − tpv dy } . (6.2)The first integrals for ˆ V and ˇ V are ˆ I = x, ˆ I = vp, ˆ I = vr + v p, ˇ I = y, ˇ I = tq − y (6.3)and we easily calculate that ˆ V ( ∞ ) ∩ ˇ V = { ˆ η } and ˆ V ∩ ˇ V ( ∞ ) = { } , where ˆ η = d ˆ I + (( ˆ I ) − ˆ I ) d ˆ I = vα − v pα . (6.4)A 1-adapted coframe (2.6) is therefore given by θ = α , θ = α , ˆ σ = d ˆ I , ˆ σ = d ˆ I , ˆ η , ˇ σ = d ˇ I , ˇ σ = d ˇ I . (6.5) xamples ˆ π = σ , ˆ π = σ , ˆ π = ˆ η , ˇ π = ˇ σ , ˇ π = ˇ σ and calculate d ˆ π = − ˆ I ˆ π ∧ ˆ π + ˆ π ∧ ˆ π ,dθ = ( u − x ) ˆ π ∧ ˆ π + ˆ I ˆ π ∧ θ + ˇ π ∧ θ ,dθ = q ˆ π ∧ ˆ π + q ˇ π ∧ ˇ π + ˆ I ˆ π ∧ θ + ˇ I ˇ π ∧ θ . (6.6)This coframe satisfies the structure equations (4.13) and is therefore 2-adapted.The next step, described in Section 4.2, is to eliminate the ˇ π α ∧ θ i termsfrom (6.6). We calculate the distributions ˆ U and ˇ U (see (4.15)) and their derivedflags to be ˆ U = { ∂ ˆ π + (( ˆ I ) − ˇ I ) ∂ ˆ π , ∂ ˆ π , ∂ ˆ π } , ˇ U = { ∂ ˇ π , ∂ ˇ π } , (6.7) ˆ U ( ∞ ) = ˆ U ∪ { ( u − x ) ∂θ + q∂θ , ∂θ } , ˇ U ( ∞ ) = ˇ U ∪ { q ∂θ , q ∂θ } . and then, in accordance with (4.19)-(4.21), define X = ( u − x ) ∂θ + q∂θ , X = ∂θ , Y = q ∂θ , Y = ∂θ , (6.8)The coframes dual to the vector fields { X i , ˆ U , ˇ U } and { Y i , ˆ U , ˇ U } are the 3-adapted coframes θ X = 1 q θ , θ X = θ − u − xq θ , θ Y = 1 q θ , θ Y = 1 q θ with (6.9) dθ X = ˆ π ∧ ˆ π + ˇ π ∧ ˇ π = dθ Y ,dθ X = − ( u − x ) ˇ π ∧ ˇ π + θ X ∧ θ X + ˆ π ∧ θ X + ˆ I ˆ π ∧ θ X ,dθ Y = u − xq ˆ π ∧ ˆ π − θ Y ∧ θ Y + ˇ π ∧ θ Y − ˇ I ˇ π ∧ θ Y . (6.10)The coframe (6.9) is in fact 4-adapted and hence the Vessiot algebra for (6.1) isa 2 dimensional non-abelian Lie algebra.We may skip the adaptations given in Section 4.3 and move on to the finaladaptations given in Section 4.4. The Vessiot algebra is 1-step solvable andthe structure equations (6.10) are precisely of the form (4.68). The change ofcoframe ˆ θ = θ X + ˆ I π transforms the structure equations (6.10) to the form(4.70). The change of coframe ˆ θ = θ X − x ˆ θ leads to the ˆ5-adapted coframe { ˆ θ , ˆ θ } with structure equations d ˆ θ = ˇ π ∧ ˇ π , d ˆ θ = − u ˇ π ∧ ˇ π + ˆ θ ∧ ˆ θ . (6.11) xamples ˇ θ = θ Y + ˇ I ˇ π , ˇ θ = θ Y − y ˇ θ satisfies d ˇ θ = ˆ π ∧ ˆ π , d ˇ θ = u − x − yqq ˆ π ∧ ˆ π − ˇ θ ∧ ˇ θ . (6.12)Before continuing we remark that the vector fields X , X , defined by (6.8),are given in terms of the dual vector fields ˆ X and ˆ X , computed from the ˆ5-adapted coframe, by X = ˆ X − x ˆ X and X = ˆ X . These vector field systemshave the same orbits and structure equations but the actions are evidentlydifferent and it is the latter action that is needed to properly construct thesuperposition formula.The forms (5.30) are ˆ ω = ˆ θ + ˇ I ˇ π = dqq , ˆ ω = ˆ θ − u ˇ I ˇ π = du − qdy − u dqq , ˇ ω = ˇ θ + ˆ I ˆ π = dqq , ˇ ω = ˇ θ + ( pq + yvp ) ˆ π = duq − dy − y dqq . (6.13)The Vessiot group for (6.1) is the matrix group " b a with Maurer-Cartanforms ω L = daa , ω L = db − ba da, ω R = daa , ω R = dba . The map ρ : M → G defined by a = q and b = u − yq satisfies (5.4).Finally, if we introduce coordinates y = y , u = u , q = q , t = t on the ˆ V ( ∞ ) integral manifold M = { ˆ I = 0 , ˆ I = 0 , ˆ I = 0 } and x = x , u = u , p = p , q = u , r = r on the ˇ V ( ∞ ) integral manifold M = { ˇ I = 0 , ˇ I = 0 } ,then the superposition formula (5.60) is x = x , pu − x = p u − x , ru − x + p ( u − x ) = r u − x + p ( u − x ) y = y , tq = t q , q = q q , u − yq = u + ( u − y q ) q , or, explicitly in terms of the original coordinates { x, y, u, p, q, r, t } on M, x = x , y = y , u = u + q u , p = (1 + u q u − x ) p , q = q q ,r = (1 + u q u − x ) r + u p q ( u − x ) , t = t q . (6.14)It remains to find the integral manifolds for ˆ W and ˇ W . Restricted to M ,the Pfaffian system ˆ V becomes ˆ W = { du − q dy , dq − t dy } with integralmanifolds y = β, u = f ( β ) , q = f ′ ( β ) , t = f ′′ ( β ) . (6.15) xamples M , the Pfaffian system ˇ V becomes ˇ W = { du − p dx , dp − r dx , dq − p q u − x dx } . To find the integral manifolds of ˇ W we calculate thesecond derived Pfaffian system to be ˇ W (2) = { dq − q u − x du } which leadsto the equation q du − u dq + x dq = 0 or d (cid:18) u q (cid:19) − x d (cid:18) q (cid:19) = 0 or d (cid:18) u − x q (cid:19) + 1 q dx = 0 . The integral manifolds for ˇ W are therefore given by x = α, u = x − g ( α ) /g ′ ( α ) , q = − /g ′ ( α ) , (6.16)with p and r determined algebraically from the vanishing of the first andsecond forms in ˇ W . The substitution of (6.15) and (6.16) into the superpositionformula (6.14) leads to the closed form general solution u = − f ( y ) − g ( x ) g ′ ( x ) + x (6.17)for (6.1). xamples Example 6.2.
In this example we shall construct the superposition formula forthe Pfaffian system I = { α , α , α } , where α = du − p dx − q dy, α = dp + 1 b (tan bτ − bτ ) dx − s dy,α = dq − s dx − b ( bτ + cot bτ ) dy. (6.18)The coordinates for this example are ( x, y, u, p, q, s, τ ) and b is a parameter.This example nicely illustrates the various coframe adaptations in Sections 4and has a surprising connection with some of Cartan’s results in the celebrated5 variables paper [7]. The values b = 1, b = √− b = 0 give the three Pfaffian systems forthe equations u xx = f ( u yy )which are Darboux integrable at the 2-jet level ([2], pages 373-374) and [4],pages 400–411). The case b = 0 is treated in Goursat([15] Vol 2, page 130-132.One easily calculates ˆ V ∞ ∩ ˇ V = ˆ V ∩ ˇ V ∞ = { } and that the first integralsfor ˆ V and ˇ V are ˆ I = s + τ, ˆ I = − ( x + b y ) ˆ I + q + b p, ˇ I = s − τ, ˇ I = − ( x − b y ) ˇ I + q − b p. (6.19)We immediately arrive at the 2-adapted coframe ˆ π = d ˆ I , ˆ π = d ˆ I , ˇ π = d ˆ I , ˇ π = d ˇ I , θ = 2 α ,θ = − b cot bτ α + 1 b tan bτ α , θ = − b cot bτ α − b tan bτ α , (6.20)with structure equations d θ = ( x + b y ) ˆ π ∧ θ + ˆ π ∧ θ − ( x − b y ) ˇ π ∧ θ − ˇ π ∧ θ ,d θ = ˆ π ∧ ˆ π − b cot 2 bτ ˆ π ∧ θ + b csc 2 bτ ˇ π ∧ θ ,d θ = ˇ π ∧ ˇ π − b csc 2 bτ ˆ π ∧ θ + b cot 2 bτ ˇ π ∧ θ . (6.21)To compute the 3-adapted coframe θ iX , we simply calculate the derived flagfor the 2 dimensional distribution ˆ U = { ∂ ˆ π , ∂ ˆ π } (see Section 4.2). From thestructure equations (6.21) we find ˆ U (1) = ˆ U ∪ { − ∂ θ } and ˆ U (2) = ˆ U ∪ { − ∂ θ , ( x + b y ) ∂ θ − b cot 2 bτ ∂ θ − b csc 2 bτ ∂ θ , ∂ θ } . (6.22) xamples { X , X , X } to be the last 3 vectors in ˆ U (2) and calculate[ X , X ] = b X . (6.23)Therefore the Vessiot algebra for the Pfaffian system (6.18) is abelian if b = 0and nilpotent otherwise. The 3-adapted coframe θ X (see Theorem 4.4) is θ X = − θ + cos 2 bτ θ , θ X = − b sin 2 bτ θ , θ X = θ + 1 b ( x + b y ) sin 2 bτ θ , and the structure equations (4.22) are d θ X = − ˆ π ∧ ˆ π + cos 2 bτ ˇ π ∧ ˇ π + b ˆ π ∧ θ X ,d θ X = − b sin 2 bτ ˇ π ∧ ˇ π − ˆ π ∧ θ X ,d θ X = 1 b ( x + b y ) sin 2 bτ ˇ π ∧ ˇ π − ˆ π ∧ θ X − b θ X ∧ θ X . (6.24)This coframe is actually 4-adapted.We labeled the vectors in the derived flag (6.22) so that the last vector X spans the derived algebra of the Vessiot algebra. By doing so we are assuredthat the structure equations (6.24) are of the precise form (4.68a)-(4.68b). Theanalysis at this point refers back to Case II and the structure equations (4.59),as applied to just the first two equations in (6.24). The matrices M and R (seeequations (4.60) and (4.61)) are found to be M = " b ˆ π − ˆ π and R = cos b ˆ I − b sin b ˆ I b sin b ˆ I cos b ˆ I . (6.25)We then compute the 2-forms χ i and the 1-forms φ i (see (4.63)) to be χ = − cos b ˆ I ˆ π ∧ ˆ π , χ = − b sin b ˆ I ˆ π ∧ ˆ π ,φ = − ˆ I cos b ˆ I ˆ π , φ = − b ˆ I sin b ˆ I ˆ π , (6.26)so that the forms (4.69) are ˆ θ = cos b ˆ I θ X − b sin b ˆ I θ X − ˆ I cos b ˆ I ˆ π , ˆ θ = 1 b sin b ˆ I θ X + cos b ˆ I θ X − b ˆ I sin b ˆ I ˆ π . (6.27)The structure equations (6.24) are now reduced to the form (4.70). The finalrequired frame change (4.74) leads to the ˆ5-adapted coframe ˆ θ = ˆ θ , ˆ θ = ˆ θ , ˆ θ = θ X + ˆ I cos b ˆ I ˆ θ + b ˆ I sin b ˆ I θ + 12 ( ˆ I ) ˆ π (6.28) xamples d ˆ θ = cos b ˇ I ˇ π ∧ ˇ π , d ˆ θ = 1 b sin b ˇ I ˇ π ∧ ˇ π ,d ˆ θ = 1 b (cid:0) ( x + b y ) sin 2 bτ + b ˆ I cos 2 bτ (cid:1) ˇ π ∧ ˇ π − b ˆ θ ∧ ˆ θ . (6.29)The diffeomorphism Φ( x, y, u, p, q, s, τ ) = ( x, − y, u, p, − q, − s, τ ) is an invo-lution for (6.18) in the sense of Remark 2.13 and can therefore be used to findthe ˇ5 adapted coframe.The forms (5.30) are ˆ ω = ˆ θ + ˇ I cos b ˇ I ˇ π , ˆ ω = ˆ θ + ˇ I b sin b ˇ I ˇ π , ˆ ω = ˆ θ + A ˇ π , (6.30)where A = ˇ I ( ˆ I cos 2 bτ + 1 b ( x + b y ) sin 2 bτ − ˇ I ).The Vessiot group for this example is the matrix group z z + b z z /
20 1 b z (6.31)with multiplication z = z + z , z = z + z , z = z + z − b z z − z z ) (6.32)and left invariant forms ˆ ω = dz , ˆ ω = dz , ˆ ω = dz + b z dz − z dz ) . (6.33)The map ρ (Theorem 5.1) is then found to be z = 1 b ( − x sin bs sin bτ + 2 b y cos bs cos bτ − b ˆ I sin b ˆ I + b ˇ I sin b ˇ I ) ,z = 1 b ( 2 bx cos bs sin bτ + 2 yb sin bs cos bτ + ˆ I b cos b ˆ I − ˇ I b cos ˇ I ) , (6.34) z = 2 u − b (sin 2 bτ − bτ cos 2 bτ )( x − b y ) − px cos bτ − qy sin bτ − b ˆ I ˇ I sin 2 bτ. Finally, the combination of equations (6.19), (6.32) and (6.34) leads to the xamples x = 12 ( x + x + b ( y − y )) − sin b ( τ − τ )sin b ( τ + τ ) ξ,y = 12 b ( x − x + b ( y + y )) − cos b ( τ − τ ) b cos b ( τ + τ ) ξ,u = u + u + 2 p sin 2 bτ − p sin 2 bτ )sin 2 b ( τ + τ ξ + 1 b (cid:18) τ sin (2 bτ )sin (2 b ( τ + τ )) + 2 τ sin (2 bτ )sin (2 b ( τ + τ )) − sin 2 bτ sin 2 bτ b sin 2 b ( τ + τ ) (cid:19) ξ ,p = p + p + 2 τ sin 2 bτ − τ sin 2 bτ b sin 2 b ( τ + τ ) ξ,q = − b p + b p − τ sin 2 bτ + τ sin 2 bτ sin 2 b ( τ + τ ) ξ,s = − τ + τ , τ = τ + τ , where ξ = 12 ( x − x − b ( y + y )) . (6.35)The restriction of ˆ V to the manifold M = { ˆ I = ˆ I = 0 } gives ˆ W = { du − p dx + b dy , dp − /b (tan bτ − bτ ) dx + τ dy , (6.36) τ dx − b ( bτ + cot( bτ )) dy − b dp } . (6.37)This is a rank 3 Pfaffian system on a 5 manifold whose derived flag has dimen-sions [3 , , T in two variables. For b = 0, thistensor vanishes while for b = 0 we find that T is the 4-th symmetric power ofa 1-form. In accordance with Cartan’s result the symmetry algebra of ˆ W when b = 0 is the 14 dimensional exceptional Lie algebra g and, indeed, it is notdifficult to transform ˆ W to the canonical Pfaffian system for the Hilbert-Cartanequation z ′ = ( y ′′ ) . For b = 0 the symmetry algebra of ˆ W is the 7 dimensionalsolvable Lie algebra with infinitesimal generators { ∂ x , ∂ y , ∂ u , x ∂ x + y ∂ y + 2 u ∂ u + p ∂ p , ( x − b y ) ∂ u − ∂ p , Y , Y } , (6.38)where Y = [ ∂ y , Y ] and Y =( x + b y ) (cid:0) b cot bτ ∂ x − b tan bτ ∂ y + 2 p b csc bτ ∂ u + 2 τ csc 2 bτ ∂ p (cid:1) + 2 x y ∂ u + ( y − x b ) ∂ p − ∂ τ . (6.39) xamples b = 0 may be transformed into Cartan [7], page170, equation (5’). Example 6.3.
For our next example, let G be an n -parameter matrix groupand, for the mapping ( x, y ) → U ( x, y ) ∈ G , consider the system of differentialequations U xy = U x U − U y . (6.40)The general solution to these equations is well-known to be U ( x, y ) = A ( x ) B ( y ),with A ( x ) , B ( y ) ∈ G . In the case when U is a 1 × v xy = 0 under the change of variable u = exp( v ). We showhow our integration method leads directly to the general solution and, in theprocess, we calculate the Vessiot algebra of (6.40) to be the Lie algebra of G .The Pfaffian system for (6.40) is I = { Θ , Θ , Θ } , whereΘ = dU − U x dx − U y dy, Θ = dU x − U xx dx − U x U − U y dy, Θ = dU y − U x U − U y dx − U yy dy. (6.41)The first integrals for the singular systems are ˆ I = y, ˆ I = U − U y , ˆ I = D y ( ˆ I ) = U − U yy − U − U y U − U y , ˇ I = x, ˇ I = U x U − , ˇ I = D x ( ˇ I ) = U xx U − − U x U − U x U − , (6.42)and our 0-adapted coframe for I is { Θ , d ˆ I , d ˆ I , ˆ η, d ˇ I , d ˇ I , ˇ η } , where ˆ η = d ˆ I − ˆ I d ˆ I = U − Θ − U − Θ ˆ I , and ˇ η = d ˇ I − ˇ I d ˇ I = Θ U − − ˇ I Θ U − . The structure equations are d Θ = d ˆ I ∧ ( U ˆ η + Θ ˆ I ) + d ˇ I ∧ ( ˇ η U + ˇ I Θ) . (6.43)This coframe satisfies (4.13) and is therefore 2-adapted.The next step is to eliminate either the d ˇ I ∧ ( ˆ I Θ) or the d ˆ I ∧ (Θ ˇ I ) termsin (6.43). By inspection, we see that the forms Θ X = U − Θ and Θ Y = Θ U − provide us with the required 4-adapted coframes, with structure equations d Θ X = d ˆ I ∧ ˇ η + d ˇ I ∧ ( U − ˆ η U ) − Θ X ∧ Θ X + d ˆ I ∧ (Θ X ˆ I − ˆ I Θ X ) ,d Θ Y = d ˆ I ∧ ( U ˇ η U − ) + d ˇ I ∧ ˇ η + Θ Y ∧ Θ Y − d ˇ I ∧ (Θ Y ˇ I − ˇ I Θ Y ) . (6.44) xamples X and Θ Y are Lie algebra valued, these structure equationsshow that the Vessiot algebra for (6.40) is the Lie algebra of G (Theorem 4.6).The final coframe adaptation in Section 4.4 is given by ˆ Θ = Θ X + ˆ I d ˆ I = U − dU − U − U x dx and ˇ Θ = Θ Y + ˇ I d ˇ I = dU U − − U y U − dy, (6.45)with structure equations d ˆ Θ = d ˇ I ∧ ( U − ˇ η U ) − ˆ Θ ∧ ˆ Θ and d ˇ Θ = d ˆ I ∧ ( U ˆ η U − ) + ˇ Θ ∧ ˇ Θ . (6.46)The form (5.30) are then found to be precisely the left and right invariant formson G , that is, ˆ ω = U − dU and ˇ ω = dU U − , (6.47)so that the map ρ constructed in Theorem 5.1 is simply ρ ( x, y, U, . . . ) = U .With respect to coordinates x, U , U x , U xx on the M = { ˆ I a = 0 } andcoordinates y, U y , U yy on the level set M = { ˇ I a = 0 } , the Pfaffian systems ˆ W and ˇ W are ˆ W = { dU − U x dx, dU x − U xx dx } and ˇ W = { dU − U y dy, dU y − U yy dy } and the superposition formula is U = U U , U x = U x U , U y = U U y , U xx = U xx U , U yy = U U yy . Example 6.4.
It is an open problem to determine which scalar Darboux inte-grable equations admit generalizations wherein the dependent variable U takesvalues in an arbitrary non-commutative, finite dimensional algebra A . Here aretwo such examples which provide us with many Darboux integrable systemsamenable to the methods presented in this paper. I . U xy = ( U x + I ) U − U y II . U xy = U x ( U − y ) − U y + U y ( U − x ) − U x . The first integrals for the singular systems (excluding x and y ) and generalsolutions are We remark that the solution to II given by Vessiot [28] (equations C II and (70), pages 6and 45) in the scalar case is incorrect. xamples I . ˆ I = ( U x + I ) U − , ˆ I = D x ( ˆ I ) , ˇ I = U − y U yy ,U = ( F ′ ) − ( − F + G ) , II . ˆ I = ( U − x ) − [ U xx U − x ( U − x ) − U x + I ] , ˇ I = ( U − y ) − [ U yy U − y ( U − y ) − U y + I ] ,U = ( xF ′ + yG ′ − F − G ) ( F ′ + G ′ ) − , where F = F ( x ) and G = G ( y ) take values in A . For both systems I and II the Vessiot algebra is the tensor product of A with the Vessiot algebra for thecorresponding scalar equation. We conjecture that all equations of Moutard type([15] Volume II, page 250, equation 19) admit non-commutative generalizations. Example 6.5.
Some of the simplest examples of Darboux integrable systemscan be constructed by the coupling of a Darboux integrable scalar equation toa linear or Moutard-type equation. As examples, we give I . u xy = e u , v xy = n ( n + 1) e u v, II . u xy = e u u y , v xy + (( n − α ) e u + αu x ) v x = 0 , III . u xy = e u u y , v xy − e u v y + ( n + 1) u y v x + ( n + 1)! e u u y = 0 , IV . u xy = e u u x v xy + ( e v ) x − ( nBe − v ) y + ( n − B = 0 , where B = e u u x and n is a positive integer. The system I appears in [19](page116); systems II – IV do not seem to have appeared in the literature.For each of these systems the restricted Pfaffian systems ˆ W and ˇ W are jetspaces for two functions of a single variable ( x or y ). The Vessiot algebra for I isthe semi-direct product of sl (2) and an Abelian Lie algebra of dimension 2 n + 1,as determined by the (unique) (2 n + 1)-dimensional irreducible representationof sl (2). The infinitesimal action of the Vessiot group for I , restricted to ˆ W or ˇ W , is the action listed in [14] as number 27 (where now the variables x , y in[14] serve as the dependent variables for the jet spaces ˆ W and ˇ W ).For II the Vessiot algebra is a semi-direct product of the 2-dimensionalsolvable algebra with an ( n + 1)-dimensional Abelian algebra. The infinitesimalaction of the Vessiot group, restricted to ˆ W or ˇ W , is number 24 in [14]. Theinfinitesimal Vessiot groups for III and IV have dimensions n + 3 and n + 4 andcoincide, respectively, with numbers 25 and 26 in [14]. xamples n = 1 the general solutions to these systems are I . u = 12 ln F ′ G ′ ( F + G ) , v = 2 F − G F + G − F ′ F ′ + G ′ G ′ II . u = ln F ′ G − F , v = 1 F α (cid:0) F − G − ( F − G ) G ′ G ′ (cid:1) , III . u = ln F ′ G − F , v = F − G ( F − G ) − G ′ ( F − G ) G ′ − ln( G − F ) , IV . u = ln G ′ F − G ,v = ln (cid:0) ( G ′ ( F − G ) − G ′ ( F − G ) ) F ′ ( F ′ ( F − G ) − F ′ ( F − G )) ( F − G ) (cid:1) . The general solutions for arbitrary n can be obtained in closed compact formby the method of Laplace.In addition, any non-linear Darboux integrable system can be coupled to itsformal linearization to obtain another Darboux integrable system. For example,if we prolong the partial differential equations V . u xx u yy + 1 = 0 , v xx − u yy v yy = 0to order 3 in the derivatives of u , we obtain a rank 8 Pfaffian system on a 14-dimensional manifold which is Darboux integrable, with 4 first integrals for eachassociated singular Pfaffian system.The Vessiot algebra is Abelian and of dimension 6. The two Lie algebras ofvector fields dual to the forms ˆ θ and ˇ θ for the 5-adapted coframe coincide andare given by { ∂ y , ∂ u , x∂ u + ∂ u x , ∂ v , x∂ v + ∂ v x , u y ∂ v + u xy ∂ v x + u yy ∂ v y + u xyy ∂ v xy + u yyy ∂ v yy } . In accordance with Remark 4.10, these vector fields are also infinitesimal sym-metries for V . In terms of the arbitrary functions φ ( α ) and ψ ( β ) appearing inthe general solution to 3 u xx u yy + 1 = 0 (Goursat([15] Vol. 2, page 130), thegeneral solution for v is given implicitly as x = 12 φ ′′ − ψ ′′ α − β , y = 12 ( β − α )( φ ′′ + ψ ′′ ) + φ ′ − ψ ′ , t = 1 α − β , (6.48) v = F + G − φ ′′ − ψ ′′ + ( β − α ) φ ′′′ ( β − α ) φ ′′′′ F ′ − φ ′′ − ψ ′′ + ( β − α ) ψ ′′′ ( β − α ) ψ ′′′′ G ′ , (6.49)where F = F ( α ) and G = G ( β ). xamples Example 6.6.
Although we are unaware of an explicit general proof it is gen-erally acknowledged that the Toda lattice systems (see, for example, [19], [22])are Darboux integrable. In this example we shall check that the B Toda latticeequations u xy = 2 e u − e v , v xy = − e u + 2 e v (6.50)are Darboux integrable and find the closed-form, general solution. We use thisexample to illustrate a slightly different computational approach, one basedupon the symmetry reduction interpretation of the superposition formula givenin Sections 3 and 5.3.The canonical Pfaffian system for (6.50) satisfies our definition of Darbouxintegrable upon prolongation to 4-th order, that is, as a rank 14 Pfaffian system I on a 20 dimensional manifold. The diffeomorphism x ↔ y interchanges thesingular Pfaffian systems ˆ V and ˇ V . The first integrals for the singular Pfaffiansystem ˆ V (containing dx ) are ˆ I = x, ˆ I = v xx + 23 u xx − u x v x − u x − v x , ˆ I = D x ˆ I , ˆ I = D x ˆ I , and ˆ I = u xxxx + 2 v xxxx − v x v xxx − u xx u x − u x u xx v x + 18 u x + 12 u xx +12 v x u x − v x u xx − v xx u x + v xx u xx + 12 u x v x − u x v xxx . Let ˆ W be the restriction of ˆ V (or, equivalently, I ) to M = { ˆ I a = 0 } . Wefind that ˆ W is a rank 12 Pfaffian system on a 15 manifold. The derived flagof ˆ W has dimensions [12 , , , , , , ,
0] while the dimensions of the space ofCauchy characteristics for these derived Pfaffian sytems are [0, 2, 4, 6, 8, 10, 12,15]. By using the invariants of these Cauchy characteristics as new coordinates,we are able to write ˆ W in the canonical form ˆ W = { du − ˙ u du , d ˙ u − ¨ u du , . . . , du − u ′ dx, du ′ − du ′′ dx, . . . } . (6.51)Here there are 7 contact forms for u and 5 for u . In these coordinates theintegral manifolds of ˆ W are given by u = F ( x ), u ′ = F ′ ( x ), . . . and u = F ( F ( x )), ˙ u = ( ˙ F )( F ( x )), . . . . xamples M , now takes aremarkably simple and well-known form – it is the infinitesimal conformal group o (3 ,
2) acting on the 3-dimensional space ( u , u , ˙ u ) by contact transforms.(See, for example, Olver [21] page 473.) Explicitly, the generating functions forthe infinitesimal action of the Vessiot group on M are Q = [ u , − u + u ˙ u , − u ˙ u + 2 u ˙ u , , ˙ u ,
12 ˙ u , u , − u ˙ u − u + 2 u u ˙ u , − u u + u ˙ u , u ] . (6.52)To obtain the infinitesimal generator ˆ X q corresponding to a function q ∈ Q , firstconstruct the vector field X q = − q ˙ u ∂ u + ( q − ˙ u q ˙ u ) ∂ u + ( q u + ˙ u q u ) ∂ u andthen prolong X q to the vector field ˆ X q on M by requiring it to be a symmetryof ˆ W . We remark that the basis for o (3 ,
2) so obtained is the canonical Chevalleybasis in the sense that the first 2 vectors define the Cartan subalgebra, the next4 correspond to the positive roots, and the last 4 to the negative roots.By Theorem 5.13, the superposition formula for the B Toda lattice cantherefore be constructed from the joint invariants for the diagonal action of theconformal algebra o (3 ,
2) on M × M . We use coordinates [ y, v , v ′ , v ′′ . . . , v , ˙ v , ¨ v , . . . ]on M . To compactly describe these joint invariants we first calculate the jointdifferential invariants in the variables { u , u ′ , v , v ′ , ˙ u , ˙ v , ¨ u , ¨ v , ...u , ...v } for the 7 dimensional subalgebra of o (3 ,
2) generated by Q , Q , Q , Q , Q , Q , Q . These are J = v ′ ( ...u ) / ( ...v ) / / (¨ u − ¨ v ) , J = u ′ ( ...u ) / ( ...v ) / / (¨ u − ¨ v ) ,J = (¨ u v − ¨ u u + ˙ u − ˙ v )( ...u ) / ( ...v ) / / (¨ v − ¨ u ) ,J = (¨ v u − ¨ v v + ˙ v − ˙ u )( ...u ) / ( ...v ) / / (¨ v − ¨ u ) , and (6.53) J = − (¨ u ¨ v v − u ¨ v v u + ¨ u ¨ v u − u ˙ u u + 2¨ u ˙ u v + 2¨ u u − u v − v ˙ v v + 2¨ v ˙ v u + 2¨ v v − v u (6.54)+ ˙ u − u ˙ v + ˙ v ) ...u ...v / (2(¨ v − ¨ u ) ) . Then, in terms of these partial invariants the (lowest) order joint differentialinvariants for o (3 ,
2) are K = − J J ( J J − J ) ( J J − J ) and K = − J J ( J J − J )( J J − J ) (6.55) xamples B Toda lattice are u = ln( K /
4) and v = ln(2 K ) , where (6.56) u = F ( x ) , u = F ( F ( x )) , v = G ( y ) , v = G ( G ( y )) . (6.57)It is hoped that a more transparent representation of these solutions, similar tothat available for the A n Toda lattice will be be obtained.
Example 6.7.
Let P denote the 2-dimensional Minkowski plane with metric dx ⊙ dy and let N be a pseudo-Riemannian manifold with metric g . A map-ping ϕ : P → N which is a solution to the Euler-Lagrange equations for theLagrangian L = g ( ∂ϕ∂x , ∂ϕ∂y ) dx ∧ dy (6.58)is called a wave map. There are precisely two inequivalent, non-flat metrics (upto constant scaling) in 2 dimensions, namely g = 11 + e − u ( du + dv ) and g = 11 − e − u ( du + dv ) (6.59)which define Darboux integrable wave maps at the 2-jet level (that is, withoutprolongation). Surprizingly, these metrics are not constant curvature. It is notdifficult to check that under the change of coordinates x = x − y, t = x + y, θ = arctan( √ e u − , χ = v/ g . The Vessiot algebras for the wave map equations for g and g are sl (2) × R and so (3) × R respectively.The wave map equations for g are u xy = v x v y − u x u y e u + 2 , v xy = − u x v y + u y v x e u + 2 . (6.61)The standard encoding of these equations as a Pfaffian system results in a rank6 Pfaffian system on a 12 manifold. There are 4 first integrals for each singularPfaffian system – for the singular Pfaffian system ˆ V containing dx the firstintegrals are ˆ I = x , ˆ I = e u ( u x + v x )1 + e u , ˆ I = D x ˆ I , and ˆ I = v x u x + v x u xx − u x u x + v x v xx − (1 + 2 e u ) v x e u . (6.62) xamples F ( x ), F ( x ), G ( y ), G ( y ) to be 2 e u = − p A p B + AB sin(∆) (6.63)and v = F ( x ) − F ( x ) + G ( y ) − G ( y ) (6.64)+ arctan AF ′ √ A A ′ ! + arctan BG ′ √ B B ′ ! + arctan AB ′ cos(∆) + G ′ B √ A √ B + G ′ AB (1 + B ) sin(∆) G ′ AB √ B cos(∆) − BB ′ √ A − AB ′ √ B sin(∆) ! + arctan A ′ B cos(∆) − F ′ A √ A √ B − F ′ AB (1 + A ) sin(∆) F ′ AB √ A cos(∆) + AA ′ √ B + A ′ B √ A sin(∆) ! , where A ( x ) = s(cid:18) F ′ F ′ (cid:19) − F ′ F ′ , B ( y ) = s(cid:18) G ′ K ′ (cid:19) − G ′ G ′ , ∆ = F − G . (6.65)Note that F ′ ( x ) = F ′ ( x )(1 + p A ( x ) ) , G ′ ( y ) = G ′ ( y )(1 + p B ( y ) ) . (6.66) xamples Example 6.8.
We turn now to some simple examples of overdetermined sys-tems for a single unknown function of 3 independent variables, beginning withthe system u xz = uu x , u yz = uu y . (6.67)The structure equations for the canonical encoding of this system as a rank 4Pfaffian system I = { α , α , α , α } on an 11-manifold are (modulo I ), dα ≡ dα ≡ ˆ π ∧ ˆ π + π ∧ π , dα ≡ ˆ π ∧ ˆ π + π ∧ π , dα ≡ ˇ π ∧ ˇ π , (6.68)where ˆ π = dx , ˆ π = dy , ˇ π = dz , ˆ π = du xx − ( u xx u + u x ) dz, ˆ π = du xy − ( u xy u + u y u x ) dz, ˆ π = du yy − ( u yy u + u y ) dz, ˇ π = du zz − ( u z + u )( u x dx + u y dy ) . (6.69)The first integrals for the singular Pfaffian systems ˆ V = I ∪ { ˆ π , . . . , ˆ π } and ˇ V = I ∪ { ˇ π , ˇ π } are ˆ I = x , ˆ I = y , ˇ I = z , ˆ I = u y u x , ˆ I = D x ˆ I , ˆ I = D y ˆ I , ˇ I = u z − u / , ˇ I = D z ˇ I . (6.70)The form ˆ π is not in ˆ V ∞ + ˇ V and therefore (6.67) is not Darboux integrableon the 2-jet. The prolongation of (6.67) defines a decomposable rank 8 Pfaffiansystem I [1] = { α , . . . , α } on a 16 dimensional manifold. In addition to thefirst integrals (6.70), we now also have ˆ I = D x ˆ I , ˆ I = D xy ˆ I , ˆ I = D y ˆ I , ˆ I = u xxx u x − u xx u x , ˇ I = D z ˇ I (6.71)and the conditions (1.8)–(1.10) for Darboux integrability of I [1] are now satisfied.A 1-adapted coframe is ˆ σ = d ˆ I , ˆ σ = d ˆ I , ˆ σ = d ˆ I , ˆ σ = d ˆ I , ˆ σ = d ˆ I , ˆ σ = d ˆ I , ˇ σ = d ˇ I , ˇ σ = d ˇ I , ˆ η = 1 u x α − u y u x α = d ˆ I − ˆ I d ˆ I − ˆ I d ˆ I , ˆ η = 1 u x α − u y u x α − u xx u x α − u xy u x − u y u xx u x α = d ˆ I − ˆ I d ˆ I − ˆ I d ˆ I , ˆ η = 1 u x α − u y u x α − u xy u x α − − u y u xy + u yy u x u x α = d ˇ I − ˆ I d ˆ I − ˆ I d ˆ I , ˇ η = α − u α = d ˇ I − ˇ I d ˇ I , ˇ η = α − u α − u z α = d ˇ I − ˇ I d ˇ I ,θ = α = du − u x dx − u y dy − u z dz,θ = α = du x − u xx dx − u xy dy − uu x dz,θ = α = du xx − u xxx dx − u xxy dy − ( uu xx + u x ) dz. xamples ˆ U ( ∞ ) = ˆ U ∪ { X , X , X } and ˇ U ( ∞ ) = ˇ U ∪ { Y , Y , Y } , where X = − u x ˆ I ∂ θ − ( u xx ˆ I + u x ˆ I ) ∂ θ − u xx ˆ I + 4 u xx u x ˆ I − u x ˆ I ˆ I u x ∂ θ ,X = − u x ∂ θ , X = − u x ∂ θ − u xx ∂ θ − u xx + 2 u x ˆ I u x ∂ θ , Y = − ∂ θ , (6.72) Y = u∂ θ + u x ∂ θ + u xx ∂ θ , Y = ( − u ˇ I ) ∂ θ − uu x ∂ θ − ( u x + uu xx ) ∂ θ . For the computations of Section 4.3 we use the base point defined by setting u = 0, u x = 1, u xx = 0, ˆ I = 1, and all other first integrals (6.70)–(6.71) to 0,The matrices (4.33) and (4.36) are then given by P = ˆ I ˆ I ˆ I ˆ I − ˆ I ˆ I − ˆ I and R = − ˇ I . (6.73)From these matrices we calculate the 4 adapted coframes θ X = u xx u x θ − u x θ , θ Y = − u x + uu xx u x θ + uu x θ ,θ X = − u xx u x θ + 2 u xx u x θ − u x θ , θ Y = u xx u x θ − u x θ , (6.74) θ X = − u x θ , θ Y = − θ + u (2 u x + uu xx )2 u x θ − u u x θ . The Vessiot algebra is sl (2). Because this algebra is semi-simple, we can useCase I of Section 4.4, Theorem 5.1, and (6.74) to directly determine the map ρ : M → Aut( sl (2) as ρ ( u, u x , u xx ) = λ = u x − uu xx u x − uu x − u xx ( uu xx − u x )2 u x u xx u x u x u xx u x u ( uu xx − u x )2 u x u u x ( uu xx − u x ) u x . (6.75)To obtain the superposition formula we introduce local coordinates ( z , w , w x , w xx , w z , w zz , w zzz ) for M and ( x , y , v , v x , v y , v z , v xx , v xy , v yy , v yz , v xxx , xamples v xxy , v xyy , v yyy , v yyz ) for M . The inclusions ι : M → M and ι : M → M are fixed by ι ( w I ) = u I , ι ∗ ( ˆ I a ) = 0 , ι ( v I ) = u I , ι ∗ ( ˇ I a ) = 0 . The superposition formula is then found by solving the equations ι ∗ ( ˇ I a ) = ˇ I a , ι ∗ ( ˆ I a ) = ˆ I a , ρ ( u, u x , u xx ) = ρ ( w, w x , w xx ) · ρ ( v, v x , v xx ) (6.76)for the coordinates of M . We find that u = w − vw x − w x + vw xx . (6.77)Finally, we calculate the integral manifolds for ˆ W and ˇ W . It is immediatethat ˇ W is the canonical Pfaffian system on J ( R , R ) and hence the integralmanifolds are defined by v = V ( x, y ). The last non-zero form in the derived flagfor ˆ W = { dw − w z dz, dw z − w zz dz, dw zz − w zzz dz, dw x − w x wdz,dw xx + ( − w x − ww xx ) dz } yields the Pfaffian equation dw xx − w xx w x dw x − w x dz = 0from which it follows that w x = G ′ ( z ) and w xx = G ( z ) G ′ ( z ). One of theremaining equations in ˆ W then gives w = G ′′ ( z ) /G ′ ( z ). On replacing V ( x, y )by − /F ( x, y ), the superposition formula (6.77) becomes u ( x, y, z ) = G ′′ ( z ) G ′ ( z ) − G ′ ( z ) F ( x, y ) + G ( z ) , (6.78)which gives the general solution to (6.67).We continue this example by considering two variations on (6.67). First weobserve that to (6.67) we may add any equation of the form F ( x, y, u y u x , u y u xx − u x u xy u x , u y u xy − u x u tyy u x ) = 0 (6.79)to obtain a rank 4 Pfaffian system I = { α , α , α , α } on a 10 dimensionalmanifold. The structure equations become ( mod I ) dα ≡ dα ≡ ˆ π ∧ ˆ π , dα ≡ ˆ π ∧ ˆ π , dα ≡ ˇ π ∧ ˇ π (6.80) xamples s = 3.For example, consider the system of 3 equations u y u xy − u x u yy = 0 , u xz = uu x , u yz = uu y (6.81)The foregoing calculations can be repeated, almost without modification, toarrive at the same superposition formula (6.77) – the only difference is that now ˇ W is the (prolonged) canonical Pfaffian system for the equation v y v xy − v x v yy =0, an equation which is itself Darboux integrable. Thus, in more complicatedsituations, the method of Darboux, can be used to integrate the Pfaffian systems ˆ W and ˇ W and the superposition formula for the original system is given by acomposition of superposition formulas. In the case of the present example,the calculation of the first integrals for ˇ W reveals that this system is contactequivalent to the wave equation (via X = x , Y = v , V = y , V X = − v x /v y , V Y = 1 /v y ) and leads to the parametric solution x = σ, y = f ( σ ) + g ( τ ) , u = G ′′ ( z ) G ′ ( z ) − G ′ ( z ) τ + G ( z ) . (6.82)Our second variation of (6.67) is obtained by the differential substitution u x = exp( v ). This leads to the equations v xz = exp( v ) and v yzz = v yz v z . (6.83)It is surprising that the canonical Pfaffian system for these equations (obtainedby the restriction of the contact ideal on J ( R , R )) does not define a decom-posable Pfaffian system. The following theorem resolves this difficulty. Theorem 6.9.
The system of differential equations u xz = F ( x, y, z, u, u z , u yz , u zz ) ,u yzz = G ( x, y, z, u, u x , u y , u z , u yz , u zz , u yyz , u zzz ) (6.84) determines the rank 4 Pfaffian system α = du − u x dx − u y dy − u z dz, α = du z − F dx − u yz dy − u zz dz,α = du yz − D y ( F ) dx − u yyz dy − Gdz,α = du zz − D z ( F ) dx − Gdy − u zzz dz, (6.85) on an 11 dimensional manifold. If the compatiblity conditions for (6.84) hold,then this Pfaffian system is decomposable and involutive with Cartan characters s = 1 and s = 1 . xamples v ( x, y, z ) = ln (cid:0) F x ( x, y ) G ′ ( z )( F ( x, y ) + G ( z )) (cid:1) . (6.86) References [1] I. M. Anderson and M. E. Fels,
Exterior Differential Systems with Symmetry , Acta. Appl.Math. (2005), 3–31.[2] I. M. Anderson and M. Jur´aˇs, Generalized Laplace Invariants and the Method of Darboux ,Duke J. Math (1997), 351–375.[3] B. M. Barbashov, V. V. Nesterenko, and A. M. Chervyakov, General solutions of nonlin-ear equations in the geometric theory of the relativistic string , Commun. Math Physics (1982), 471–481.[4] F. De Boer, Application de la m´ethode de Darboux `a l’int´egration de l’´equationdiff´erentielle s = f(r,t). , Archives Neerlandaises (1893), 355–412.[5] R. L Bryant and P. A. Griffiths, Characteristic cohomology of differential systems (I) ,Selecta Math. (N.S.) (1995), 21–112.[6] R. L. Bryant, P. A. Griffiths, and L. Hsu, Hyperbolic exterior differential systems andtheir conservation laws, Parts I and II , Selecta Math., New series (1995), 21–122 and265–323.[7] ´E Cartan, Les syst`emes de Pfaff `a cinq variables et les ´equations aux d´eriv´ees partiellesdu second ordre , Ann. Sci. ´Ecole Norm. (1910), no. 27, 109–192.[8] , Sur les syst`emes en involution d’´equations aux d´eriv’ees partielles du secondordre `a une fonction inconnue de trois variable ind`ependantes , Bull Soc. Math France (1911), 352-443.[9] G. Darboux, Le¸cons sur la th´eorie g´en´erale des surfaces et les applications g´eom´etriquesdu calcul infinit´esimal , Gauthier-Villars, Paris, 1896.[10] P. T. Eendebak,
Contact Structures of Partial Differential Equations , 2007.[11] H. Flanders,
Differential Forms with Applications to the Physical Sciences , Dover, NewYork, 1963.[12] M. E. Fels and P. J. Olver,
Moving coframes I. A practical algorithm , Acta. Appl. Math. (1998), 161-213.[13] A. Forsyth, Theory of Differential Equations, Vol 6 , Dover Press, New York, 1959.[14] A. Gonz´alez-L´opez, N. Kamran, and P. J. Olver,
Lie algebras of vector fields in the realplane , Proc. London Math. Soc. (1992), 339-368.[15] E. Goursat, Lecon sur l’int´egration des ´equations aux d´eri´ees partielles du second ordre´a deux variables ind´ependantes, Tome 1, Tome 2 , Hermann, Paris, 1897. xamples [16] E. E. Goursat, Recherches sur quelques ´equations aux d´eri´ees partielles du second ordre ,Ann. Fac. Sci. Toulouse (1899), 31–78 and 439–464.[17] P. Griffiths, On Cartan’s method of Lie groups and moving frames as applied to unique-ness and existence questions in differential geometry , Duke Math. J. (1974), 775–814.[18] S.S. Kobyashi and K. Nomizu, Foundations of Differential Geometry , John Wiley, 1963.[19] A. N. Leznov and M. V. Saleliev,
Representation theory and integration of nonlinearspherically symmetric equations to gauge theories , Commun. Math. Phys. (1980),111-118.[20] , Two-dimensional exactly and completely integrable dynamical systems , Com-mun. Math. Phys. (1983), 59–75.[21] P. J. Olver, Equivalence, Invariants, and Symmetry (1995).[22] V. V. Sokolov and A. V. Ziber,
On the Darboux integrable hyperbolic equations , PhysLett. A , 303–308.[23] S Sternberg,
Lectures on Differential Geometry , Second, Chelsa, New York, 1984.[24] O. Stormark,
Lie’s structural approach to PDE systems (2000).[25] V. S. Varadarajan, Lie Groups, Lie Algebras and Their Representations , Springer-Verlag,New York, 1984.[26] P. J. Vassiliou,
Vessiot structure for manifolds of ( p, q ) -hyperbolic type: Darboux integra-bility and symmetry , Trans. Amer. Math. Soc. (2001), 1705–1739.[27] E. Vessiot, Sur les ´equations aux d´eriv´ees partielles du second ordre, F(x,y,z,p,q,r,s,t)=0,int´egrables par la m´ethode de Darboux , J. Math. Pure Appl. (1939), 1–61.[28] , Sur les ´equations aux d´eriv´ees partielles du second ordre, F(x,y,z,p,q,r,s,t)=0,int´egrables par la m´ethode de Darboux , J. Math. Pure Appl.21