On integrable structures for a generalized Monge-Ampere equation
Paul Kersten, Iosif Krasil'shchik, Alexander Verbovetsky, Raffaele Vitolo
aa r X i v : . [ n li n . S I] J un ON INTEGRABLE STRUCTURES FOR A GENERALIZEDMONGE-AMP`ERE EQUATION
P. KERSTEN, I. KRASIL ′ SHCHIK, A. VERBOVETSKY, AND R. VITOLO
Abstract.
We consider a 3rd-order generalized Monge-Amp`ere equa-tion u yyy − u xxy + u xxx u xyy = 0 (which is closely related to the asso-ciativity equation in the 2-d topological field theory) and describe allintegrable structures related to it (i.e., Hamiltonian, symplectic, and re-cursion operators). Infinite hierarchies of symmetries and conservationlaws are constructed as well. Introduction
Monge-Amp`ere equations [13] is one of the most interesting objects to ap-ply methods of geometrical theory of differential equation. Generalizationsof classical Monge-Amp`ere equations are discussed, e.g., in [2]. One of suchgeneralizations is the equation u yyy − u xxy + u xxx u xyy = 0 . (1)This is a third-order Monge-Amp`ere equation ([2, 13]), but this does nothelp too much in understanding its integrability properties.Equation (1) is closely related to the associativity equation in 2-d topolog-ical field theory [4] and was studied in a number of papers ([5, 6, 8, 9, 14]) andits integrability (existence of a bi-Hamiltonian structure) was established.Note though that in these papers the equation was not considered in theinitial form (1), but was rewritten as a three-component system a y = b x , b y = c x , c y = ( b − ac ) x (2)of hydrodynamical type. Of course, equations (1) and (2) are closely related,but not the same and even not equivalent being associated to each other bythe differential substitution a = u xxx , b = u xxy , c = u xyy (just like the KdV and mKdV are related by the Miura map or the Burgersand heat equations by the Cole-Hopf transformation).The aim of this paper is to attack Equation (1) directly, not reducingit to the evolutionary form, and to study the structures that arise on thisequation. To this end, we use geometrical and cohomological methods de-scribed initially in [10] and discussed in detail in a recent review paper [12]. Key words and phrases.
Monge-Amp`ere equations, integrability, Hamiltonian opera-tors, symplectic structures, symmetries, conservation laws, jet spaces, WDVV equations,2-d topological field theory.This work was supported in part by the NWO-RFBR grant 047.017.015 (PK, IK & AV),RFBR-Consortium E.I.N.S.T.E.IN grant 09-01-92438 (IK, AV & RV) and RFBR-CNRSgrant 08-07-92496 (IK & AV). ′ SHCHIK, A. VERBOVETSKY, AND R. VITOLO
These methods has been successfully applied to a number of equation (see,e.g., [7, 11]).In Section 1 we briefly recall basic notions from the geometry of jet spaces.Section 2 contains main results on the Monge-Amp`ere equation (1) (includ-ing description of Hamiltonian, symplectic, and recursion operators, as wellhierarchies of symmetries and conservation laws). In particular, we showthat Equation (1) admits a symplectic structure of the form D x (this is theonly local operator that is responsible for the integrability of the equationand it corresponds to the symplectic structure described in [3, 18]). A non-local Hamiltonian structure D − x corresponds to this operator. Other oper-ators are quite complicated and are described in Sections 2.4.1, 2.4.2, 2.5.1,and 2.5.2.All computations were done using CDIFF , a
REDUCE package for computa-tions in geometry of differential equations (see http://gdeq.org ).1.
Theoretical background
Jets and equations.
Recall that geometric approach to PDEs [1] as-sumes that an equation E together with all its prolongations (i.e., differentialconsequences) is a submanifold in the manifold J ∞ ( π ) of infinite jets of somebundle π : E → M , where M and E are smooth manifolds of dimensions n and n + m , respectively.The first manifold is the one that contains independent variables, whilethe sections of π play the role of unknown functions (fields) in E . If U ⊂ M is a coordinate neighborhood such that π | U is trivial then we choose localcoordinates x , . . . , x n in U and u , . . . , u m in the fiber of π | U . Then thecorresponding adapted coordinates u jσ , σ being a multi-index, in J ∞ ( π ) aredefined as follows. For a local section f = ( f , . . . , f m ) we set f ∗ ( u jσ ) = ∂ | σ | f j ∂x σ . Functions on J ∞ ( π ) may depend on x i and finite number of u jσ only.The vector fields D i = ∂∂x i + X j,σ u jσi ∂∂u jσ i = 1 , . . . , n are called total derivatives and differential operators in total derivatives arecalled C -differential operators .If an equation is given by the system F = 0, where F = ( F , . . . , F r ) is avector-function on J ∞ ( π ), then its infinite prolongation E is given by D σ ( F ) = 0 , | σ | ≥ , where D σ = D σ ◦ · · · ◦ D σ s for σ = σ . . . σ s . Total derivatives can berestricted to E (we preserve the same notation for these restrictions) andgenerate the Cartan distribution C . This distribution is integrable in aformal sense, i.e., [ X, Y ] ∈ C for any X , Y ∈ C , and its n -dimensionalintegral manifolds are solutions of E . NTEGRABLE STRUCTURES FOR A MONGE-AMP`ERE EQUATION 3
Symmetries.
Denote by π ∞ : E → M the natural projection. A π ∞ -vertical vector field X on E is called a symmetry of E if it preserves theCartan distribution, i.e., if [ X, C ] ⊂ C . Every symmetry of E is of the form З ϕ = X D σ ( ϕ j ) ∂∂u jσ , where summation is taken over internal coordinates on E and the vector-function ϕ = ( ϕ , . . . , ϕ m ) satisfies the equation ℓ E ( ϕ ) = 0 . Here ℓ E is the linearization operator of the vector function F restricted to E .To be more precise, we take the functions F α that define the equation E and construct the matrix C -differential operator ℓ F = P σ ∂F ∂u σ D σ . . . P σ ∂F ∂u mσ D σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P σ ∂F r ∂u σ D σ . . . P σ ∂F r ∂u mσ D σ . Since thus defined operator is a C -differential operator, it can be restrictedto E and we set ℓ E = ℓ F | E . The function ϕ is called the generating function (or section , or character-istic ) of the corresponding symmetry З ϕ and we usually make no distinctionbetween symmetries and their generating functions. The set of symmetriesis a Lie algebra over R with respect to the commutator. We denote this alge-bra by sym( E ). The bracket of vector fields induces a bracket of generatingfunctions by З { ϕ ,ϕ } = [ З ϕ , З ϕ ] , which is called the Jacobi bracket and is presented by { ϕ , ϕ } j = X α,σ (cid:16) D σ ( ϕ α ) ∂ϕ j ∂u ασ − D σ ( ϕ α ) ∂ϕ j ∂u ασ (cid:17) in coordinates.1.3. Conservation laws.
Consider the space Λ ( E ) of differential 1-formson E . It consists of finite sums ω = X i A i dx i + X j,σ B σj du jσ ,A i and B σj being smooth functions on E . The space Λ ( E ) splits naturallyinto the direct sum Λ ( E ) = Λ h ( E ) ⊕ Λ v ( E ) , (3)where Λ h ( E ) = { ω ∈ Λ ( E ) | ω = X A i dx i } is the subspace of horizontal forms, while Λ v ( E ) is generated by the differen-tial forms ω jσ = du jσ − P i u jσi dx i and is the subspace of vertical (or Cartan )forms.
P. KERSTEN, I. KRASIL ′ SHCHIK, A. VERBOVETSKY, AND R. VITOLO
Splitting (3) generates the splittingΛ s ( E ) = X p + q = s Λ qh ( E ) ⊗ Λ pv ( E ) , where Λ qh ( E ) = Λ h ( E ) ∧ · · · ∧ Λ h ( E ) | {z } q times , Λ pv ( E ) = Λ v ( E ) ∧ · · · ∧ Λ v ( E ) | {z } p times . Let us introduce the notation Λ p,q ( E ) = Λ qh ( E ) ⊗ Λ pv ( E ). Consequently,the de Rham differential d : Λ s ( E ) → Λ s +1 ( E ) splits into the sum of thehorizontal d h : Λ p,q ( E ) → Λ p,q +1 ( E )and vertical d v : Λ p,q ( E ) → Λ p +1 ,q ( E )parts and one has [ d h , d v ] = 0 . (4)Due to (4), we have a bi-complex structure on Λ ∗ ( E ) which is a particularcase of Vinogradov’s C -spectral sequence [15, 16, 17]. Denote by E p,q ( E )the cohomology of d h at the term Λ p,q ( E ). Then d v induces the differentials δ : E p,q ( E ) → E p +1 ,q ( E ) . The group E ,n − ( E ) plays a special role in the theory. Its elements arecalled conservation laws of E , while the group itself is denoted by Cl( E ).We also shall need the group E ,n − whose elements are called cosymmetries and which is denoted by cosym( E ).To proceed, we shall need additional constructions. Let P and Q be thespaces of sections of vector bundles over E . Let ˆ P = Hom( P, Λ n ( E )) andsimilar for Q . Then for any C -differential operator ∆ : P → Q its formallyadjoint ∆ ∗ : ˆ Q → ˆ P is defined by the Green formula h ∆ ∗ (ˆ q ) , p i − h ˆ q, ∆( p ) i = d h ω (ˆ q, p ) , where ω : ˆ Q × P → Λ n − ( E ) is a map which is a C -differential operator inboth arguments and h· , ·i denotes the natural pairing. If ∆ is given by thematrix (∆ ij ), where ∆ ij = P σ a σij D σ , then ∆ ∗ = (∆ ∗ ji ), where∆ ∗ ji = X σ ( − | σ | D σ ◦ a σji . In what follows, we shall assume that equation at hand satisfies the fol-lowing conditions:(1) The differentials dF j of the functions that define E are linear inde-pendent at all points of E .(2) If ∆ is a C -differential operator such that ∆ ◦ ℓ E = 0 then ∆ = 0.(3) If ∆ is a C -differential operator such that ∆ ◦ ℓ ∗ E = 0 then ∆ = 0.If an equation enjoys these conditions then the following statements arevalid : They follow from Vinogradov’s 2-Line Theorem [17].
NTEGRABLE STRUCTURES FOR A MONGE-AMP`ERE EQUATION 5 (1) The differential δ : Cl( E ) → cosym( E ) is monomorphic, i.e., δ ( ω ) =0 if and only if ω = 0.(2) The group of cosymmetries coincides with the kernel of ℓ ∗ E , i.e., ψ ∈ cosym( E ) if and only if ℓ ∗ E ( ψ ) = 0.If ω ∈ Cl( E ) is a conservation law then the cosymmetry δ ( ω ) is called its generating function (or generating section ).1.4. Differential coverings.
Let E and ˜ E be two equations. A smoothmap τ : ˜ E → E is called a morphism if it takes the Cartan distribution on ˜ E to that on E . A surjective morphism τ is said to be a covering if for anypoint θ ∈ ˜ E the differential dτ | θ maps the Cartan plane C θ ( ˜ E ) to C τ ( θ ) ( E )isomorphically. Coordinates along the fibers of τ are called nonlocal variables in the covering under consideration. Let τ ′ : ˜ E ′ → E be another covering.We say that it is equivalent to τ if there exists a morphism f : ˜ E → ˜ E ′ whichis a diffeomorphism and such that τ = τ ′ ◦ f .If D , . . . , D n are total derivatives on E and w , . . . , w r , . . . are nonlocalvariables then the covering structure is given by vector fields˜ D i = D i + X i , i = 1 , . . . , n, (5)where X i = P α X αi ∂/∂w α are τ -vertical fields that satisfy the condition D i ( X j ) − D j ( X i ) + [ X i , X j ] = 0 , ≤ i < j ≤ n. (6)A covering is Abelian if the coefficients X αi do not depend on nonlocal vari-ables. In this case, (6) amounts to D i ( X j ) − D j ( X i ) = 0 , ≤ i < j ≤ n. (7)In the particular case of one-dimensional coverings, conditions (7) definea d h -closed horizontal 1-form ω τ = P i X i dx i on E and two coverings ofthis type are equivalent if and only if the corresponding forms are in thesame cohomology class, i.e., ω τ − ω τ ′ = d h ( g ) for some function g . When n (the number of independent variables) equals two, this establishes a one-to-one correspondence between the group Cl( E ) and the equivalence classes ofone-dimensional Abelian coverings over E .If τ : ˜ E → E is a covering then symmetries of ˜ E are called nonlocal τ -symmetries of E . Note also that any C -differential operator ∆ on E can belifted to a C -differential operator ˜∆ on ˜ E . This is being done by changingtotal derivatives D i in the local representation of ∆ to ˜ D i using (5). Inparticular, the linearization operator ℓ E can be lifted in such a way andsolutions of the equation ˜ ℓ E ( ϕ ) = 0are called (nonlocal) shadows of symmetries in the covering τ . In a similarway, solutions of ˜ ℓ ∗ E ( ψ ) = 0are (nonlocal) shadows of cosymmetries in the covering τ . P. KERSTEN, I. KRASIL ′ SHCHIK, A. VERBOVETSKY, AND R. VITOLO
The ℓ -covering. Let E be an equation. Consider a new set of de-pendent variables q = ( q , . . . , q m ) (in many respects it is convenient toconsider q as an odd variable), where m is the number of unknown functionsin E , and augment the initial equation with ℓ E ( q ) = 0 . (8)The resulting system, consisting of E and equation (8), is called the ℓ -covering of E . It is an analog of the tangent bundle for the equation E . The ℓ -covering is important for the subsequent computations due to the followingproperties.1.5.1. Recursion operators for symmetries.
Consider a vector-function Φ =(Φ , . . . , Φ m ), where Φ j = P α,σ Φ jα,σ q ασ , which is a symmetry shadow in the ℓ -covering. This means that it satisfies the equation˜ ℓ E (Φ) = 0 , (9)where ˜ ℓ E is the linearization operator lifted to the ℓ -covering. Then it canbe shown that the matrix C -differential operator R Φ = ( P σ Φ jα,σ D σ ) takessymmetries of E to symmetries. In other words, R Φ is a recursion operatorfor symmetries.A recursion operator R is called hereditary if { R ϕ , R ϕ } − R (cid:0) { R ϕ , ϕ } + { ϕ , R ϕ } − R { ϕ , ϕ } (cid:1) = 0Hereditary operators possess the following property important for integra-bility: let ϕ be a symmetry such that З ϕ ( R ) − [ ℓ ϕ , R ] = 0 . Then all symmetries R i ϕ pair-wise commute, i.e., form a commutative hi-erarchy.1.5.2. Symplectic operators.
In a similar way, let us consider now a vector-function Ψ = (Ψ , . . . , Ψ r ), where Ψ j = P α,σ Ψ jα,σ q ασ (recall that r is thenumber of the functions F j that define E ), that satisfies the equation˜ ℓ ∗ E (Ψ) = 0 , (10)where ˜ ℓ ∗ E is the lift of the operator ℓ ∗ E to the ℓ -covering. Then the opera-tor S Ψ = ( P σ Ψ jα,σ D σ ) takes symmetries of the equation E to its cosym-metries.Let an operator S satisfy the condition S ∗ ◦ ℓ E = ℓ ∗ E ◦ S . (11)Then S is identified with a variational -form Ω S on E whose values onsymmetries is given by Ω S ( ϕ , ϕ ) = h S ϕ , ϕ i . This form can be considered as an element of the group E ,n − ( E ) in theterm E of the C -spectral sequence.Let now ω , ω be two conservation laws such that δω i = S ϕ i , i = 1 , , NTEGRABLE STRUCTURES FOR A MONGE-AMP`ERE EQUATION 7 for some ϕ , ϕ ∈ sym E . Then the bracket { ω , ω } S = Ω S ( ϕ , ϕ )is defined. This bracket is skew-symmetric by (11) and satisfies the Jacobiidentity if δ Ω S = 0 . (12)Operators that enjoy properties (11) and (12) are called symplectic .1.5.3. Nonlocal covectors.
Solving equations (9) or (10) leads often to trivialresults only. The reason is that symplectic and especially recursion operatorsare in many cases nonlocal, i.e., contain terms like D − x . Such terms areincorporated into solution by introducing nonlocal variables that amountsto constructing appropriate coverings. One of the ways to construct thelatter is based on the following fact (see [12]): to any cosymmetry of theequation E there corresponds a conservation law on the ℓ -covering . Wecall these conservation laws nonlocal covectors . Consequently, if n = 2 anAbelian covering corresponds to a cosymmetry. Numerous computations [7,10, 11] show that nonlocal variables arising in such a way are sufficient tofind necessary structures.1.6. The ℓ ∗ -covering. Let again E be an equation. Consider a anothernew set of dependent variables p = ( p , . . . , p r ), where r is the numberof functions F j that determine the equation E , and augment the initialequation with ℓ ∗ E ( p ) = 0 . (13)The resulting system, consisting of E and equation (13), is called the ℓ ∗ -covering of E . It is an analog of the cotangent bundle for the equation E .The ℓ ∗ -covering is also important for the subsequent computations due tothe following properties. Like the variable q in the ℓ -covering, it is convenientto consider the variable p to be odd.1.6.1. Hamiltonian operators.
Consider a vector-function Φ = (Φ , . . . , Φ m ),where Φ j = P α,σ Φ jα,σ p ασ , and assume that it is a symmetry shadow in the ℓ ∗ -covering. This means that it satisfies the equation˜ ℓ E (Φ) = 0 , (14)where ˜ ℓ E is the linearization operator lifted to the ℓ ∗ -covering. Then it canbe shown that the the matrix C -differential operator H Φ = ( P σ Φ jα,σ D σ )takes cosymmetries of E to symmetries.Solutions to (14) of special type are identified with variational bivec-tors Λ H on E . These solutions must satisfy the condition ℓ E ◦ H = H ∗ ◦ ℓ ∗ E . In this case, the operation { ω , ω } H = h H ( δω ) , δω i , ω , ω ∈ Cl( E ) , defines a skew-symmetric bracket on the space of conservation laws. Thisbracket satisfies the Jacobi identity if and only if [[Λ H , Λ H ]] = 0, where [[ · , · ]] P. KERSTEN, I. KRASIL ′ SHCHIK, A. VERBOVETSKY, AND R. VITOLO is the variational Schouten bracket (see [10, 12]) on the space of variationalmulti-vectors. In this case, one has H δ ( { ω , ω } H ) = { H δω , H δω } . Recursion operators for cosymmetries.
Finally, let us now consider avector-function Ψ = (Ψ , . . . , Ψ r ), where Ψ j = P α,σ Ψ jα,σ p ασ , and assumethat it is a cosymmetry shadow in the ℓ ∗ -covering. This means that itsatisfies the equation ˜ ℓ ∗ E (Ψ) = 0 , (15)where ˜ ℓ E is the linearization operator lifted to the ℓ ∗ -covering. Then it canbe shown that the the matrix C -differential operator ˆ R Ψ = ( P σ Ψ jα,σ D σ )takes cosymmetries of E to cosymmetries. In other words, ˆ R is a recursionoperator for cosymmetries of E .1.6.3. Nonlocal vectors.
Similar to Section 1.5, equations (14) and (15) leadoften to trivial results only. The reason is the same: Hamiltonian and re-cursion operators are in many cases nonlocal. Such terms are incorporatedinto solution by introducing nonlocal variables that amounts to constructingappropriate coverings. One of the ways to construct the latter is based onthe following fact: to any symmetry of the equation E there corresponds aconservation law on the ℓ ∗ -covering . We call these conservation laws non-local vectors . Consequently, if n = 2 an Abelian covering corresponds to asymmetry. As computations [7, 10, 11] show, nonlocal variables arising insuch a way are sufficient to find necessary structures.1.7. General computational scheme.
In all the computations we didto analyze particular equations (Equation (1) included) we adhered to thefollowing scheme: • Extension of the initial equation with a minimal set of nonlocalvariables (usually associated with conservation laws) to ensure exis-tence of nontrivial solutions to the main equations defining integrablestructures. • Computation of a minimal set of (local and nonlocal) symmetriesand cosymmetries necessary to (a) hierarchy generation and (b) con-struction of nonlocal vectors and covectors. • Extension of the ℓ -covering and and construction of symplectic struc-tures and recursion operators for symmetries. • Extension of the ℓ ∗ -covering and and construction of Hamiltonianstructures and recursion operators for cosymmetries.2. Main results
For internal coordinates on E we choose the functions u k,i = ∂ k + i u∂x k ∂y i , i = 0 , , , k = 0 , , . . . , (16)and then the total derivatives on E acquire the form D x = ∂∂x + X k ≥ ( u k +1 , ∂∂u k, + u k +1 , ∂∂u k, + u k +1 , ∂∂u k, ) , (17) NTEGRABLE STRUCTURES FOR A MONGE-AMP`ERE EQUATION 9 D y = ∂∂y + X k ≥ ( u k, ∂∂u k, + u k, ∂∂u k, + D kx ( u , − u , u , ) ∂∂u k, ) . Equation (1) is homogeneous with respect to the following weights | u | = 0 , | x | = − , | y | = − . Conservation laws and Abelian coverings.
In the sequel, we shallneed nonlocal variables that will be denoted by Q i,j . The second subscripthere indicates the weight of the variable, while first the one corresponds tothe level of nonlocality . By the latter we mean the following. The variablesof zero level are determined by local functions on E : ∂Q , ∂x = − u , u , + u , , ∂Q , ∂y = − u , u , − u , u , + u , u , ,∂Q , ∂x = 2 u , u , + u , − u , u , , ∂Q , ∂y = 2 u , u , − u , u , , and ∂Q , ∂x = u , ( u , − u , u , ) , ∂Q , ∂y = 12 ( u , − , u , , u , u , ) . The variables of level 1 are determined by local functions and by variablesof zero level: ∂Q , ∂x = 2 u , − u , , ∂Q , ∂y = 2( Q , + u , u , − u , u , ) ,∂Q , ∂x = Q , + u , u , , ∂Q , ∂y = 12 u , , and ∂Q , ∂x = Q , − u , u , u , − u , u u + 2 u , u , ,∂Q , ∂y = 4 Q , − u , u , u , − u , u , u , − u , u , u , − u , u , u , − u , u , + 2 u , u , u , + 2 u , u , . There exist deeper level nonlocalities, such as ∂Q , ∂x = − Q , − u , u , + 4 u , u , + u , ,∂Q , ∂y = − Q , − u , u , + 4 u , u , and ∂Q , ∂x = − Q , u , − Q , − u , − u , u , u , + u , u , u , − u ,∂Q , ∂y = − Q , u − Q , − u , u , + 20 u , u , u , + 10 u , u , u , − u , u , + u , u , u , − u , u , , ′ SHCHIK, A. VERBOVETSKY, AND R. VITOLO as well as ∂Q , ∂x = 13 Q , + Q , u , − u , u , ,∂Q , ∂y = 2 Q , u , + Q , u , − Q , − u , − u , u , . Remark . The zero level nonlocal variables are associated to conservationlaws of equation (1). For example, to Q , there corresponds the conservationlaw ω , = ( − u , u , + u , ) dx + ( − u , u , − u , u , + u , u , ) dy. The first level nonlocal variables are associated to conservation laws of theAbelian coverings determined by the zero level variables, etc.
Remark . Of course, the list of nonlocal variables above is not exhaustiveat all. We described only those ones that are used to construct the necessarynonlocal symmetries ( § § § Symmetries.
This direct computation is needed by the two reasons:to construct nonlocal vectors (see Subsection 1.6.3) and to use the obtainedsymmetries as ‘seeds’ for the hierarchies.The linearization of equation (1) has the form D y ( ϕ ) − u , D x D y ( ϕ ) + u , D x ( ϕ ) + u , D x D y ( ϕ ) = 0 , (18)where the total derivatives D x and D y are given by (17).Solving (18), we get the following solutions .2.2.1. Symmetries of degree ϕ = 1 , ϕ = u , , ϕ = u , , ϕ = Q , + 8 u , u , ,ϕ = − Q , u , + Q , − u , + u , u , . Symmetries of degree ϕ − = y, ϕ − = x,ϕ , = xu , − u, ϕ , = yu , + u,ϕ = 4 xu , − Q , ,ϕ = x ( Q , + 8 u , u , ) − Q , − Q , u , + 16 uu , ) . Symmetries of degree ϕ − = y, ϕ − = y , ϕ − = xy, ϕ − = x ,ϕ − = x u , + 4 xyu , − xu − yQ , ,ϕ = 2 x u , − xQ , − u , . In the notation for symmetries, the superscript indicates the polynomial order of asymmetry with respect to x and y , the first subscript equals the weight, while the secondone, if any, is the number of a symmetry in the set given by particular weight and order. NTEGRABLE STRUCTURES FOR A MONGE-AMP`ERE EQUATION 11
Symmetries of degree ϕ − = x − yu , ,ϕ = 12 x u , − x Q , − xu , − y ( Q , + 8 u , u , ) + 24 uu , . Symmetries of degree
4: We shall also need one symmetry of order 4,which is of the form ϕ − = x − xyu , − y u , + 16 yu. Cosymmetries.
The reasons to compute cosymmetries explicitly aresimilar to those indicated in Subsection 2.2.To find cosymmetries, we are to solve the equation adjoint to (18), i.e., D y ( ψ ) − D x D y ( u , ψ ) + D x ( u , ψ ) + D x D y ( u , ψ ) = 0 . (19)Using the notation similar to the one from Section 2.2, let us some compu-tational results needed below.2.3.1. Cosymmetries of degree ψ = 1 , ψ = u , , ψ = u , ,ψ = − Q , + 6 u , u , + 12 u , u , + u , ,ψ = 2 Q , u , − Q , + 4 u , u , + 2 u , u , u , − u , u , . Cosymmetries of degree ψ − = y, ψ − = x, ψ , = xu , − u , ,ψ , = yu , + u , , ψ = 4 xu , + 2 u , + u , . Cosymmetries of degree ψ − = 3 x − yu , ,ψ = x u , + 4 xyu , − xu , + y (2 u , + u , ) − u,ψ = 2 x u , + x (2 u , + u , ) − Q , − u , u , . Cosymmetries of degree ψ − = 2 u , y − u , y − u , xy + x ,ψ = x u , + x (cid:18) u , + 34 u , (cid:19) − x (cid:18) Q , + 3 u , u , (cid:19) + y (cid:18) Q , − u , u , − u , u , − u , (cid:19) + (2 uu , + 12 u , ) . The ℓ -covering. This covering is determined by the system of equa-tions u yyy − u xxy + u xxx u xyy = 0 ,q yyy − u xxy q xxy + u xyy q xxx + u xxx q xyy = 0 , where q is an odd variable along the fiber of the covering.Internal coordinates on the space of the ℓ -covering are functions (16)together with q k,i = ∂ k + i q∂x k ∂y i , i = 0 , , , k = 0 , , . . . , (20) ′ SHCHIK, A. VERBOVETSKY, AND R. VITOLO while the total derivatives in these coordinates take the form D x = ∂∂x + X k ≥ (cid:16) u k +1 , ∂∂u k, + u k +1 , ∂∂u k, + u k +1 , ∂∂u k, + q k +1 , ∂∂q k, + q k +1 , ∂∂q k, + q k +1 , ∂∂q k, (cid:17) ,D y = ∂∂y + X k ≥ (cid:16) u k, ∂∂u k, + u k, ∂∂u k, + D kx ( u , − u , u , ) ∂∂u k, + q k, ∂∂q k, + q k, ∂∂q k, + D kx (2 u , q , − u , q , − u , q , ) ∂∂q k, (cid:17) . By the general theory [12], to any cosymmetry ψ of the initial equationthere corresponds a conservation law on the ℓ -covering (a nonlocal form).Denote by Ω ψ the corresponding nonlocal variable. In the case of Equa-tion (1), this variable is defined by the relations ∂ Ω ψ ∂x = ψq , + a , q , + a , q,∂ Ω ψ ∂y = b , q , + b , q , + b , q , + b , q , + b , q , + b , q, (21)where b , = − u , ψ, b , = 2 u , ψ, b , = − u , ψ, (22) b , = − D x ( b , ) , b , = − D x ( b , ) ,b , = − D x ( b , )and a , = D x ( b , ) − D y ( ψ ) , a , = D x ( b , ) − D y ( a , ) . (23)Below we use the notation Ω ki,j = Ω ψ ki,j Recursion operators for symmetries.
To find recursion operators forsymmetries of Equation (1), we solve the equation˜ ℓ E (Φ) = 0 , where ˜ ℓ E is the linearization operator (18) in the ℓ -covering extended bynonlocal forms and Φ is a function on this extension.The simplest nontrivial solution is of the formΦ = Ω − − x Ω − + 4 y Ω , + 2 y Ω , + 3 x Ω − − u , Ω − − y Ω − xy Ω + (2 u , y − x ) Ω . (24)Using the first of Equations (21), we put into correspondence to everynonlocal form Ω ψ the operator D ψ = D − x ◦ ( ψD y + a , D y + a , ) (25)where the coefficients are determined using relations (22) and (23), i.e., a , = − D x ( u , ψ ) − D y ( ψ ) ,a , = − D x ( u , ψ ) + D x D y ( u , ψ ) + D y ( ψ ) . NTEGRABLE STRUCTURES FOR A MONGE-AMP`ERE EQUATION 13
Then the recursion operator R = D ψ − − x D ψ − + 4 y D ψ , + 2 y D ψ , + 3 x D ψ − − u , D ψ − − y D ψ − xy D ψ + (2 u , y − x ) D ψ (26)corresponds to solution (24). Remark . Since the variables x and y in Equation (1) “enjoy equal rights”,one can put into correspondence to the nonlocal form Ω ψ , using the secondequality in (21), the operator D ′ ψ = D − x ◦ ( b , D y + b , D x D y + b , D x + b , D y + b , D x + b , )and construct a recursion operator R ′ similar to operator (26).The action of operators R and R ′ on symmetries of Equation (1) is thesame.2.4.2. Symplectic structures.
To find symplectic structures, we solve theequation ˜ ℓ ∗ E (Ψ) = 0 , where, similar to Section 2.4.1, ˜ ℓ ∗ E is the operator (19) on the ℓ -coveringextended by the nonlocal forms, while Ψ is a function on this extension.Below we present the first two solutions of this equation. The simplestone is Ψ = Ω , , and the corresponding symplectic structure S = D x . (27)The next solution is nonlocal:Ψ = Ω − − x Ω − + 2 u , Ω − + 2 y Ω − (2 u , y − x ) Ω . The corresponding symplectic operator S : sym E → cosym E is S = D ψ − − x D ψ − + 2 u , D ψ − + 2 y D ψ − (2 u , y − x ) D ψ , where the operators D ψ are defined by Equation (25).2.5. The ℓ ∗ -covering. As it was indicated above, this covering is obtainedby adding to the initial equation another one, which is adjoint to the lin-earization. In other words, this covering is described by the equations u yyy − u xxy + u xxx u xyy = 0 ,u xxyy p xx − u xxxy p xy + u xxxx p yy + u xyy p xxx − u xxy p xxy + u xxx p xyy + p yyy = 0 , where p is a new odd variable.Internal coordinates in the space of the ℓ ∗ -covering are functions (16)together with the functions p k,i = ∂ k + i p∂x k ∂y i , i = 0 , , , k = 0 , , . . . , while the total derivatives in these coordinates are of the form D x = ∂∂x + X k ≥ (cid:16) u k +1 , ∂∂u k, + u k +1 , ∂∂u k, + u k +1 , ∂∂u k,
24 P. KERSTEN, I. KRASIL ′ SHCHIK, A. VERBOVETSKY, AND R. VITOLO + p k +1 , ∂∂p k, + p k +1 , ∂∂p k, + p k +1 , ∂∂p k, (cid:17) ,D y = ∂∂y + X k ≥ (cid:16) u k, ∂∂u k, + u k, ∂∂u k, + D kx ( u , − u , u , ) ∂∂u k, + p k, ∂∂p k, + p k, ∂∂p k, − D kx ( u , p , − u , p , + u , p , + u , p , − u , p , + u , p , ) ∂∂p k, (cid:17) . Let ϕ be a symmetry of Equation (1). Then (see [12]) a conservationlaw on the ℓ ∗ -covering corresponds to this symmetry and, consequently anonlocal variable, which we denote by Π ϕ and call a nonlocal vector . ForEquation (1), the correspondence ϕ Π ϕ is given by the relations ∂ Π ϕ ∂x = ϕp , + a , p , + a , p,∂ Π ϕ ∂y = b , p , + b , p , + b , p , + b , p , + b , p , + b , p, (28)where b , = − u , ϕ, b , = 2 u , ϕ, b , = − u , ϕ,b , = − D x ( b , ) + 2 u , ϕ, b , = − D x ( b , ) − u , ϕ,b , = − D x ( b , ) (29)and a , = D x ( b , ) − D y ( ϕ ) + u , ϕ, a , = D x ( b , ) − D y ( a , ) . We use the notation Π ki,j = Π ϕ ki,j below.To describe the subsequent results, let us put into correspondence to anonlocal vector Π ϕ the operator D ϕ = D − x ◦ ( ϕD y + a , D y + a , ) , (30)see the first of Equations (28). Here, due to relations (29) and (30), thecoefficients a , and a , are of the form a , = − u , D x ( ϕ ) + u , D x D y ( ϕ ) + D y ( ϕ ) − u , D x ( ϕ ) ,a , = − u , D x ( ϕ ) − D y ( ϕ ) . Hamiltonian structures.
Similar to Sections 2.4.1 and 2.4.2, Hamil-tonian structures are solutions of the equation˜ ℓ E (Φ) = 0 , (31)where the operator ˜ ℓ E is the linearization lifted to the ˜ ℓ ∗ -covering extendedby nonlocal vectors.The simplest solution of Equation (31) isΦ = Π − − y Π − + y Π , to which the operator H : cosym E → sym EH = D ϕ − − y D ϕ − + y D ϕ NTEGRABLE STRUCTURES FOR A MONGE-AMP`ERE EQUATION 15 corresponds. The next solution is much more complicated and has the formΦ = Π − − x Π − + 6 x Π − − u , Π − − u , Π − + 16 y Π , + (8 xu , + 16 yu , − u ) Π − + 8 y Π , − (4 x − yu , ) Π − − y Π − xy Π + ( x − xyu , − y u , + 16 yu ) Π and the operator H = D ϕ − − x D ϕ − + 6 x D ϕ − − u , D ϕ − − u , D ϕ − + 16 y D ϕ , + (8 xu , + 16 yu , − u ) D ϕ − + 8 y D ϕ , − (4 x − yu , ) D ϕ − − y D ϕ − xy D ϕ + ( x − xyu , − y u , + 16 yu ) D ϕ . corresponds to this solution. Here and below the operators D ϕ are definedby Equality (30). Remark . The form of the above presented solutions is determined by thechoice of nonlocal vectors in the ℓ ∗ -covering. They, in turn, are due to abasis in the space of symmetries. Actually, Equation (31) has a simplersolution Φ ′ = Π ′ , where the nonlocal variable Π ′ is defined by the system ∂ Π ′ ∂x = p, ∂ Π ′ ∂y = Π ′ ,∂ Π ′ ∂x = p , , ∂ Π ′ ∂y = Π ′ ,∂ Π ∂x = p , , ∂ Π ′ ∂y = − u , p , + 2 u , p , − u , p , . To this solution there corresponds the Hamiltonian operator H ′ = D − x , which is the inverse to the symplectic operator (27). This operator cor-responds to the Hamiltonian operator J from [9]. The operator H thatexplicitly depends on x and y seems to be new. Remark . There is a relation H = R ◦ H , between the two Hamiltonian structures, where R is the recursion operatorgiven by Equality (26).2.5.2. Recursion operators for cosymmetries.
To conclude our study, we con-sider finally the equation ˜ ℓ ∗ E (Ψ) = 0on the ℓ ∗ -covering extended by nonlocal vectors. Here is its simplest non-trivial solutionΨ = Π − − x Π − + 2 u , Π − + 2 u , Π − + 2( u , + 2 u , y − u , x ) Π − − (2 u , y − x ) Π − + 2 y Π − (2 u , y − u , y − u , xy + x ) Π . ′ SHCHIK, A. VERBOVETSKY, AND R. VITOLO
To this solution there corresponds the recursion operator ˆ R : cosym E → cosym E of the formˆ R = D ϕ − − x D ϕ − + 2 u , D ϕ − + 2 u , D ϕ − + 2( u , − u , y − u , x ) D ϕ − − (2 u , y − x ) D ϕ − + 2 y D ϕ − (2 u , y − u , y − u , xy + x ) D ϕ . Remark . In the case of evolution equations, existence of a commutativehierarchy means that the initial equation possesses “higher analogs” thatare obtained by the action of a recursion operator. In the general case,there exists no well defined action of recursion operators on the equationand thus such a construction is impossible. Nevertheless, one can considerthe following alternative scheme: (1) to pass, if possible, to the evolutionarypresentation of the equation at hand, (2) to construct “higher analogs” inthis presentation and (3) return back to the initial variables. We did not setthe question whether such a scheme is invariant (i.e., whether the result iscompletely determined by the initial equation or depends on the procedure).Probably, an answer to this question will lead to a deeper understanding ofintegrability for general equations.
References [1] A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khor ′ kova, I.S. Krasil ′ shchik,A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky, and A.M. Vinogradov, Symme-tries and conservation laws for differential equations of mathematical physics , Amer.Math. Soc., 1999.[2] G. Boillat,
Sur l´equation g´en´erale de Monge-Amp`ere d´ordre sup´erieur , C. R. Acad.Sci. Paris S´er. I. Math., (1992), 1211–1214.[3] C. Crnkowi´c and E. Witten,
Covariant description of canonical formalism in ge-ometrical theories . In: S.W. Hawking W. Israel, editors,
Three Hundred Years ofGravitation , Cambridge University Press, 1989.[4] B. Dubrovin,
Geometry of 2-D topological field theories , in: Integrable Systems andQuantum Groups (Montecatini Terme, 1993), Lecture Notes in Math. , Springer,Berlin, 1996, 120–348, hep-th/9407018 .[5] E.V. Ferapontov, C.A.P. Galv˜ao, O.I. Mokhov, and Y. Nutku,
Bi-Hamiltonian struc-ture of equations of associativity in 2-d topological field theory , Comm. Math. Phys. (1997), 649–669.[6] E.V. Ferapontov and O.I. Mokhov,
Equations of associativity of two-dimensionaltopological field theory as integrable Hamiltonian nondiagonalisable systems ofhydrodynamic type , Funkt. Anal. and its Appl. , no. 3 (1996), 62–72, arXiv:hep-th/9505180 .[7] V.A. Golovko, I.S. Krasil ′ shchik, and A.M. Verbovetsky, On integrability of theCamassa-Holm equation and its invariants , Acta Appl. Math. (2008), 59–83, arXiv:0812.4681 .[8] J. Kalayci and Y. Nutku,
Bi-Hamiltonian structure of a WDVV equation in 2-dtopological field theory , Phys. Lett. A , Issues 3-4, 17 March 1997, 177–182.[9] J. Kalayci and Y. Nutku, “Alternative” bi-Hamiltonian structures for WDVV equa-tions in 2-d topological field theory , J. Phys. A: Math. and Gen. (1998), 723–734, arXiv:hep-th/9810076 .[10] P. Kersten, I. Krasil ′ shchik, and A. Verbovetsky, Hamiltonian operators and ℓ ∗ -coverings , J. Geom. Phys. (2004), 273–302, arXiv:math/0304245 .[11] P. Kersten, I. Krasil ′ shchik, and A. Verbovetsky, A geometric study of the dis-persionless Boussinesq type equation , Acta Appl. Math. (2006), 143–178, arXiv:nlin/0511012 . NTEGRABLE STRUCTURES FOR A MONGE-AMP`ERE EQUATION 17 [12] I. Krasil ′ shchik and A. Verbovetsky, Geometry of jet spaces and integrable systems ,J. Geom. Phys. (2011) doi:10.1016/j.geomphys.2010.10.012, arXiv:1002.0077 .[13] A. Kushner, V. Lychagin, and V. Rubtsov,
Contact geometry and non-linear differ-ential equations , Cambridge University Press, 2007.[14] I.A.B. Strachan,
On the integrability of a third-order Monge-Amp`ere type equa-tion , Phys. Lett. A , Issues 4-5, 15 January 1996, 267–272 doi:10.1016/0375-9601(95)00944-2.[15] A.M. Vinogradov,
On algebro-geometric foundations of Lagrangian field theory , SovietMath. Dokl. (1977) 1200–1204.[16] A.M. Vinogradov, A spectral sequence associated with a nonlinear differential equationand algebro-geometric foundations of Lagrangian field theory with constraints , SovietMath. Dokl. (1978) 144–148.[17] A.M. Vinogradov, The C -spectral sequence, Lagrangian formalism, and conservationlaws. I. The linear theory. II. The nonlinear theory , J. Math. Anal. Appl. (1984),1–129.[18] G.J. Zuckerman, Action principles and global geometry . In: S.T. Yau, edi-tor,
Mathematical aspects of string theory (San Diego, Calif., 1986), 259-284, Adv.Ser. Math. Phys., vol. 1, 1987.
Paul KerstenUniversity of Twente, Faculty of Mathematical Sciences, P.O. Box 217, 7500AE Enschede, The Netherlands (on retirement since 2010)
E-mail address : [email protected] Iosif Krasil ′ shchikIndependent University of Moscow, B. Vlasevsky 11, 119002 Moscow, Russia E-mail address : [email protected] Alexander VerbovetskyIndependent University of Moscow, B. Vlasevsky 11, 119002 Moscow, Russia
E-mail address : [email protected] Raffaele VitoloDept. of Mathematics “E. De Giorgi”, Universit`a del Salento, via per Arne-sano, 73100 Lecce, Italy
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