Using the result by D.Gessler (Differential Geom. Appl. 7 (1997) 303-324,
DIPS-9/98, http://diffiety.ac.ru/preprint/98/09_98abs.htm), we show that any
invariant variational bivector (resp., variational 2-form) on an evolution
equation with nondegenerate right-hand side is Hamiltonian (resp., symplectic).Comment: 5 pages, AMS-LaTeX. v2: minor correction

Kersten, Paul

Krasil'shchik, Iosif

Verbovetsky, Alexander

English

arXiv

Using the result by D. Gessler, we show that any invariant variational bivector (resp., variational 2-form) on an evolution equation with nondegenerate right-hand side is Hamiltonian (resp., symplectic)

Kersten, P.H.M.

Krasil'shchik, I.

Verbovetsky, A.

NARCIS 

On the Integrability Conditions for Some Structures Related to Evolution Differential Equations

Kersten, P.

University of Twente Research Information

Acta Applicandae Mathematicae 83: 167–173, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.167On the Integrability Conditions for Some StructuresRelated to Evolution Differential EquationsP. KERSTEN1, I. KRASIL’SHCHIK2 and A. VERBOVETSKY31University of Twente, Faculty of Electrical Engineering, Mathematics and Computer Science,P.O. Box 217, 7500 AE Enschede, The Netherlands. e-mail: kersten@math.utwente.nl2The Diffiety Institute and Independent University of Moscow, B. Vlasevsky 11, 119002 Moscow,Russia. e-mail: josephk@diffiety.ac.ru3Independent University of Moscow, B. Vlasevsky 11, 119002 Moscow, Russia.e-mail: verbovet@mccme.ruAbstract. Using the result by D. Gessler, we show that any invariant variational bivector (resp.,variational 2-form) on an evolution equation with nondegenerate right-hand side is Hamiltonian(resp., symplectic).Mathematics Subject Classifications (2000): 37K05, 35Q53.Key words: evolution equations, Hamiltonian structures, symplectic structures, integrability.IntroductionIn [4], we described a method to construct Hamiltonian and symplectic structureson nonlinear evolution equations. The method was essentially based on the notionsof variational multivector [3] and variational differential form. From technicalviewpoint, for a given evolution equation E , it consisted of two steps: (1) solvingthe linearized equation Eϕ = 0 in the so-called ∗E -covering (resp., the equa-tion ∗Eψ = 0 in the E -covering) and (2) checking the Hamiltonianity conditionϕ, ϕ = 0, where · , · denotes the variational Schouten bracket [3] (resp., thecondition for ψ to be symplectic, i.e., closed with respect to a certain differentialin the Vinogradov C-spectral sequence [6]).Surprisingly enough, it was found out in particular computations that the secondcondition always holds true ‘by default’ and we still do not know counterexam-ples (except for the case of first-order equations). On the other hand, a ratherold result by Gessler [2] states that all terms Ep,n−11 (E) of the Vinogradov C-spectral sequence vanish in the nondegenerate case for p  3 (here n is thenumber of independent variables). This fact means exactly that all variational 2-forms on nondegenerate evolution equations are closed and thus symplectic. SinceGessler’s proof almost literary works in the case of multivectors, we immediatelyobtain that all bivectors on such equations are Hamiltonian. These facts explain ourexperimental results.168 P. KERSTEN ET AL.We expose the details below. In Section 1, necessary introduction to the geom-etry of jet bundles and evolution equations is presented. Section 2 deals with thecalculus of variational multivectors and forms on evolution equations. To make ourexposition self-contained, we repeat Gessler’s proof from [2]. Finally, in Section 3we derive our main results on the integrability of Hamiltonian and symplecticstructures on nondegenerate evolution equations of order > 1.1. Generalities: Jet Bundles and Evolution EquationsLet us fix notation and recall briefly some definitions and results we will use. Forexplanations we refer to [1, 3, 5].Let π: E → M be a vector bundle over an n-dimensional base manifold Mand π∞: J ∞(π) → M be the infinite jet bundle of local sections of the bundle π .In coordinate language, if x1, . . . , xn, u1, . . . , um are coordinates on E suchthat xi are base coordinates and uj are fiber ones, then π∞: J ∞(π) → M is aninfinite-dimensional vector bundle with fiber coordinates ujσ , where σ = i1 . . . i|σ |is a symmetric multi-index.The basic geometric structure on J ∞(π) is the Cartan distribution. In coordi-nate language, the Cartan distribution is spanned by the total derivativesDi = ∂∂xi+∑j,σujσ i∂∂ujσ.A differential operator on J ∞(π) is called C-differential operator (or hori-zontal operator) if it can be written as a sum of compositions of C∞(J ∞(π))-linear maps and vector fields that belong to the Cartan distribution. In coordinates,C-differential operators are total derivatives operators.Let P and Q be C∞(J ∞(π))-modules of sections of some vector bundles overJ ∞(π). All C-differential operators from P to Q form a C∞(J ∞(π))-moduledenoted by CDiff(P,Q). More generally, a map : P1 × · · · × Pk → Q is calleda multidifferential operator (of degree k) if it is a C-differential operator in eachargument. Choose elements pi ∈ Pi , i = 1, . . . , k, and consider the operatorsi = (p1, . . . , pi−1, · , pi+1, . . . , pk): Pi → Q.Let li be the order of i . We define the symbolsmbl(): Sl1(	1(M)) ⊗ P1 × · · · × Slk (	1(M)) ⊗ Pk → Q,where Sl denotes the symmetric power, of  as follows. For any f ∈ C∞(M), letus set(δ(i)f )(p1, . . . , pk) = f (p1, . . . , pk)−(p1, . . . , pi−1, fpi, pi+1, . . . , pk) Of course, this number depends on the choice of p’s; so we define the order as the maximumover all possible choices.INTEGRABILITY CONDITIONS 169and δ(i)f1,...,fl = δ(i)f1 ◦ · · · ◦δ(i)fl . If now ωi = df i1 . . . df ili , i = 1, . . . , k, are symmetricforms on M, we set(smbl )(ω1 ⊗ p1, ω2 ⊗ p2, . . . , ωk ⊗ pk)= δ(1)f 11 ,...,f1l1◦ · · · ◦ δ(k)f k1 ,...,fklk()(p1, . . . , pk).Let ρ ∈ J ∞(π) and x = π∞(ρ) ∈ M. Then the value of the symbol at ρ is themapsmbl()|ρ: T ∗x M ⊗ P1,ρ × · · · × T ∗x M ⊗ Pk,ρ → Qρpolynomially dependent on points of T ∗x M (Pi,ρ and Qρ denote here the fibers ofthe corresponding vector bundles at the point ρ).The lift of the de Rham complex on M to J ∞(π) is called horizontal de Rhamcomplex and is denoted by0 −→ C∞(J ∞(π)) d̄−→ 	̄1(π) d̄−→ · · · d̄−→ 	̄n(π) −→ 0.The cohomology of the horizontal de Rham complex are called horizontal coho-mology and denoted by H̄ q(π).The adjoint operator to a C-differential operator : P → Q we denote by∗: Q̂ → P̂ , where P̂ = HomC∞(J∞(π))(P, 	̄n(π)).In coordinates,∥∥∥∥∑σaσijDσ∥∥∥∥∗=∥∥∥∥∑σ(−1)|σ |Dσ ◦ aσji∥∥∥∥,where aσij ∈ C∞(J ∞(π)), and Dσ = Di1 ◦ · · · ◦ Di|σ | for σ = i1 . . . i|σ |.Denote by CDiff sk−ad(k) (P,Q) the module of k-linear skew-symmetric and skew-adjoint in each argument C-differential operators P × · · · × P → Q.A π∞-vertical vector filed on J ∞(π) is called evolutionary if it preserves theCartan distribution. The Lie algebra of evolutionary fields is denoted by (π). It isknown that (π) is naturally isomorphic to the set of sections of the bundle π∗∞(π);thus (π) is endowed with a structure of C∞(J ∞(π))-module.In local coordinates, the evolutionary field that corresponds to a section ϕ =(ϕ1, . . . , ϕm) has the formϕ =∑j,σDσ (ϕj)∂∂ujσ.We shall call elements of (π) variational vectors. Elements of the moduleCDiff sk−ad(k−1) (̂, ) will be called variational k-vector, while elements ofCDiff sk−ad(k−1) (, ̂) will be called variational k-forms.One knows that standard constructions and formulas of the calculus of vectorfields and forms on manifolds (the de Rham differential, inner product, the Lie170 P. KERSTEN ET AL.derivative, the Schouten bracket) are also valid for their ‘variational’ counterparts,with elements of H̄ n(π) being regarded as ‘functions’.In particular, the Lie derivative on variational vectors Lϕ:  →  takes theform Lϕ = ϕ −ϕ, where the linearization operator p is defined by the equalityp(α) = α(p), α ∈ .The Lie derivative on variational forms Lϕ: ̂ → ̂ is of the form Lϕ =ϕ + ∗ϕ.The Lie derivative on variational k-vectors or k-forms satisfies the equalityLϕ (A)(ξ1, . . . , ξk−1)= Lϕ (A(ξ1, . . . , ξk−1)) −∑iA(ξ1, . . . , ξi−1, Lϕ (ξi), ξi+1, . . . , ξk−1),where A is a multivector or a form, while ξ1, . . . , ξk−1 are elements of ̂ in theformer case and elements of  in the latter one.Consider a determined evolution equationu1t = f 1(t, x, ujσ ),. . . . . . . . . . . . . . .umt = f m(t, x, ujσ ),where x = (x1, . . . , xn). We shall interpret it in a geometric way as the spaceE∞ = J ∞(π) × R with the Cartan distribution generated by the Cartan fields onJ ∞(π) and the vector field Dt = ∂/∂t + f , where t is the coordinate along R.The linearization of E∞ is of the form E = Dt −f , while the adjoint lineariza-tion is ∗E = −Dt − ∗f .Note, that from the above we haveE = LDt:  → ,∗E = −LDt: ̂ → ̂. (1)2. Variational Multivectors and Forms on Evolution EquationsLet A be a (possibly dependent on t) variational multivector or form on J ∞(π).If LDt (A) = 0 then A is called a variational multivector or form on the equa-tion E∞.Remark 1. Variational bivectors on evolution equations were considered in [4].Remark 2. From (1) it follows that the set of variational 1-forms on E∞ co-incides with the term E1,n−11 (E) = ker ∗E of the Vinogradov spectral sequence.Similarly, the terms Ep,n−11 (E) consist of variational p-forms.INTEGRABILITY CONDITIONS 171The set of variational multivectors and forms on E∞ is closed with respect toall operations that are defined on jet spaces whenever they are applicable: the dif-ferential on variational forms, inner product, the Schouten bracket, Lie derivative.PROPOSITION 1. Let E∞ be an evolution equation ut = f . For an operator Ato be a variational k-vector or k-form on E∞ it is necessary and sufficient to have∇(A(ξ1, . . . , ξk−1)) +∑iA(ξ1, . . . , ξi−1,∇∗(ξi), ξi+1, . . . , ξk−1) = 0, (2)where ∇ = E if A is a multivector and ∇ = ∗E if A is a form; here ξ1, . . . , ξk−1are elements of ̂ in the case of multivectors and elements of  in the case of forms.Proof. We haveLDt (A)(ξ1, . . . , ξk−1) =LDt (A(ξ1, . . . , ξk−1))−−∑iA(ξ1, . . . , ξi−1, LDt (ξi), ξi+1 . . . , ξk−1) = 0.Using (1), we get the result. PROPOSITION 2. Let E∞ be an evolution equation ut = f . If operators A, A1,. . . , Ak−1 satisfy the equation∇(A(ξ1, . . . , ξk−1)) +∑iAi(ξ1, . . . , ξi−1,∇∗(ξi), ξi+1, . . . , ξk−1) = 0, (3)where ∇ = E or ∗E , then A1 = A2 = · · · = Ak−1 = A.Proof. Denote the left-hand side of (3) by (ξ1, . . . , ξk−1). Then we get(ξ1, . . . , ξi−1, tξi , ξi+1 . . . , ξk−1) − t(ξ1, . . . , ξk−1)= ±(A(ξ1, . . . , ξk−1) − Ai(ξ1, . . . , ξk−1)) = 0. Remark 3. The last proposition shows that computing variational multivectorsand forms on an equation amounts to solving equation ∇(s) = 0 on the ∇∗-covering (see [4] for the definition of -coverings associated to a C-differentialoperator ).THEOREM 1. Suppose that the symbol of the C-differential operator f is non-singular on a dense open subset of E∞ and the order of the operator f is greaterthan 1. Then there are no nonzero operators A that satisfy Equation (2) for k  3.Proof ([2, Th. 3]). Equation (2) can be written in the form± Dt(A)(ξ1, . . . , ξk−1) + ∇′(A(ξ1, . . . , ξk−1))++∑iA(ξ1, . . . , ξi−1,∇′∗(ξi), ξi+1, . . . , ξk−1) = 0, (4)172 P. KERSTEN ET AL.where ∇′ = f if A is a multivector and ∇′ = ∗f if A is a form. Take a pointρ ∈ E∞ such that the symbol λ = smbl(∇′)|ρ is nondegenerate at ρ. Let θ =∑ni=1 θi dxi |ρ be a covector, so that, in coordinates, λ is an m×m matrix λ = ‖λij‖,where λij are homogeneous polynomials in θ’s of degree l = ord(∇′). Denotethe components of the symbol a = smbl(A)|ρ by aji1...ik−1(θ1, . . . , θk−1), θp =(θp1 , . . . , θpn ). Then the symbol of Equation (4) takes the formm∑j=1λij (θ1 +· · ·+ θk−1)aji1...ik−1 + (−1)lk−1∑p=1m∑j=1aii1...ip−1jip+1ik−1λipj (θp) = 0, (5)where 1  i, i1, . . . , ik−1  m.System (5) can be considered as a linear system of algebraic equations withpolynomial coefficients over C. Let us show that the determinant of this systemdoes not vanish.Since λ = λ(θ) is nonsingular, there exists v ∈ Cm such that det λ(v) = 0. Onecan assume that λ(v) has an upper triangular form, λij (v) = 0 if i  j and λii(v) =0. Then for any α ∈ C the matrix λ(αv) = αlλ(v) has also an upper triangularform. Since l = ord f  2 and k  3, there exist αp ∈ C, p = 1, . . . , k − 1, suchthat for any 1  i, i1, . . . , ik−1  mAii1...ik−1 = λii(v)(α1 + · · · + αk−1)l + (−1)lk−1∑p=1λipip(αp)l = 0. (6)Put θ i = αiv. Then system (4) is upper triangular with respect to the lexicographicorder of indexes, with diagonal entries Aii1...ik−1 = 0. Hence, the determinant ofsystem (4) does not equal to zero, thus a = 0. Therefore, the symbol of A vanisheson a dense subset of E∞, so that A = 0. 3. Hamiltonian and Symplectic StructuresRecall that a bivector A on equation E∞ is said to be Hamiltonian if [[A,A]] = 0,where [[· , ·]] is the variational Schouten bracket [3, 4]. Two structures A and Bare compatible (or constitute a pencil) if [[A,B]] = 0. Respectively, a symplecticstructure on E∞ is a closed variational form.Now, our main result is obtained by reformulating the previous theorem.THEOREM 2. Assume that E = {ut = f } is an evolution equation and fsatisfies the hypothesis of Theorem 1. Then any variational bivector on E is Hamil-tonian, any two Hamiltonian structures are compatible and any variational 2-formis symplectic.Proof. Indeed, let A and B be bivectors. Then [[A,B]] is a 3-vector and thus van-ishes. Similarly, the differential of any variational 2-form is a 3-form and thereforeequals zero. INTEGRABILITY CONDITIONS 173Remark 4. The hypothesis of Theorem 1 comprises two conditions: (1) the or-der of E is to be > 1; (2) the symbol of f is to be nondegenerate. The first one isreally essential. As for the second condition, it seems that it may be weakened. Atleast in some computation (e.g., for the Boussinesq equation, see [4]) all bivectorsare automatically Hamiltonian.Remark 5. The proof of Theorem 1 does not use the fact that the operator Ais skew-symmetric and holds also for symmetric C-differential operators. Thismeans that equations satisfying the hypothesis of the theorem may admit linearHamiltonian and symplectic structures only.References1. Bocharov, A. V., Chetverikov, V. N., Duzhin, S. V., Khor’kova, N. G., Krasil’shchik, I. S.,Samokhin, A. V., Torkhov, Y. N., Verbovetsky, A. M. and Vinogradov, A. M.: Symmetries andConservation Laws for Differential Equations of Mathematical Physics, Amer. Math. Soc.,Providence, RI, 1999.2. Gessler, D. M.: On the Vinogradov C-spectral sequence for determined systems of differentialequations, Differential Geom. Appl. 7 (1997), 303–324.3. Igonin, S., Verbovetsky, A. and Vitolo, R.: On the formalism of local variational dif-ferential operators, Memorandum 1641, Faculty of Mathematical Sciences, University ofTwente, The Netherlands, 2002, URL http://www.math.utwente.nl/publications/2002/1641abs.html.4. Kersten, P., Krasil’shchik, I. and Verbovetsky, A.: Hamiltonian operators and ∗-coverings,J. Geom. Phys. 50 (2004), 273–302.5. Krasil’shchik, J. and Verbovetsky, A.: Homological Methods in Equations of MathematicalPhysics, Advanced Texts in Mathematics, Open Education & Sciences, Opava, 1998, math.DG/9808130.6. Vinogradov, A. M.: The C-spectral sequence, Lagrangian formalism, and conservation laws. I.The linear theory. II. The nonlinear theory, J. Math. Anal. Appl. 100 (1984), 1–129.

On the integrability conditions for some structures related to evolution
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On the integrability conditions for some structures related to evolution differential equations

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