The Diffiety Institute Preprint Series

Preprint DIPS–07 October 29, 2003

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The Diffiety Institute

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On the integrability conditions for some structures related to evolution differential

equations

by

P. Kersten, I. Krasil′shchik, A. Verbovetsky

DIPS-7/2003 math.DG/0310451

ON THE INTEGRABILITY CONDITIONS FOR SOME STRUCTURES RELATED TO EVOLUTION DIFFERENTIAL EQUATIONS

P. KERSTEN, I. KRASIL′SHCHIK, A. VERBOVETSKY

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

Introduction

In [4], we described a method to construct Hamiltonian and symplectic structures on nonlinear evolution equations. The method was essentially based on the notions of variational multivector [3] and variational differential form. From technical viewpoint, for a given evolution equation ℰ, it consisted of two steps: (1) solving the linearized equation ℓ_{ℰ} φ = 0 in the so-called ℓ_{ℰ}^{∗} -covering (resp., the equation ℓ_{ℰ}^{∗} ψ = 0 in the ℓ_{ℰ} -covering) and (2) checking the Hamiltonianity condition [[φ, φ]] = 0, where [[· , ·]] denotes the variational Schouten bracket [3] (resp., the condition for ψ to be symplectic, i.e., closed with respect to a certain differential in the Vinogradov 𝒞 -spectral sequence [6]).

Surprisingly enough, it was found out in particular computations that the second condition always holds true “by default” and we still do not know counterexamples (except for the case of first order equations). On the other hand, a rather old result by D. Gessler [2] states that all terms E_{1}^{p,n−1}(ℰ) of the Vinogradov 𝒞 -spectral sequence vanish in the nondegenerate case for p ≥ 3 (here n is the number of independent variables). This fact means exactly that all variational 2-forms on nondegenerate evolution equations are closed and thus symplectic. Since Gessler’s proof almost literary works in the case of multivectors, we immediately obtain that all bivectors on such equations are Hamiltonian. These facts explain our experimental results.

We expose the details below. In Section 1, necessary introduction to the geometry of jet bundles and evolution equations is presented. Section 2 deals with the calculus of variational multivectors and forms on evolution equations. To make our exposition self-contained, we repeat Gessler’s proof from [2]. Finally, in Section 3 we derive our main results on the integrability of Hamiltonian and symplectic structures on nondegenerate evolution equations of order > 1.

1. Generalities: Jet bundles and evolution equations

Let us fix notation and recall briefly some definitions and results we will use. For explanations we refer to [1, 5, 3].

Let π: E → M be a vector bundle over an n-dimensional base manifold M and π_{∞}: J^{∞}(π) → M be the infinite jet bundle of local sections of the bundle π.

In coordinate language, if x_{1}, …, x_{n}, u^{1}, …, u^{m} are coordinates on E such that x_{i} are base coordinates and u^{j} are fiber ones, then π_{∞}: J^{∞}(π) → M is an infinite-dimensional vector bundle with fiber coordinates u_{σ}^{j}, where σ = i_{1} … i_{|σ|} is a symmetric multi-index.

The basic geometric structure on J^{∞}(π) is the Cartan distribution. In coordinate language, the Cartan distribution is spanned by the total derivatives

D_{i} = \frac{∂}{∂x_{i}} + ∑_{j,σ} u_{σi}^{j} \frac{∂}{∂u_{σ}^{j}} .

Key words and phrases. Evolution equations, Hamiltonian structures, symplectic structures, integrability.

2 P. KERSTEN, I. KRASIL′SHCHIK, A. VERBOVETSKY

A differential operator on J^{∞}(π) is called 𝒞 -differential operator (or horizontal 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 over J^{∞}(π). All 𝒞 - differential operators from P to Q form a C^{∞}(J^{∞}(π))-module denoted by 𝒞 Diff (P, Q). More generally, a map Δ: P_{1} × ⋯ × P_{k} → Q is called a multidifferential operator (of degree k) if it is a 𝒞 -differential operator in each argument. Choose elements p_{i} ∈ P_{i}, i = 1, …, k, and consider the operators

Δ_{i} = Δ(p_{1}, …, p_{i−1}, · , p_{i+1}, …, p_{k}): P_{i} → Q.

Let l_{i} be the order^{1} of Δ_{i}. We define the symbol

smbl(Δ): S^{l_{1}}(Λ^{1}(M)) ⊗ P_{1} × ⋯ × S^{l_{k}}(Λ^{1}(M)) ⊗ P_{k} → Q,

where S^{l} denotes the symmetric power, of Δ as follows. For any f ∈ C^{∞}(M), let us set

(δ_{f}^{(i)}Δ)(p_{1}, …, p_{k}) = fΔ(p_{1}, …, p_{k}) − Δ(p_{1}, …, p_{i−1}, fp_{i}, p_{i+1}, …, p_{k})

and δ_{f_{1},...,f_{l}}^{(i)} = δ_{f_{1}}^{(i)} ∘ ⋯ ∘ δ_{f_{l}}^{(i)}. If now ω^{i} = df_{1}^{i} … df_{l_{i}}^{i}, i = 1, …, l, are symmetric forms on M, we set

(smbl Δ)(ω^{1} ⊗ p_{1}, ω^{2} ⊗ p_{2}, …, ω^{k} ⊗ p_{k}) = δ_{f_{1}^{1},...,f_{l_{1}}^{1}}^{(1)} ∘ ⋯ ∘ δ_{f_{1}^{k},...,f_{l_{k}}^{k}}^{(k)} (Δ)(p_{1}, …, p_{k}).

Let ρ ∈ J^{∞}(π) and x = π_{∞}(ρ) ∈ M. Then the value of the symbol at ρ is the map

smbl(Δ)|_{ρ} : T_{x}^{∗}M ⊗ P_{1,ρ} × ⋯ × T_{x}^{∗}M ⊗ P_{k,ρ} → Q_{ρ}

polynomially dependent on points of T_{x}^{∗}M (P_{i,ρ} and Q_{ρ} denote here the fibers of the corresponding vector bundles at the point ρ).

The lift of the de Rham complex on M to J^{∞}(π) is called horizontal de Rham complex and is denoted by

0 ⟶ C^{∞}(J^{∞}(π)) ⟶^{d̄} Λ̄^{1}(π) ⟶^{d̄} ⋯ ⟶^{d̄} Λ̄^{n}(π) ⟶ 0.

The cohomology of the horizontal de Rham complex are called horizontal cohomology and denoted by H̄^{q}(π).

The adjoint operator to a 𝒞 -differential operator Δ: P → Q we denote by Δ^{∗}: Q̂ → P̂, where P̂ = Hom_{C^{∞}(J^{∞}(π))}(P, Λ̄^{n}(π)).

In coordinates,

∥ ∑_{σ} a_{ij}^{σ}D_{σ}∥^{∗} = ∥ ∑_{σ}(−1)^{|σ|}D_{σ} ∘ a_{ji}^{σ}∥,

where a_{ij}^{σ} ∈ C^{∞}(J^{∞}(π)), and D_{σ} = D_{i_{1}} ∘ ⋯ ∘ D_{i_{|σ|}} for σ = i_{1} … i_{|σ|}.

Denote by 𝒞 Diff _{(k)}^{sk-ad}(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 the Cartan distribution. The Lie algebra of evolutionary field is denoted by κ(π). It is known that κ(π) is naturally isomorphic to the set of sections of the bundle π_{∞}^{∗}(π); thus κ(π) is endowed with a structure of C^{∞}(J^{∞}(π))-module.

In local coordinates, the evolutionary field that corresponds to a section φ = (φ^{1}, …, φ^{m}) has the form

�_{φ} = ∑_{j,σ} D_{σ}(φ^{j})\frac{∂}{∂u_{σ}^{j}} .

We shall call elements of κ(π) variational vectors. Elements of the module 𝒞 Diff _{(k−1)}^{sk-ad}(κ̂, κ) will be called variational k-vector, while elements of 𝒞 Diff _{(k−1)}^{sk-ad}(κ, κ̂) will be called variational k-forms.

One knows that standard constructions and formulas of the calculus of vector fields and forms on manifolds (the de Rham differential, inner product, the Lie 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 L_{�_{φ}} : κ → κ takes the form L_{�_{φ}} = �_{φ} − ℓ_{φ}, where the linearization operator ℓ_{p} is defined by the equality ℓ_{p}(α) = �_{α}(p), α ∈ κ.

1Of course, this number depends on the choice of p’s; so we define the order as the maximum over all possible choices.

The Lie derivative on variational forms L_{�_{φ}} : κ̂ → κ̂ is of the form L_{�_{φ}} = �_{φ} + ℓ_{φ}^{∗}.

The Lie derivative on variational k-vectors or k-forms satisfies the equality

L_{�_{φ}} (A)(ξ_{1}, …, ξ_{k−1}) = L_{�_{φ}} (A(ξ_{1}, …, ξ_{k−1})) − ∑_{i} A(ξ_{1}, …, ξ_{i−1}, L_{�_{φ}} (ξ_{i}), ξ_{i+1}, …, ξ_{k−1}),

where A is a multivector or a form, while ξ_{1}, …, ξ_{k−1} are elements of κ̂ in the former case and elements of κ in the latter one.

Consider a determined evolution equation

u_{t}^{1} = f^{1}(t, x, u_{σ}^{j}),
. . . . . . . . . . . . . . . .
u_{t}^{m} = f^{m}(t, x, u_{σ}^{j}),

where x = (x_{1}, …, x_{n}). We shall interpret it in a geometric way as the space ℰ^{∞} = J^{∞}(π) × ℝ with the Cartan distribution generated by the Cartan fields on J^{∞}(π) and the vector field D_{t} = ∂/∂t + �_{f}, where t is the coordinate along ℝ.

The linearization of ℰ^{∞} is of the form ℓ_{ℰ} = D_{t} −ℓ_{f}, while the adjoint linearization is ℓ_{ℰ}^{∗} = −D_{t} −ℓ_{f}^{∗}. Note, that from the above we have

ℓ_{ℰ} = L_{D_{t}}: κ → κ,
ℓ_{ℰ}^{∗} = −L_{D_{t}}: κ̂ → κ̂.  (1)

2. Variational multivectors and forms on evolution equations

Let A be a (possibly dependent on t) variational multivector or form on J^{∞}(π). If L_{D_{t}}(A) = 0 then A is called a variational multivector or form on the equation 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^{∞} coincides with the term E_{1}^{1,n−1}(ℰ) = ker ℓ_{ℰ}^{∗} of the Vinogradov spectral sequence. Similarly, the terms E_{1}^{p,n−1}(ℰ) consist of variational p-forms.

The set of variational multivectors and forms on ℰ^{∞} is closed with respect to all operations that are defined on jet spaces whenever they are applicable: the differential on variational forms, inner product, the Schouten bracket, Lie derivative.

Proposition 1. Let ℰ^{∞} be an evolution equation u_{t} = f. For an operator A to be a variational k-vector or k-form on ℰ^{∞} it is necessary and sufficient to have

∇(A(ξ_{1}, …, ξ_{k−1})) + ∑_{i} A(ξ_{1}, …, ξ_{i−1}, ∇^{∗}(ξ_{i}), ξ_{i+1}, …, ξ_{k−1}) = 0,  (2)

where ∇ = ℓ_{ℰ} if A is a multivector and ∇ = ℓ_{ℰ}^{∗} if A is a form; here ξ_{1}, …, ξ_{k−1} are elements of κ̂ in the case of multivectors and elements of κ in the case of forms.

Proof. We have

L_{D_{t}}(A)(ξ_{1}, …, ξ_{k−1}) = L_{D_{t}}(A(ξ_{1}, …, ξ_{k−1})) − ∑_{i} A(ξ_{1}, …, ξ_{i−1}, L_{D_{t}}(ξ_{i}), ξ_{i+1} …, ξ_{k−1}) = 0.

Using (1), we get the result. □

Proposition 2. Let ℰ^{∞} be an evolution equation u_{t} = f. If operators A, A_{1}, …, A_{k−1} satisfy the equation

∇(A(ξ_{1}, …, ξ_{k−1})) + ∑_{i} A_{i}(ξ_{1}, …, ξ_{i−1}, ∇^{∗}(ξ_{i}), ξ_{i+1}, …, ξ_{k−1}) = 0,  (3)

where ∇ = ℓ_{ℰ} or ℓ_{ℰ}^{∗}, then A_{1} = A_{2} = ⋯ = A_{k−1} = A.

Proof. Denote the left-hand side of (3) by Ω(ξ_{1}, …, ξ_{k−1}). Then we get

Ω(ξ_{1}, …, ξ_{i−1}, tξ_{i}, ξ_{i+1} …, ξ_{k−1}) − tΩ(ξ_{1}, …, ξ_{k−1}) = ±(A(ξ_{1}, …, ξ_{k−1}) − A_{i}(ξ_{1}, …, ξ_{k−1})) = 0.

□

Remark 3. The last proposition shows that computing variational multivectors and forms on an equation amounts to solving equation ∇(s) = 0 on the ∇^{∗}-covering (see [4] for the definition of Δ-coverings associated to a 𝒞 -differential operator Δ).

Theorem 1. Suppose that the symbol of the 𝒞 -differential operator ℓ_{f} is nonsingular on a dense open subset of ℰ^{∞} and the order of the operator ℓ_{f} is greater than 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

±D_{t}(A)(ξ_{1}, …, ξ_{k−1}) + ∇′(A(ξ_{1}, …, ξ_{k−1})) + ∑_{i} A(ξ_{1}, …, ξ_{i−1}, ∇^{′∗}(ξ_{i}), ξ_{i+1}, …, ξ_{k−1}) = 0, (4)

where ∇′ = ℓ_{f} if A is a multivector and ∇′ = ℓ_{f}^{∗} if A is a form. Take a point ρ ∈ ℰ^{∞} such that the symbol λ = smbl(∇′)|_{ρ} is nondegenerate at ρ. Let θ = ∑_{i=1}^{n} θ_{i} dx_{i}|_{ρ} be a covector, so that, in coordinates, λ is an m × m matrix λ = ∥λ_{j}^{i}∥, where λ_{j}^{i} are homogeneous polynomials in θ’s of degree l = ord(∇′). Denote the components of the symbol a = smbl(A)|_{ρ} by a_{i_{1}...i_{k−1}}^{j}(θ^{1}, …, θ^{k−1}), θ^{p} = (θ_{1}^{p}, …, θ_{n}^{p}). Then the symbol of equation (4) takes the form

∑_{j=1}^{m} λ_{j}^{i}(θ^{1} + ⋯ + θ^{k−1})a_{i_{1}...i_{k−1}}^{j} + (−1)^{l} ∑_{p=1}^{k−1} ∑_{j=1}^{m} a_{i_{1}...i_{p−1}ji_{p+1}i_{k−1}}^{i}λ_{j}^{i_{p}}(θ^{p}) = 0,  (5)

where 1 ≤ i, i_{1}, …, i_{k−1} ≤ m.

System (5) can be considered as a linear system of algebraic equations with polynomial coefficients over ℂ. Let us show that the determinant of this system does not vanish.

Since λ = λ(θ) is nonsingular, there exists v ∈ ℂ^{m} such that det λ(v) ≠ 0. One can assume that λ(v) has an upper triangular form, λ_{j}^{i}(v) = 0 if i ≥ j and λ_{i}^{i}(v) ≠ 0. Then for any α ∈ ℂ the matrix λ(αv) = α^{l}λ(v) has also an upper triangular form. Since l = ord ℓ_{f} ≥ 2 and k ≥ 3, there exist α_{p} ∈ ℂ, p = 1, …, k − 1, such that for any 1 ≤ i, i_{1}, …, i_{k−1} ≤ m

A_{ii_{1}...i_{k−1}} = λ_{i}^{i}(v)(α_{1} + ⋯ + α_{k−1})^{l} + (−1)^{l} ∑_{p=1}^{k−1} λ_{i_{p}}^{i_{p}}(α_{p})^{l} ≠ 0.  (6)

Put θ^{i} = α_{i}v. Then system (4) is upper triangular with respect to the lexicographic order of indexes, with diagonal entries A_{ii_{1}...i_{k−1}} ≠ 0. Hence, the determinant of system (4) does not equal to zero, thus a = 0. Therefore, the symbol of A vanishes on a dense subset of ℰ^{∞}, so that A = 0. □

3. Hamiltonian and symplectic structures

Recall 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 B are compatible (or constitute a pencil) if [[A, B]] = 0. Respectively, a symplectic structure on ℰ^{∞} is a closed variational form.

Now, our main result is obtained by reformulating the previous theorem.

Theorem 2. Assume that ℰ = {u_{t} = f} is an evolution equation and ℓ_{f} satisfies the hypothesis of Theorem 1. Then any variational bivector on ℰ is Hamiltonian, any two Hamiltonian structures are compatible and any variational 2-form is symplectic.

Proof. Indeed, let A and B be bivectors. Then [[A, B]] is a 3-vector and thus vanishes. Similarly, the differential of any variational 2-form is a 3-form and therefore equals zero. □

Remark 4. The hypothesis of Theorem 1 comprises two conditions: (1) the order of ℰ is to be > 1;

(2) the symbol of ℓ_{f} is to be nondegenerate. The first one is really essential. As for the second condition, it seems that it may be weakened. At least in some computation (e.g., for the Boussinesq equation, see [4]) all bivectors are automatically Hamiltonian.

Remark 5. The proof of Theorem 1 does not use the fact that the operator A is skew-symmetric and holds also for symmetric 𝒞 -differential operators. This means that equations satisfying the hypothesis of the theorem may admit linear Hamiltonian and symplectic structures only.

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, Symmetries and conservation laws for differential equations of mathematical physics, Monograph, Amer. Math. Soc., 1999.

[2] D. M. Gessler, On the Vinogradov 𝒞 -spectral sequence for determined systems of differential equations, Differential

   Geom. Appl. 7 (1997), 303–324, URL http://diffiety.ac.ru/preprint/98/09_98abs.htm.

[3] S. Igonin, A. Verbovetsky, and R. Vitolo, On the formalism of local variational differential operators, Memorandum 1641, Faculty of Mathematical Sciences, University of Twente, The Netherlands, 2002, URL http://www.math.

   utwente.nl/publications/2002/1641abs.html.

[4] P. Kersten, I. Krasil′shchik, and A. Verbovetsky, Hamiltonian operators and ℓ^{∗}-coverings, J. Geom. Phys. (2004), to

   appear, arXiv:math.DG/0304245.

[5] J. Krasil′shchik and A. M. Verbovetsky, Homological methods in equations of mathematical physics, Advanced Texts in Mathematics, Open Education & Sciences, Opava, 1998, arXiv:math.DG/9808130.

[6] A. M. Vinogradov, 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.

Paul Kersten, Faculty of Mathematical Sciences, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

   E-mail address: kersten@math.utwente.nl

Iosif Krasil′shchik, The Diffiety Institute and Independent University of Moscow, B. Vlasevsky 11, 119002 Moscow, Russia

   E-mail address: josephk@diffiety.ac.ru

Alexander Verbovetsky, Independent University of Moscow, B. Vlasevsky 11, 119002 Moscow, Russia

   E-mail address: verbovet@mccme.ru