Lobachevskii Journal of Mathematics Vol. 3, 1999, 127–145 \begin{matrix}c \\ ◯ \end{matrix}I.S. Krasil’shchik

I.S. Krasil’shchik

COHOMOLOGICAL APPROACH TO POISSON STRUCTURES ON NONLINEAR EVOLUTION EQUATIONS

(submitted by B.N.Shapukov)

Abstract. Let E be a differential equation, and let F = F(E) be the function algebra on the infinite prolongation E^{∞}. Consider the algebra A = Λ^{∗}(F) of differential forms on F endowed with the horizontal differential d_{h}: A → A. A Poisson structure P on E is understood as the homotopy equivalence class (with respect to d_{h}) of a skew-symmetric super bidifferential operator P in A satisfying the condition [[P, P]]^{s} = 0, [[•, •]]^{s} being the super Schouten bracket.

A description of Poisson structures for an evolution equation with an arbitrary number of space variables is given. It is shown that the computations, in essence, reduce to solving the operator equation P ∘ ℓ̂_{E}+ℓ_{E}∘P = 0. We demonstrate that known structures for some evolution equations (e.g., the KdV equation) are special cases of those considered here.

1. Introduction.

This text is based on my old preprint [6], never published elsewhere. In [5] we introduced the concept of Poisson structure for any differential

equation E ⊂ J^{k}(π), where π: E → M is a smooth vector bundle and J^{k}(π) is the total space of the bundle π_{k}: J^{k}(π) → M of its k-jets. The general outline of the theory is as follows.

Let A be a commutative ℤ^{n}-graded algebra with a differential δ: A → A, δ∘ δ = 0. We shall always assume that δ is odd, i.e., 1 + δ · δ ∈ 2ℤ. Then δinvariant calculus over A can be constructed. Namely, we consider differential operators Δ: A → A satisfying [[δ, Δ]]^{s} = 0 and classes of such operators modulo operators of the form Δ = [[δ, ∇]]^{s}. Here [[•, •]]^{s} is the super Schouten bracket. Denote the set of these classes by Diff(A; δ). Then Diff(A; δ) acts on

The research described in this publication was made possible in part by the INTAS grant 96-0793..

1991 Mathematical Subject Classification. 35A30, 58F05, 58G05, 16W55, 81T60. Key words and phrases. nonlinear evolution differential equations, Poisson struc-

tures, Hamiltonian formalism.

the homology group H(A; δ) = ker δ/ im δ and in fact it is this action which really matters in the theory, cf. A.M. Vinogradov’s approach to the so-called “Secondary Calculus” in [14].

The set Diff(A; δ) inherits the associative algebra structure with respect to the composition of differential operators. One can extend the action of δ to Diff(A; δ) by setting δ([Δ]) = [δ∘Δ], where Δ: A → A is a differential operator and [·] denotes its class. Obviously, δ ∘ δ = 0 again, which makes it possible to define the modules Diff^{i}(A; δ) =^{def} Diff(A, Diff^{i−1}(A; δ); δ). Thus the whole conceptual approach of differential calculus in commutative algebras [12] can be carried over to the δ-invariant setting. In particular, one can define modules of δ-invariant polyderivations D_{i}(A; δ) as well as of symmetric Diff_{i}^{sym}(A; δ) and skew-symmetric Diff_{i}^{alt}(A; δ) polydifferential operators respectively and define the evaluation maps

D_{i}(A; δ) ⊗ H(A; δ) → D_{i−1}(A; δ),
Diff_{i}^{sym}(A; δ) ⊗ H(A; δ) → Diff_{i−1}^{sym}(A; δ),
Diff_{i}^{alt}(A; δ) ⊗ H(A; δ) → Diff_{i−1}^{alt}(A; δ).

The construction of the super Schouten bracket [4] is obviously δ-invariant and consequently one obtains the brackets

[[D_{i}(A; δ), D_{j}(A; δ)]]^{s} ⊂ D_{i+j}(A; δ),
[[ Diff_{i}^{alt}(A; δ), Diff_{j}^{alt}(A; δ)]]^{s} ⊂ Diff_{i+j}^{alt}(A; δ).

A δ-invariant Poisson structure in A is defined as an element P ∈ D_{2}(A; δ) satisfying [[P, P]]^{s} = 0 and such that P · P ∈ 2ℤ. In a similar manner one can define higher order Poisson structures as elements of Diff_{2}^{alt}(A; δ) complying with the same conditions.

These general algebraic considerations are specified for nonlinear differential equations as follows. For an equation E we consider its infinite prolongation^{1} [7] E^{∞} ⊂ J^{∞}(π) ⟶^{π_{∞}} M and define A =^{def} Λ^{∗}(E^{∞}), δ =^{def} d_{h}, where d_{h} is the horizontal differential on E^{∞}. In [4] it was shown that higher Poisson structures in this case can be identified with pairs of homotopy classes of super differential operators acting from Λ^{∗}(E^{∞}) to Λ^{∗}(E^{∞}) ⊗ D^{v}(E^{∞}) and satisfying certain additional conditions (see below). Here D^{v}(E^{∞}) is the module of π_{∞}- vertical vector fields on E^{∞}, while homotopy classes are considered with respect to the differentials d_{h} and ∂_{C}, the latter being the differential associated to the Cartan connection on E^{∞} [3]. In [4] we have also described Poisson structures for the case E^{∞} = J^{∞}(π).

Here the results obtained in [4] are used to describe Poisson structures for systems of evolution equations E with time t and space variables x_{1}, …, x_{n}. In Section 2 preparatory material for the reduction to E is presented. Section 3 contains a description of Poisson structures. Finally, in Section 4 we consider an illustrative example of the KdV equation.

1The equation E is supposed to be formally integrable.

129 COHOMOLOGICAL APPROACH TO POISSON STRUCTURES

Let us finally note that, in view of a recent paper by A.M Verbovetsky [15], the results developed below seem to be valid for an arbitrary formally integrable equation.

2. Reduction.

Let E ⊂ J^{k}(π), π: E → M, dim M = n + 1, dim π = m, be a differential equation of the form

\frac{∂u}{∂t} = Ψ(t, x, …, \frac{∂^{|σ|}u}{∂x^{σ}}, …), |σ| ≤ k,  (2.2.1)

where (t, x) are local coordinates in U ⊂ M with x = (x_{1}, …, x_{n}) and u = (u^{1}, …, u^{m}) are coordinates along the fiber of π corresponding to a trivialization of π over U. Here σ = (σ_{1}, …, σ_{n}), |σ| = σ_{1} + ⋯ + σ_{n} and

\frac{∂^{|σ|}u}{∂x^{σ}} = \frac{∂^{|σ|}u}{∂x_{1}^{i_{1}} … ∂x_{n}^{i_{n}}} .

Let (…, u_{σ}^{α}, …) be adapted coordinates on E^{∞}.

Denote by

D_{i} = \frac{∂}{∂x_{i}} + ∑_{σ,α} u_{σ+1_{i}}^{α} \frac{∂}{∂u_{σ}^{α}}

the total derivatives corresponding to x_{i}, i = 1, …, n and let

D_{t} = \frac{∂}{∂t} + ∑_{σ,α} D_{σ}(Ψ^{α})\frac{∂}{∂u_{σ}^{α}}

be the total derivative corresponding to t. We use the notations D_{σ} = D_{1}^{σ_{1}} ∘ ⋯ ∘ D_{n}^{σ_{n}} and σ + 1_{i} = (σ_{1}, …, σ_{i} + 1, …, σ_{n}). To any equation E the horizontal differential d_{h}: Λ_{h}^{q}(E^{∞}) → Λ_{h}^{q+1}(E^{∞}) is associated, where Λ_{h}^{∗}(E^{∞}) = ∑_{q≥0} Λ_{h}^{q}(E^{∞}) is the algebra of horizontal forms on E^{∞}. For (2.2.1) the horizontal differential is represented as

d_{h} = dt ∧ D_{t} + ∑_{i=1}^{n} dx_{i} ∧ D_{i}.  (2.2.2)

Hence, (2.2.2) can be rewritten as

d_{h} = dt ∧ D_{t} + d_{x},  (2.2.3)

where d_{x} is the horizontal differential for the “empty equation”. To be more precise, we consider the bundle π′: ℝ^{n+m} → ℝ^{n} with coordinates (x_{1}, …, x_{n}) in the base and (u^{1}, …, u^{m}) along the fiber and set d_{x} = d_{h} for the horizontal differential d_{h} on J^{∞}(π′).

Let now τ: E_{τ}^{∞} → E^{∞} be a linear covering over E^{∞} [8, 9]. In particular, τ is a vector bundle and we denote by Γ(τ) the module of its sections. Then horizontal cohomologies of E with coefficients in Γ(τ) are defined. In fact, let C: D(M) → D(E^{∞}) be the Cartan connection on E^{∞} (see [7]). Denote by Der(τ) the module of first order differential operators Δ: Γ(τ) → Γ(τ) such that

Δ(φf) = X(φ)f + φΔ(f), f ∈ Γ(τ), φ ∈ C^{∞}(E^{∞}),

for some X = X_{Δ} ∈ D(E^{∞}). Then by the definition of a covering, for any vector field X ∈ D(M) there exists an operator X^{τ} ∈ Der(τ) such that

X^{τ}(φf) = CX(φ)f + φX^{τ}(f)

and

[X, Y]^{τ} = [X^{τ}, Y^{τ}], X, Y ∈ D(M).  (2.2.4)

Consider two vector bundles ξ_{i}: E_{i} → M, i = 1, 2, over M and a C-differen-tial operator^{2} ∇: Γ_{1} =^{def} Γ(π_{∞}^{∗}(ξ_{1})) → Γ_{2} =^{def} Γ(π_{∞}^{∗}(ξ_{2})). Then from (2.2.4) it follows that such an operator determines an operator ∇^{τ}: Γ_{1} ⊗Γ(τ) → Γ_{2} ⊗Γ(τ). In particular, since d_{h} is a C-differential operator (see (2.2.2)), for any covering τ one obtains the operator

d_{h}^{τ}: Λ_{h}^{q}(E^{∞}) ⊗ Γ(τ) → Λ_{h}^{q+1}(E^{∞}) ⊗ Γ(τ)

satisfying

d_{h}^{τ} ∘ d_{h}^{τ} = 0.

Consequently, one can define the cohomology group

H_{h}^{q}(E; τ) =^{def} ker d_{h}^{τ}/im d_{h}^{τ},

where image and kernel of d_{h}^{τ} are taken at the term Λ_{h}^{q}(E^{∞}) ⊗ Γ(τ). Similarly to (2.2.3) we can express d_{h}^{τ} as

d_{h}^{τ} = dt ∧ D_{t}^{τ} + d_{x}^{τ}.  (2.2.5)

Let us introduce the notation Λ_{h}^{∗}(E^{∞}; τ) =^{def} Λ_{h}^{∗}(E^{∞}) ⊗ Γ(τ) and decompose the module Λ_{h}^{∗}(E^{∞}; τ) into the direct sum

Λ_{h}^{∗}(E^{∞}; τ) = Λ_{h}^{∗}(π′; τ) ⊕ dt ∧ Λ_{h}^{∗}(π′; τ),  (2.2.6)

where Λ_{h}^{∗}(π′; τ) are Γ(τ)-valued horizontal forms on J^{∞}(π′) for π′: ℝ^{n+m} → ℝ^{n} (see above). Now, for (2.2.5) one has

0 = d_{h}^{τ} ∘ d_{h}^{τ} = (dt ∧ D_{t}^{τ} + d_{x}^{τ}) ∘ (dt ∧ D_{t}^{τ} + d_{x}^{τ}) = (dt ∧ D_{t}^{τ}) ∘ (dt ∧ D_{t}^{τ}) + (dt ∧ D_{t}^{τ}) ∘ d_{x}^{τ} + d_{x}^{τ} ∘ (dt ∧ D_{t}^{τ}) + d_{x}^{τ} ∘ d_{x}^{τ}.

But d_{x}^{τ} ∘ d_{x}^{τ} = 0 as well as

(dt ∧ D_{t}^{τ}) ∘ (dt ∧ D_{t}^{τ}) = dt ∧ dt ∧ (D_{t}^{τ} ∘ D_{t}^{τ}) = 0.

Consequently

d_{x}^{τ} ∘ (dt ∧ D_{t}^{τ}) + (dt ∧ D_{t}^{τ}) ∘ d_{x}^{τ} = 0  (2.2.7)

and since dt ∧ D_{t}^{τ} acts trivially on the second summand of (2.2.6), we obtain the bicomplex

\begin{matrix}dt ∧ Λ_{h}^{∗}(π′; τ) & \overline{→^{−dt∧d_{x}^{τ}}} & dt ∧ Λ_{h}^{∗}(π′; τ) \\ Λ_{h}^{∗}(π′; τ) & ^{^{dt∧D_{t}^{τ}} ↑} & \overline{^{d_{x}^{τ}} →} & Λ_{h}^{∗}(π′; τ) & ^{↑^{dt∧D_{t}^{τ}}} \end{matrix}

2Recall that a C-differential operator is an operator admitting restrictions to the graphs of formal solutions, [7]. In local coordinates, C-differential operators are expressed in total derivatives.

whose total differential coincides with d_{h}^{τ}. Denote by H_{h}^{q}(π′; τ) cohomology of the differential d_{x}^{τ} at the q-th term. Then due to (2.2.7), D_{t}^{τ} determines the map

∂_{t}^{τ,[q]} : H_{h}^{q}(π′; τ) → H_{h}^{q}(π′; τ)

of cohomology groups and by standard reasoning [11] we obtain

Theorem 2.1. Let E be an evolution equation of the form (2.2.1) and τ: E_{τ}^{∞} → E^{∞} be a linear covering. Then horizontal cohomology of E with coefficients in τ is of the form

H_{h}^{q}(E; τ) = ker ∂_{t}^{τ,[q]} ⊕ dt ∧ coker ∂_{t}^{τ,[q−1]},

where ∂_{t}^{τ,[q]} is the cohomology map induced by D_{t}^{τ}.

In what follows we shall need two particular coverings naturally associated to any equation E ⊂ J^{k}(π). Denote by C^{1}Λ(E^{∞}) the module of Cartan one-forms over E^{∞}:

C^{1}Λ(E^{∞}) = { ω ∈ Λ^{1}(E^{∞}) | CX⌋ω, ∀X ∈ D(M) } .

and by D^{v}(E^{∞}) the Lie algebra of vertical vector fields:

D^{v}(E^{∞}) = { Z ∈ D(E^{∞}) | Z(f) = 0, ∀f ∈ C^{∞}(M) } .

Then there exists the direct sum decompositions

D(E^{∞}) = CD(E^{∞}) ⊕ D^{v}(E^{∞}),

where CD(E^{∞}) is spanned in D(E^{∞}) by the fields CX, X ∈ D(M), and

Λ^{1}(E^{∞}) = Λ_{h}^{1}D(E^{∞}) ⊕ C^{1}Λ(E^{∞}).  (2.2.8)

Note that from (2.2.8) it follows that

Λ^{i}(E^{∞}) = ∑_{p+q=i} Λ_{h}^{q}(E^{∞}) ⊗ C^{p}Λ(E^{∞}),

where

C^{p}Λ(E^{∞}) = \underbrace{C^{1}Λ(E^{∞}) ∧ … ∧ C^{1}Λ(E^{∞})}_{p times} .

Denote by τ^{p}: E_{τ^{p}}^{∞} → E^{∞} the vector bundle determined by

Γ(τ^{p}) = C^{p}Λ(E^{∞})

and by τ^{p,v}: E_{τ^{p,v}}^{∞} → E^{∞} the bundle satisfying

Γ(τ^{p,v}) = C^{p}Λ(E^{∞}) ⊗ D^{v}(E^{∞}).

Let X ∈ D(M). For any ω ∈ C^{p}Λ(E^{∞}) and Z ∈ D^{v}(E^{∞}) we set

X^{τ^{p}}(ω) = L_{CX}(ω)  (2.2.9)

and

X^{τ^{p,v}}(ω ⊗ Z) = L_{CX}(ω) ⊗ Z + ω ⊗ [CX, Z].  (2.2.10)

Proposition 2.2. Formulas (2.2.9) and (2.2.10) determine covering structures in τ^{p} and τ^{p,v} respectively.

Proof. Obviously, X^{τ^{p}} ∈ Der(τ^{p}) and X^{τ^{p,v}} ∈ Der(τ^{p,v}) for any X ∈ D(M). Let φ ∈ C^{∞}(M). Then

L_{C(φX)}ω = L_{φCX}ω = φL_{CX}ω + dφ ∧ (CX⌋ω)

and

[C(φX), Z] = [φCX, Z] = φ[CX, Z] − Z(φ)CX.

But CX⌋ω = 0 and Z(φ) = 0 by the definitions of C^{p}Λ(E^{∞}) and D^{v}(E^{∞}). Finally, the relations [X^{τ^{p}}, Y^{τ^{p}}] = [X, Y]^{τ^{p}} and [X^{τ^{p,v}}, Y^{τ^{p,v}}] = [X, Y]^{τ^{p,v}} are consequences of

L_{[CX,CY]} = [L_{CX}, L_{CY}]

and

[CX, [CY, CZ]] − [CY, [CX, CZ]] = [[CX, CY], CZ].

Proposition 2.3. Let E be an equation of the form (2.2.1). Then:

(i) The horizontal complex of E^{∞} with coefficients in τ^{p} coincides with the p-th column of the Vinogradov C-spectral sequence term E_{0} (see [13]):

d_{h}^{τ^{p}} = d_{0}^{p,q}: Λ_{h}^{q}(E^{∞}) ⊗ C^{p}Λ(E^{∞}) → Λ_{h}^{q+1}(E^{∞}) ⊗ C^{p}Λ(E^{∞}).

(ii) The horizontal complex of E^{∞} with coefficients in τ^{p,v} coincides with the C-complex for E^{∞} (see [3]) restricted onto Λ_{h}^{∗}(E^{∞}) ⊗ C^{p}Λ(E^{∞}) ⊗ D^{v}(E^{∞}):

d_{h}^{τ^{p,v}} = ∂_{C}^{p,q}: Λ_{h}^{q}(E^{∞}) ⊗ C^{p}Λ(E^{∞}) ⊗ D^{v}(E^{∞}) →

→ Λ_{h}^{q+1}(E^{∞}) ⊗ C^{p}Λ(E^{∞}) ⊗ D^{v}(E^{∞}).

Proof. It suffices to compare corresponding expressions for the above mentioned differentials.

Recall that in the case E^{∞} = J^{∞}(π), π: ℝ^{m+n} → ℝ^{n}, there exists an exact description of the groups H_{h}^{q}(π′; τ) for τ = τ^{p} and τ = τ^{p,v}.

Theorem 2.4. Let E be an “empty” equation in the bundle π: ℝ^{m+n} → ℝ^{n}. Then (see [13] for (i) and [3] for (ii)):

(i) Let κ = Γ((π′)^{∗}(π′)) and κ̂ = hom(κ, Λ_{h}^{n}(π′)). Then

H_{h}^{q}(π′; τ^{p}) = \begin{cases}ℝ & for p = q = 0, \\ H_{h}^{q}(π′) & for p = 0, q = n, \\ CA Diff_{p−1}^{alt}(κ, κ̂) & for p ≥ 0, q = n, \\ 0, & otherwise, \end{cases}

where CA Diff_{i}^{alt}(κ, κ̂) is the module of anti self-adjoint skew-symmetric C-differential operators in i variables.

(ii)

H_{h}^{q}(π′; τ^{p,v}) = \begin{cases}C^{p}Λ(π′) ⊗ κ & for p ≥ 0, q = 0, \\ 0, & otherwise, \end{cases}

and for any ω ∈ C^{p}Λ(π′) ⊗ κ corresponding Λ^{∗}(π′)-valued vertical derivation is of the form

�_{ω} = ∑_{α=1}^{m} ∑_{|σ|≥0} D_{σ}(ω^{α})\frac{∂}{∂u_{σ}^{α}} .

Remark 2.1. Correspondence between E_{0}^{i,n}(π′) and C Diff_{i}^{alt}(κ, Λ_{h}^{n}(π′)) is established by the relation

E_{0}^{i,n}(π′) ∋ ω ↦ Δ_{ω} ∈ C Diff_{i}^{alt}(κ, Λ_{h}^{n}(π′)),

where we set

Δ_{ω}(φ_{1}, …, φ_{i}) =^{def} �_{φ_{1}}⌋(… �_{φ_{i}}⌋ω), φ_{1}, …, φ_{i} ∈ κ.

We also formally set CA Diff_{0}^{alt}(κ, Λ_{h}^{n}(π′)) = C Diff_{0}^{alt}(κ, Λ_{h}^{n}(π′)) = κ̂ (see also Remark 4.1 on page 143).

Corollary 2.5 (see [3, 13]). Let E be a determined evolution equation of the form (2.2.1). Then

(i) For any Δ ∈ CA Diff_{p−1}^{alt}(κ, κ̂) and φ_{1}, …, φ_{p−1} ∈ κ one has

(∂_{t}^{τ^{p},[n]})(φ_{1}, …, φ_{p−1})

= ℓ̂_{E}(Δ(φ_{1}, …, φ_{p−1})) + ∑_{i=1}^{p−1} Δ(φ_{1}, …, ℓ_{E}(φ_{i}), …, φ_{p−1}) (2.2.11)

where ℓ_{E}: κ → κ is the universal linearization operator for E restricted onto E^{∞} and “hat” denotes formally adjoint.

(ii) Cohomology groups H_{h}^{q}(E; τ^{p}) are nontrivial only in the cases p = q = 0 and q ≥ n:

H_{h}^{q}(π′; τ^{p}) = \begin{cases}ℝ & for p = q = 0, \\ ker ∂_{t}^{τ^{p},[n]} & for q = n, \\ dt ∧ coker ∂_{t}^{τ^{p},[n]} & for q = n + 1, \\ 0, & otherwise. \end{cases}

(iii) For any ω ∈ C^{p}Λ(π′) ⊗ κ one has

∂_{t}^{τ^{p,v},[0]}ω = ℓ_{E}^{[p]}ω,

where ℓ_{E}^{[p]} is a natural extension of ℓ_{E} onto C^{p}Λ(π′) ⊗ κ. (iv) Cohomology groups H_{h}^{q}(E; τ^{p,v}) are nontrivial only in the cases q ≤ 1:

H_{h}^{q}(π′; τ^{p,v}) = \begin{cases}ker ℓ_{E}^{[p]} & for q = 0, \\ dt ∧ coker ℓ_{E}^{[p]} & for q = 1, \\ 0, & otherwise. \end{cases}

3. Poisson structures.

Here we use the description of Poisson structures obtained in [5] for the case E^{∞} = J^{∞}(π) to describe such structures over determined systems of evolution equations. The techniques below are based on reduction procedures of Section 2.

First recall the main definition [5]:

Definition 3.1. Let E be a formally integrable equation. A Poisson structure on E is a map

P: E_{1}(E) → H_{C}^{∗}(E) ⊕ H_{C}^{∗}(E),

where E_{1}(E) is the first term of the C-spectral sequence and H_{C}^{∗}(E) is the Ccohomology group of E. It is assumed that P is generated by a pair P = (P_{0}, P_{1}) of Λ^{∗}(E^{∞})-differential operators

P_{i}: Λ^{∗}(E^{∞}) → Λ^{∗}(E^{∞}) ⊗ D^{v}(E^{∞})

satisfying

(i) deg P_{0} ∈ 2ℤ, deg P_{1} = deg P_{0} + 1, (ii) P_{i} ∘ d_{0} + ∂_{C} ∘ P_{i} = 0, (iii) {ω, θ}_{P} + (−1)^{ωθ}{θ, ω}_{P} = 0, (iv) P {ω, θ}_{P} = [P ω, P θ],

where ω, θ ∈ Λ^{∗}(E^{∞}), {ω, θ}_{P} =^{def} (P ω)θ, while for any pair X = (X_{0}, X_{1}), X_{i} ∈ Λ^{∗}(E^{∞}) ⊗ D^{v}(E^{∞}), its action on Λ^{∗}(E^{∞}) is defined by

Xω = L_{X_{0}}ω + i_{X_{1}}ω  (3.3.1)

and the right-hand side in (iv) is understood as the commutator of such actions.

In what follows we often identify Poisson structures P with their representative pairs P = (P_{0}, P_{1}).

Let now E be a determined equation (2.2.1) and τ_{i}: E_{τ_{i}}^{∞} → E^{∞}, i = 1, 2, be two coverings over E. Then any ℝ-linear map Δ: Λ_{h}^{∗}(E^{∞}; τ_{1}) → Λ_{h}^{∗}(E^{∞}; τ_{2}) can be represented as a 2 × 2-matrix

Δ = (\begin{matrix}Δ_{x}^{x} & Δ_{x}^{t} \\ Δ_{t}^{x} & Δ_{t}^{t} \end{matrix})

The components of this matrix are maps Λ_{h}^{∗}(π′; τ_{1}) → Λ_{h}^{∗}(π′; τ_{2}) such that

Δ(ω_{x} + dt ∧ ω_{t}) = Δ_{x}^{x}ω_{x} + Δ_{x}^{t}ω_{t} + dt ∧ (Δ_{t}^{x}ω_{x} + Δ_{t}^{t}ω_{t}).  (3.3.2)

Our task is to express determining properties (i)–(iv) of Poisson structures in terms of decomposition (3.3.2), when

τ_{1} = τ^{∗} = τ^{0} ⊕ ⋯ ⊕ τ^{p} ⊕ …, τ_{2} = τ^{∗,v} = τ^{0,v} ⊕ ⋯ ⊕ τ^{p,v} ⊕ …

(i) Obviously,

deg Δ_{x}^{x} = deg Δ,  deg Δ_{x}^{t} = deg Δ + 1,
deg Δ_{t}^{x} = deg Δ − 1,  deg Δ_{t}^{t} = deg Δ.  (3.3.3)

(ii) Let ω = ω_{x} + dt ∧ ω_{t} ∈ Λ_{h}^{∗}(E^{∞}; τ_{1}). Then, due to (2.2.5) and (3.3.2), one has (Δ ∘ d_{h}^{τ_{1}})ω = Δ(d_{x}^{τ_{1}}ω_{x} + dt ∧ (D_{t}^{τ_{1}}ω_{x} − d_{x}^{τ_{1}}ω_{t}))

= Δ_{x}^{x}d_{x}^{τ_{1}}ω_{x} + Δ_{x}^{t}(D_{t}^{τ_{1}}ω_{x} − d_{x}^{τ_{1}}ω_{t})
+ dt ∧ (Δ_{t}^{x}d_{x}^{τ_{1}}ω_{x} + Δ_{t}^{t}(D_{t}^{τ_{1}}ω_{x} − d_{x}^{τ_{1}}ω_{t}))

while

(d_{h}^{τ_{2}} ∘ Δ)ω = d_{h}^{τ_{2}}(Δ_{x}^{x}ω_{x} + Δ_{x}^{t}ω_{t} + dt ∧ (Δ_{t}^{x}ω_{x} + Δ_{t}^{t}ω_{t})) = d_{x}^{τ_{2}}(Δ_{x}^{x}ω_{x} + Δ_{x}^{t}ω_{t})

+ dt ∧ (D_{t}^{τ_{2}}(Δ_{x}^{x}ω_{x} + Δ_{x}^{t}ω_{t}) − d_{x}^{τ_{2}}(Δ_{t}^{x}ω_{x} + Δ_{t}^{t}ω_{t})).

Hence Δ ∘ d_{h}^{τ_{1}} + d_{h}^{τ_{2}} ∘ Δ = 0 if and only if

Δ_{x}^{x} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ Δ_{x}^{x} + Δ_{x}^{t} ∘ D_{t}^{τ_{1}} = 0,  (3.3.4)
−Δ_{x}^{t} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ Δ_{x}^{t} = 0,  (3.3.5)
Δ_{t}^{x} ∘ d^{τ_{1}} − d_{x}^{τ_{2}} ∘ Δ_{t}^{x} + Δ_{t}^{t} ∘ D_{t}^{τ_{1}} + D_{t}^{τ_{2}} ∘ Δ_{x}^{x} = 0,  (3.3.6)
−Δ_{t}^{t} ∘ d_{x}^{τ_{1}} − d_{x}^{τ_{2}} ∘ Δ_{t}^{t} + D_{t}^{τ_{2}} ∘ Δ_{x}^{t} = 0.  (3.3.7)

Since Λ^{∗}(E^{∞})-differential operators

Δ: Λ^{∗}(E^{∞}) = Λ_{h}^{∗}(E^{∞}; τ^{∗}) → Λ^{∗}(E^{∞}) ⊗ D^{v}(E^{∞}) = Λ_{h}^{∗}(E^{∞}; τ^{∗,v})

are bigraded^{3}, one can split all operators Δ satisfying (3.3.4)–(3.3.7) into three types.

\underline{Type I} consists of operators Δ = Δ_{x}^{t} ∘i_{D_{t}}: dt∧Λ_{h}^{∗}(π′; τ_{1}) → Λ_{h}^{∗}(π′; τ_{2}) which satisfy

Δ_{x}^{t} ∘ d_{x}^{τ_{1}} − d_{x}^{τ_{2}} ∘ Δ_{x}^{t} = 0,
Δ_{x}^{t} ∘ D_{t}^{τ_{1}} = 0,
D_{t}^{τ_{2}} ∘ Δ_{x}^{t} = 0.

The corresponding cohomology map acts as

dt ∧ coker ∂_{t}^{τ^{∗},[n]} → ker ∂_{t}^{τ^{∗,v},[0]}.

\underline{Type II}. Operators of this type are of the form Δ = (dt∧)∘Δ_{t}^{x}: Λ_{h}^{∗}(π′; τ_{1}) → dt ∧ Λ_{h}^{∗}(π′; τ_{2}) satisfying

Δ_{t}^{x} ∘ d_{x}^{τ_{1}} − d_{x}^{τ_{2}} ∘ Δ_{t}^{x} = 0.

The cohomology map

ker ∂_{t}^{τ^{∗},[n]} → coker ∂_{t}^{τ^{∗,v},[0]}

in this case reduces to the composition of restriction onto ker ∂_{t}^{τ^{∗},[n]} with the quotient map dt ∧ H_{h}^{0}(π′; τ^{∗,v}) → dt ∧ coker ∂_{t}^{τ^{∗,v},[0]}.

\underline{Type III}. In this case Δ = (Δ_{x}^{x}, dt ∧ Δ_{t}^{t}), where the differential operators Δ_{x}^{x}, Δ_{t}^{t}: Λ_{h}^{∗}(π′; τ_{1}) → Λ_{h}^{∗}(π′; τ_{2}) satisfy

Δ_{x}^{x} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ Δ_{x}^{x} = 0,  (3.3.8)
Δ_{t}^{t} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ Δ_{t}^{t} = 0,  (3.3.9)
Δ_{t}^{t} ∘ D_{t}^{τ_{1}} + D_{t}^{τ_{2}} ∘ Δ_{x}^{x} = 0,  (3.3.10)

3With respect to ℤ^{2}-graded structure Λ^{∗}(E^{∞}) = ∑_{p,q} Λ_{h}^{q}(E^{∞}) ⊗ C^{p}Λ(E^{∞}), see above.

which means that we have cohomology maps

ker ∂_{t}^{τ^{∗},[n]} → ker ∂_{t}^{τ^{∗,v},[0]}

and

dt ∧ coker ∂_{t}^{τ^{∗},[n]} → dt ∧ coker ∂_{t}^{τ^{∗,v},[0]}

induced by Δ_{x}^{x} and Δ_{t}^{t} respectively. In what follows we confine ourselves to the last type of operators only. We call Poisson structures ot this type normal type Poisson structures.

(iii) Let P = (P_{0}, P_{1}) be a Poisson structure on E and

P_{0} = P_{0,x}^{x} + dt ∧ P_{0,t}^{t}, P_{1} = P_{1,x}^{x} + dt ∧ P_{1,t}^{t}.

Denote

P_{x}^{x} =^{def} (P_{0,x}^{x}, P_{1,x}^{x}), P_{t}^{t} =^{def} (P_{0,t}^{t}, P_{1,t}^{t})

and express the Poisson bracket {•, •}_{P} in terms of P_{x}^{x} and P_{t}^{t}. For any ω, θ ∈ Λ^{∗}(E^{∞}) one has

{ω, θ}_{P} = (P ω)θ = L_{P_{0}ω}θ + i_{P_{1}ω}θ = L_{P_{0,x}^{x}ω_{x}}θ_{x} + L_{P_{0,x}^{x}ω_{x}}(dt ∧ θ_{t}) + L_{dt∧P_{0,t}^{t}ω_{t}}θ_{x} + L_{dt∧P_{0,t}^{t}ω_{t}}(dt ∧ θ_{t})

+ i_{P_{1,x}^{x}ω_{x}}θ_{x} + i_{P_{1,x}^{x}ω_{x}}(dt ∧ θ_{t}) + i_{dt∧P_{1,x}^{x}ω_{t}}θ_{x} + i_{dt∧P_{1,x}^{x}ω_{t}}(dt ∧ θ_{t})

= {ω_{x}, θ_{x}}_{P_{x}^{x}} + dt ∧ ({ω_{t}, θ_{x}}_{P_{t}^{t}} + (−1)^{ω}{ω_{x}, θ_{t}}_{P_{x}^{x}}),

from where it follows that

{ω, θ}_{P} + (−1)^{ωθ}{θ, ω}_{P} = {ω_{x}, θ_{x}}_{P_{x}^{x}} + (−1)^{ωθ}{θ_{x}, ω_{x}}_{P_{x}^{x}}

+ dt ∧ ({ω_{t}, θ_{x}}_{P_{t}^{t}} + (−1)^{ω}{ω_{x}, θ_{t}}_{P_{x}^{x}}
+ (−1)^{ωθ}{θ_{t}, ω_{x}}_{P_{t}^{t}} + (−1)^{(ω+1)θ}{θ_{x}, ω_{t}}_{P_{x}^{x}}).

Therefore

{ω_{x}, θ_{x}}_{P_{x}^{x}} + (−1)^{ωθ}{θ_{x}, ω_{x}}_{P_{x}^{x}} = 0  (3.3.11)

and

{•, •}_{P_{x}^{x}} = {•, •}_{P_{t}^{t}}.  (3.3.12)

(iv) Let again P = (P_{0}, P_{1}) be a Poisson structure of normal type. Then

P {ω, θ}_{P} = L_{P_{0}{ω,θ}_{P}} + i_{P_{1}{ω,θ}_{P}}

and, as it was shown above,

{ω, θ}_{P} = {ω_{x}, θ_{x}}_{P_{x}^{x}} + dt ∧ ({ω_{t}, θ_{x}}_{P_{t}^{t}} + (−1)^{ω}{ω_{x}, θ_{t}}_{P_{x}^{x}}).

Consequently, one has

P_{0}{ω, θ}_{P} = P_{0,x}^{x}{ω_{x}, θ_{x}}_{P_{x}^{x}} + dt ∧ P_{0,t}^{t}({ω_{t}, θ_{x}}_{P_{t}^{t}} + (−1)^{ω}{ω_{x}, θ_{t}}_{P_{x}^{x}})

and

P_{1}{ω, θ}_{P} = P_{1,x}^{x}{ω_{x}, θ_{x}}_{P_{x}^{x}} + dt ∧ P_{1,t}^{t}({ω_{t}, θ_{x}}_{P_{t}^{t}} + (−1)^{ω}{ω_{x}, θ_{t}}_{P_{x}^{x}}),

from where it follows that

P {ω, θ}_{P} = P_{x}^{x}{ω_{x}, θ_{x}}_{P_{x}^{x}} + dt ∧ (P_{t}^{t}{ω_{t}, θ_{x}}_{P_{t}^{t}} + (−1)^{ω}P_{t}^{t}{ω_{x}, θ_{t}}_{P_{x}^{x}}).

On the other hand

[P ω, P θ] = [L_{P_{0}ω} + i_{P_{1}ω}, L_{P_{0}θ} + i_{P_{1}θ}] = [L_{P_{0,x}^{x}ω_{x}}, L_{P_{0,x}^{x}θ_{x}}] + [L_{P_{0,x}^{x}ω_{x}}, L_{dt∧P_{0,t}^{t}θ_{t}}]

+ [L_{P_{0,x}^{x}ω_{x}}, i_{P_{1,x}^{x}θ_{x}}] + [L_{P_{0,x}^{x}ω_{x}}, i_{dt∧P_{1,t}^{t}θ_{t}}] + [L_{dt∧P_{0,t}^{t}ω_{t}}, L_{P_{0,x}^{x}θ_{x}}] + [L_{dt∧P_{0,t}^{t}ω_{t}}, i_{P_{1,x}^{x}θ_{x}}]

+ [i_{P_{1,x}^{x}ω_{x}}, L_{P_{0,x}^{x}θ_{x}}] + [i_{P_{1,x}^{x}ω_{x}}, L_{dt∧P_{0,t}^{t}θ_{t}}] + [i_{P_{1,x}^{x}ω_{x}}, i_{P_{1,x}^{x}θ_{x}}] + [i_{P_{1,x}^{x}ω_{x}}, i_{dt∧P_{1,t}^{t}θ_{t}}]

+ [i_{dt∧P_{1,t}^{t}ω_{t}}, L_{P_{0,x}^{x}θ_{x}}] + [i_{dt∧P_{1,t}^{t}ω_{t}}, i_{P_{1,x}^{x}θ_{x}}] = [L_{P_{0,x}^{x}ω_{x}}, L_{P_{0,x}^{x}θ_{x}}] + [L_{P_{0,x}^{x}ω_{x}}, i_{P_{1,x}^{x}θ_{x}}]

+ [i_{P_{1,x}^{x}ω_{x}}, L_{P_{0,x}^{x}θ_{x}}] + [i_{P_{1,x}^{x}ω_{x}}, i_{P_{1,x}^{x}θ_{x}}] + dt ∧ ([L_{P_{0,t}^{t}ω_{t}}, L_{P_{0,x}^{x}θ_{x}}] + [L_{P_{0,t}^{t}ω_{t}}, i_{P_{1,x}^{x}θ_{x}}]

+[i_{P_{1,t}^{t}ω_{t}}, L_{P_{0,x}^{x}θ_{x}}]+[i_{P_{1,t}^{t}ω_{t}}, i_{P_{0,x}^{x}θ_{x}}]+(−1)^{ω}[L_{P_{0,x}^{x}ω_{x}}, L_{P_{0,t}^{t}θ_{t}}]+(−1)^{ω}[L_{P_{0,x}^{x}ω_{x}}, i_{P_{1,t}^{t}θ_{t}}]

+ (−1)^{ω}[L_{P_{1,x}^{x}ω_{x}}, L_{P_{0,t}^{t}θ_{t}}] + (−1)^{ω}[i_{P_{1,x}^{x}ω_{x}}, i_{P_{1,t}^{t}θ_{t}}])
= [P_{x}^{x}ω_{x}, P_{x}^{x}θ_{x}] + dt ∧ ([P_{t}^{t}ω_{t}, P_{x}^{x}θ_{x}] + (−1)^{ω}[P_{x}^{x}ω_{x}, P_{t}^{t}θ_{t}]).

Hence,

P_{x}^{x}{ω_{x}, θ_{x}}_{P_{x}^{x}} + dt ∧ (P_{t}^{t}{ω_{t}, θ_{x}}_{P_{t}^{t}} + (−1)^{ω}P_{t}^{t}{ω_{x}, θ_{t}}_{P_{x}^{x}})

= [P_{x}^{x}ω_{x}, P_{x}^{x}θ_{x}] + dt ∧ ([P_{t}^{t}ω_{t}, P_{x}^{x}θ_{x}] + (−1)^{ω}[P_{x}^{x}ω_{x}, P_{t}^{t}θ_{t}]).

Collecting together the results obtained we get

Proposition 3.1. Let E be a determined evolution equation and τ_{1} = τ^{∗}, τ_{2} = τ^{∗,v} be the two natural coverings over E^{∞} introduced above. A pair P = (P_{0}, P_{1}) of Λ^{∗}(E^{∞})-differential operators P_{i}: Λ_{h}^{∗}(E^{∞}; τ_{1}) → Λ_{h}^{∗}(E^{∞}; τ_{2}), i = 0, 1, determines a normal type Poisson structure on E if and only if :

(i) P_{0} = P_{0,x}^{x} + dt ∧ P_{0,t}^{t}, P_{1} = P_{1,x}^{x} + dt ∧ P_{1,t}^{t}, where P_{i,x}^{x}, P_{i,t}^{t} are differential operators acting from Λ_{h}^{∗}(π′; τ_{1}) to Λ_{h}^{∗}(π′; τ_{2}).

(ii) deg P_{0,x}^{x} = deg P_{0,t}^{t} ∈ 2ℤ, deg P_{1,x}^{x} = deg P_{1,t}^{t} = deg P_{0,x}^{x} + 1. (iii) Operators P_{i,x}^{x}, P_{i,t}^{t} satisfy the equations

P_{i,x}^{x} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ P_{i,x}^{x} = 0,  (3.3.13)
P_{i,t}^{t} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ P_{i,t}^{t} = 0,  (3.3.14)
P_{i,t}^{t} ∘ D_{t}^{τ_{1}} + D_{t}^{τ_{2}} ∘ P_{i,x}^{x} = 0.  (3.3.15)

(iv) Denote P_{x}^{x} = (P_{0,x}^{x}, P_{1,x}^{x}) and P_{t}^{t} = (P_{0,t}^{t}, P_{1,1}^{1}). Then

{ω_{x}, θ_{x}}_{P_{x}^{x}} + (−1)^{ωθ}{θ_{x}, ω_{x}}_{P_{x}^{x}} = 0, {•, •}_{P_{x}^{x}} = {•, •}_{P_{t}^{t}},

where ω_{x}, θ_{x} ∈ Λ_{h}^{∗}(E^{∞}; τ_{1}).

(v) Finally, for any ω_{x}, θ_{x}, ω_{t}, θ_{t} ∈ Λ_{h}^{∗}(E^{∞}; τ_{1}) one has

P_{x}^{x}{ω_{x}, θ_{x}}_{P_{x}^{x}} = [P_{x}^{x}ω_{x}, P_{x}^{x}θ_{x}],
P_{t}^{t}{ω_{t}, θ_{x}}_{P_{t}^{t}} = [P_{t}^{t}ω_{t}, P_{x}^{x}θ_{x}],
P_{t}^{t}{ω_{x}, θ_{t}}_{P_{x}^{x}} = [P_{x}^{x}ω_{x}, P_{t}^{t}θ_{t}].

Note now that both operators D_{t}^{τ_{1}} and D_{t}^{τ_{2}} are of the form D_{t}^{τ_{i}} = I∘D_{t} +∇_{i}, where I is the identity matrix while ∇_{i} are C-differential operators in D_{α} only. It means that any C-differential operator Δ: Λ_{h}^{∗}(π′; τ_{1}) → Λ_{h}^{∗}(π′; τ_{2}) can be presented as

Δ = ∇_{1} ∘ D_{t}^{τ_{1}} + Δ_{1}, Δ = D_{t}^{τ_{2}} ∘ ∇_{2} + Δ_{2},

where Δ_{i}, ∇_{i}: Λ_{h}^{∗}(π′; τ_{1}) → Λ_{h}^{∗}(π′; τ_{2}) and Δ_{1}, Δ_{2} are C-differential operators independent of D_{t} (see the diagram — which is not commutative!):

Λ_{h}^{∗}(π′; τ_{1})  \overline{^{D_{t}^{τ_{1}}} →} Λ_{h}^{∗}(π′; τ_{1})
[figure] Δ_{Δ1,Δ2}
Λ_{h}^{∗}(π′; τ_{2}) ↓^{∇_{2}}  \overline{_{_{D_{t}^{τ_{2}}} →}} Λ_{h}^{∗}(π′; τ_{2}) ↓^{∇_{1}}

Like in [4], we confine ourselves to the case, when Poisson structures under consideration are expressed by C-differential operators. Then one has

P_{i,x}^{x} = P_{i}^{x} + ∇_{i}^{x} ∘ D_{t}^{τ_{1}}

and

P_{i,t}^{t} = P_{i}^{x} + D_{t}^{τ_{2}} ∘ ∇_{i}^{t},

from where it follow that equations (3.3.13) and (3.3.14) transform to

(P_{i}^{x} + ∇_{i}^{x} ∘ D_{t}^{τ_{1}}) ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ (P_{i}^{x} + ∇_{i}^{x} ∘ D_{t}^{τ_{1}}) = 0

and

(P_{i}^{t} + D_{t}^{τ_{2}} ∘ ∇_{i}^{t}) ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ (P_{i}^{t} + D_{t}^{τ_{2}} ∘ ∇_{i}^{t}) = 0

respectively. Due to commutativity of the diagram on page 130, the last two equations acquire the form

P_{i}^{x} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ P_{i}^{x} + ¯∇_{i}^{x} ∘ D_{t}^{τ_{1}} = 0

and

P_{i}^{t} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ P_{i}^{t} + D_{t}^{τ_{2}} ∘ ¯∇_{i}^{x} = 0

for the operators ¯∇_{i}^{x} = ∇_{i}^{x} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}}∇_{i}^{x} and ¯∇_{i}^{t} = ∇_{i}^{t} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}}∇_{i}^{t}. But the first two summands in the left-hand sides of these equations do not depend on D_{t} and consequently one necessary has

P_{i}^{x} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ P_{i}^{x} = 0,  (3.3.16)
P_{i}^{t} ∘ d_{x}^{τ_{1}} + d_{t}^{τ_{2}} ∘ P_{i}^{x} = 0.  (3.3.17)

On the other hand, since Poisson structures are determined by cohomology maps ker D_{t}^{τ_{1}} → ker D_{t}^{τ_{2}} and coker D_{t}^{τ_{1}} → coker D_{t}^{τ_{2}}, equations (3.3.16) and (3.3.17) are sufficient. Note also that

(∇_{i}^{x} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ ∇_{i}^{x}) ∘ D_{t}^{τ_{1}} = 0,  (3.3.18)
D_{t}^{τ_{2}} ∘ (∇_{i}^{t} ∘ d_{x}^{τ_{1}} + d_{x}^{τ_{2}} ∘ ∇_{i}^{t}) = 0.  (3.3.19)

Applying the results of [5] to equations (3.3.16), (3.3.17), we see that

P_{i}^{x} = (P_{i0}^{x} ∘ E_{0} + P_{i1}^{x} ∘ E_{1}) ∘ Δ_{1} ∘ ⋯ ∘ Δ_{n},
P_{i}^{t} = (P_{i0}^{t} ∘ E_{0} + P_{i1}^{t} ∘ E_{1}) ∘ Δ_{1} ∘ ⋯ ∘ Δ_{n},

where Δ_{α} = i_{D_{α}}, E_{0}, E_{1} are the components of the super Euler operator and P_{ij}^{x}, P_{ij}^{t} are differential operators acting from the module C^{∗}Λ(π′)⊗κ̂ to C^{∗}Λ(π′)⊗κ. We assume the operators P_{ij}^{x}, P_{ij}^{t} to be C-differential ones.

To use a more compact form, we introduce the operators

P^{x} =^{def} (\begin{matrix}P_{00}^{x} & P_{01}^{x} \\ P_{10}^{x} & P_{11}^{x} \end{matrix}) , P^{t} =^{def} (\begin{matrix}P_{00}^{t} & P_{01}^{t} \\ P_{10}^{t} & P_{11}^{t} \end{matrix})

which may be understood as operators Ŝκ → Sκ, S being the superization (see [2]). Then Poisson brackets on Sκ may be rewritten as

{ω, θ}_{P^{x}} = �_{P^{x}Eω}θ, {ω, θ}_{P^{t}} = �_{P^{t}Eω}θ,  (3.3.20)

E = (E_{0}, E_{1}) being the super Euler operator.

Let us rewrite equation (3.3.15) using the new representation:

P^{t} ∘ E ∘ D_{t}^{τ_{1}} + D_{t}^{τ_{2}} ∘ P^{x} ∘ E + D_{t}^{τ_{2}} ∘ ∇ ∘ D_{t}^{τ_{1}} = 0,

where ∇ is the matrix operator depending on ∇_{i}^{x}, ∇_{i}^{t}. Now, by definition one has for any ω

D_{t}^{τ_{2}}P^{x}(ω) = [D_{t}, �_{P^{x}(ω)}](id) = �_{ℓ_{E}P^{x}(ω)}(id) = ℓ_{E}P^{x}(ω),

where ℓ_{E} is the universal linearization of E, or

D_{t}^{τ_{2}} ∘ P^{x} = ℓ_{E} ∘ P^{x}.  (3.3.21)

On the other hand, for any ω

ED_{t}^{τ_{1}}(ω) = ED_{t}(ω) = ℓ̂_{D_{t}ω}(1).

But for any θ

ℓ_{D_{t}ω}θ = �_{θ}D_{t}ω = [�_{θ}, D_{t}]ω + D_{t}�_{θ}ω = −�_{ℓ_{E}ω}θ + D_{t}ℓ_{ω}θ.

In other words,

ℓ_{D_{t}ω} = D_{t} ∘ ℓ_{ω} − ℓ_{ω} ∘ ℓ_{E}.

Consequently,

ℓ̂_{D_{t}ω}(1) = (D_{t} ∘ ℓ_{ω} − ℓ_{ω} ∘ ℓ_{E}) (1) = ℓ̂_{ω}D̂_{t}(1) − ℓ̂_{E} ℓ̂_{ω}(1),

or

E ∘ D_{t} = −ℓ̂_{E} ∘ E.  (3.3.22)

Hence, equation (3.3.15) can be rewritten as

(ℓ_{E} ∘ P^{x} − P^{t} ∘ ℓ̂_{E}) ∘ E + D_{t}^{τ_{2}} ∘ ∇ ∘ D_{t}^{τ_{1}} = 0.

But according to (3.3.18), (3.3.19), the second summand in the last equation vanishes on coker d_{h}^{τ_{i}} which means that for cohomology maps one has

(ℓ_{E} ∘ P^{x} − P^{t} ∘ ℓ̂_{E}) ∘ E = 0.

But (see [1]) for any C-differential operator Δ the equality Δ ∘ E = 0 implies Δ = 0. Thus we have

ℓ_{E} ∘ P^{x} − P^{t} ∘ ℓ̂_{E} = 0.  (3.3.23)

Note now^{4} that from the last equation by easy coordinate computations it follows that P^{x} + P^{t} = 0.

Collecting together the above said and using [5, Theorem 32], we can now formulate our main result:

4See the proof in Section 4.

Theorem 3.2 (cf. [1]). Consider E a determined evolution equation of the form (2.2.1) and let P: Ŝκ → Sκ be a C-differential operator in D_{x_{i}}, i = 1, …, n. Then P ∘ E defines a Poisson structure on E if and only if :

(i) P is an even operator, (ii)

P + P̂ = 0,  (3.3.24)

(iii) for any ω ∈ Sκ one has

[�_{P ω}, P] = ℓ_{P ω} ∘ P + P ∘ ℓ̂_{P ω}  (3.3.25)

and (iv)

ℓ_{E} ∘ P + P ∘ ℓ̂_{E} = 0.  (3.3.26)

Two Poisson structures P ∘ E and P′ ∘ E commute with respect to the Schouten bracket if and only if

[�_{P′ω}, P] + [�_{P ω}, P′] = ℓ_{P′ω} ∘ P + P ∘ ℓ̂_{P′ω} + ℓ_{P ω} ∘ P′ + P′ ∘ ℓ̂_{P ω}. (3.3.27)

Remark 3.1. Note that for any solution P of (3.3.26) the operator P̂ is a solution as well. Hence, P − P̂ is a solution satisfying (3.3.24).

4. Example.

Here we deduce coordinate formulas for special type of Poisson structures over evolution equations in one dependent variable. These Poisson structures could be called classical, since they are represented by differential operators of grading 0. We make detailed computations for the case E being the KdV equation. Basic formulas. Let

u_{t} = Ψ(t, x, u, …, u_{x}^{(k)})  (4.4.1)

be an evolution equation in (t, x). Let (t, x, u_{0}, …, u_{s}, …) be internal coordinates on E^{∞}. Then one has

ℓ_{F} = ∑_{i} \frac{∂Ψ}{∂u_{i}}D_{x}^{i} − D_{t}  (4.4.2)

and

ℓ_{F}^{∗} = ∑_{i}(−1)^{i}D_{x}^{i} ∘ \frac{∂Ψ}{∂u_{i}} + D_{t}.  (4.4.3)

Consider equation (3.3.23) and take P^{x}, P^{t} in the form

P^{x} = (\begin{matrix}Δ & 0 \\ 0 & 0 \end{matrix}) , P^{t} = (\begin{matrix}∇ & 0 \\ 0 & 0 \end{matrix})

where

Δ = ∑_{i=1}^{N} A_{i}D_{x}^{i}, ∇ = ∑_{i=1}^{N} B_{i}D_{x}^{i},  (4.4.4)

A_{i}, B_{i} being smooth functions on E^{∞}, and substitute (4.4.2), (4.4.3) and (4.4.4) into (3.3.23):

(∑_{i=0}^{k} \frac{∂Ψ}{∂u_{i}}D_{x}^{i} − D_{t}) ∘ (∑_{j=0}^{N} A_{j}D_{x}^{j})

= (∑_{j=0}^{N} B_{j}D_{x}^{j}) ∘ (∑_{i=0}^{k}(−1)^{i}D_{x}^{i} ∘ \frac{∂Ψ}{∂u_{i}} + D_{t}) ,

or

∑_{i=0}^{k} ∑_{j=0}^{N} \frac{∂Ψ}{∂u_{i}}D_{x}^{i} ∘ A_{j} ∘ D_{x}^{j} − ∑_{j=0}^{N} (D_{t}(A_{j})D_{x}^{j} + A_{j}D_{t}D_{x}^{j})

= ∑_{j=0}^{N} ∑_{i=0}^{k}(−1)^{i}B_{j}D_{x}^{i+j} ∘ \frac{∂Ψ}{∂u_{i}} + ∑_{j=0}^{N} B_{j}D_{t}D_{x}^{j}. (4.4.5)

Equating in (4.4.5) coefficients at D_{t}D_{x}^{j} we get

B_{j} = −A_{j}, j = 0, …, N,

or ∇ = −Δ. Hence, (4.4.5) transforms to

∑_{i=0}^{k} ∑_{j=0}^{N} \frac{∂Ψ}{∂u_{i}}D_{x}^{i} ∘ A_{i} ∘ D_{x}^{j} − ∑_{j=0}^{N} D_{t}(A_{j})D_{x}^{j} = ∑_{i=0}^{k} ∑_{j=0}^{N}(−1)^{i}A_{j}D_{x}^{i+j} ∘ \frac{∂Ψ}{∂u_{i}},

or after obvious computations, to

∑_{i=0}^{k} \frac{∂Ψ}{∂u_{i}} ∑_{j=0}^{n} ∑_{α=0}^{i} \binom{i}{α}D_{x}^{i−α}(A_{j})D_{x}^{j+α} − ∑_{j=0}^{N} D_{t}(A_{j})D_{x}^{j}

= ∑_{j=0}^{N} A_{j} ∑_{i=0}^{k} ∑_{β=0}^{i+j}(−1)^{i}\binom{i + j}{β}D_{x}^{i+j−β} (\frac{∂Ψ}{∂u_{i}}) D_{x}^{β}. (4.4.6)

We shall not analyze (4.4.6) in details here and make one remark only.

Consider the maximal power of D_{x} in (4.4.6) which equals N + k. Then, equating coefficients at these terms, one obtains

\binom{k}{k}\frac{∂Ψ}{∂u_{k}} A_{N} = (−1)^{k+1}\binom{N + k}{N + k}\frac{∂Ψ}{∂u_{k}} A_{N}.

It means that there are no nontrivial solutions of (3.3.26) for scalar evolution equations of even order.

We pass now to an illustrative example. Example: The KdV case. Consider the KdV equation represented in the form

u_{t} = uu_{x} + u_{xxx}  (4.4.7)

and rewrite (4.4.6) for (4.4.7):

I.S. Krasil’shchik 142

∑_{j=0}^{N} (D_{x}^{3}(A_{j}) + u_{0}D_{x}(A_{j}) + u_{1}A_{j} − D_{t}(A_{j}) + u_{0}A_{j−1}

+ 3D_{x}^{2}(A_{j−1}) + 3D_{x}(A_{j})) D_{x}^{j} = ∑_{j=0}^{N} ∑_{α=0}^{n−j+1} \binom{j + α − 1}{α}u_{α}A_{j+α−1}D_{x}^{j}. (4.4.8)

Equating in (4.4.8) coefficients at D_{x}^{j} we get the following system of equations on A_{0}, …, A_{N}:

D_{x}(A_{N}) = 0,

D_{x}(A_{N−1}) = 0,
3D_{x}(A_{N−2}) = D_{t}(A_{N}) + (N − 1)u_{1}A_{N},
u_{0}D_{x}(A_{j}) + D_{x}^{3}(A_{j}) + 3D_{x}^{2}(A_{j−1}) + 3D_{x}(A_{j−2})

= D_{t}(A_{j}) + (j − 1)u_{1}A_{j} + ∑_{α=2}^{N−j+1} \binom{i + α − 1}{α}u_{α}A_{j+α−1},

u_{0}D_{x}(A_{1}) + D_{x}^{3}(A_{1}) + 3D_{x}^{2}(A_{0}) = D_{t}(A_{1}) + ∑_{α=2}^{N} u_{α}A_{α},

u_{0}D_{x}(A_{0}) + D_{x}^{3}(A_{0}) = D_{t}(A_{0}) − u_{1}A_{0},  (4.4.9)

where j = 2, …, N − 1. Computations for n = 3. In this case (4.4.9) transforms to

D_{x}(A_{3}) = 0,  (4.4.10)
D_{x}(A_{2}) = 0,  (4.4.11)
3D_{x}(A_{1}) − D_{t}(A_{3}) = 2u_{1}A_{3},  (4.4.12)
3D_{x}(A_{0}) + 3D_{x}^{2}(A_{1}) − D_{t}(A_{2}) = u_{1}A_{2} + 3u_{2}A_{3},  (4.4.13)
3D_{x}^{2}(A_{0}) + D_{x}^{3}(A_{1}) + u_{0}D(A_{1}) − D_{t}(A_{1}) = u_{2}A_{2} + u_{3}A_{3},  (4.4.14)
D_{x}^{3}(A_{0}) + u_{0}D(A_{0}) + u_{1}A_{0} − D_{t}(A_{0}) = 0.  (4.4.15)

To solve (4.4.10)–(4.4.15), note first that from the first two equations one has

A_{3} = A_{3}(t), A_{2} = A_{2}(t).

Hence (4.4.12) is of the form

3D_{x}(A_{1}) = Ȧ_{3} + 2u_{1}A_{3},

Ȧ =^{def} dA/dt, from where it follows that

A_{1} = \frac{1}{3}xȦ_{3} + \frac{2}{3}u_{0}A_{3} + a_{1}(t).

Therefore, (4.4.13) transforms to

3D_{x}(A_{0}) = u_{2}A_{3} + Ȧ_{2},

or

A_{0} = \frac{1}{3}A_{3} + \frac{1}{3}xȦ_{2} + a_{0}(t).

Substituting expressions for A_{0}, A_{1}, A_{2}, A_{3} into (4.4.14)) we get

u_{3}A_{3} + u_{2}A_{2} + \frac{2}{3}(u_{0}u_{1} + u_{3})A_{3} + \frac{2}{3}u_{0}Ȧ_{3} + \frac{2}{3}u_{0}xÄ_{3} + ȧ_{1}

= u_{3}A_{3} + u_{2}A_{2} + \frac{2}{3}u_{3}A_{3} + u_{0} (\frac{2}{3}u_{1}A_{3} + \frac{1}{3}Ȧ_{3})

From the last equation we obtain

Ȧ_{3} = 0, ȧ_{1} = 0

and consequently

A_{0} = \frac{1}{3}u_{1}A_{3} + \frac{1}{3}u_{0}A_{2} + a_{0}(t)

with constant A_{3}. But from (4.4.15) it follows that A_{0} is a symmetry of the KdV equation; hence a_{0} = 0 and A_{2} = 0. Thus the final result is

Δ = A_{3}D_{x}^{3} + (\frac{2}{3}u_{0}A_{3} + a_{1}) D_{x} + \frac{1}{3}u_{1}A_{3},  (4.4.16)

where A_{3}, a_{1} are constants, and we obtain two linearly independent solutions

Δ_{0} = D_{x}, Δ_{1} = D_{x}^{3} + \frac{2}{3}u_{0}D_{x} + \frac{1}{3}u_{1},  (4.4.17)

coinciding with classical Hamiltonian maps for the KdV equation (see for example [10])!

Remark 4.1. Two recursion operators correspond to these maps:

R_{s} = Δ_{1} ∘ Δ_{0}^{−1} = D_{x}^{2} + \frac{2}{3}u_{0} + \frac{1}{3}u_{1}D_{x}^{−1}  (4.4.18)

and

R_{c} = Δ_{0}^{−1} ∘ Δ_{1} = D_{x}^{2} + \frac{2}{3}D_{x}^{−1} ∘ u_{0} ∘ D_{x} + \frac{1}{3}D_{x}^{−1} ∘ u_{1}  (4.4.19)

acting on symmetries and conservation laws^{5} respectively. On the other hand, we have the definition of recursion operators acting on symmetries as of solutions of the equation

ℓ_{E}^{[p]}ω = 0  (4.4.20)

(see [2]). Let us show that recursion operators obtained via bi-Hamiltonian structures satisfy (4.4.20).

In fact, let Δ be a C-differential operator over E^{∞} ⊂ J^{∞}(π). Define a vectorvalued form ω_{Δ} ∈ Γ(π) ⊗ C^{1}Λ(E) by

ω_{Δ} = Δ(ω_{∅}).

Then the correspondence Δ ↦ ω_{Δ} is an epimorphism whose kernel consists of operators ∇ ∘ ℓ_{E}. In addition, one has

ω_{Δ_{ω}} = ω

5Strictly speaking, on generating functions of conservation laws.

I.S. Krasil’shchik 144

(for the definition Δ_{ω} see remark 2.1 on page 133). Let now P, Q be two Poisson structures over E. Then

P ∘ ℓ̂_{E} + ℓ_{E} ∘ P = 0, Q ∘ ℓ̂_{E} + ℓ_{E} ∘ Q = 0.  (4.4.21)

Consider corresponding recursion operator R = Q ∘ P^{−1}. Then from (4.4.21) one has

R ∘ ℓ_{E} = ℓ_{E} ∘ R

and consequently

ℓ_{E}ω_{R} = ℓ_{E}Rω_{∅} = Rℓ_{E}ω_{∅} = 0,

i.e, ω_{R} satisfies (4.4.20). Note that only the last property of Poisson structures (see Theorem 3.2) was used in the proof.

Remark 4.2. Since all properties of Poisson structures above were expressed not in terms of conservation laws but in terms of their generating functions, it is useful to write down corresponding Poisson brackets for generating functions of conservation laws explicitly:

{φ, ψ}_{P} = �_{P φ} + ψℓ̂_{E}(P φ)  (4.4.22)

for any two generating functions φ, ψ ∈ E_{1}^{1,n}(E).

5. Acknowledgments.

I am grateful to Dr. P.H.M. Kersten who checked preliminary computations and (as always) found a number of mistakes. I want also to thank the Faculty of Applied Mathematics, University of Twente for the opportunity to finish and publish a preliminary version of this work.

References

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[2] P.H.M. Kersten, I.S.Krasil’shchik, Graded Frölicher–Nijenhuis brackets and the theory of recursion operators for super differential equations, In: Lychagin V.V. (Ed.), The Interplay between Diff. Geom. and Diff. Equations. Adv. in the Math. Sciences, 24 (1995) Amer. Math. Soc., Providence, Rhode Island, 91–130

[3] I.S. Krasil’shchik, Some new cohomological invariants for nonlinear differential equations, Diff. Geom. Appl. 2 (1992) 307–350.

[4] I.S. Krasil’shchik, Supercanonical algebras and Schouten brackets, Mat. zametki 49 (1991) 70–76 (in Russian).

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[7] I.S. Krasil’shchik, Lychagin V.V. and A.M. Vinogradov, Geometry of Jet Spaces and Nonlinear Differential Equations, Advanced Studies in Contemporary Mathematics 1 (1986) Gordon and Breach, New York, London.

[8] I.S. Krasil’shchik and A.M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math. 15 (1989) 161–209.

[9] I.S. Krasil’shchik and A.M. Verbovetsky, Homological methods in equations of mathematical physics. Advanced Texts in Mathematics. Open Education & Sciences, Opava, 1998.

[10] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978) 1156–1162.

[11] MacLane S., Homology, (1963) Springer-Verlag, Berlin.

[12] Vinogradov A.M., The logic algebra for the theory of linear differential operators, Soviet Math. Dokl., 13 (1972), 1058–1062

[13] A.M. Vinogradov, The C-spectral sequence, Lagrangian formalism and conservation laws, J. Math. Anal. Appl. 100 (1984) 2–129.

[14] A.M. Vinogradov, Introduction to Secondary Calculus, in M. Henneaux, I. S. Krasil’shchik, and A. M. Vinogradov (eds.), Secondary calculus and cohomological physics, Contemporary Mathematics, vol. 219, Amer. Math. Soc., Providence, RI, 1998, 241–272.

[15] A.M. Verbovetsky, Notes on the horizontal cohomology, in M. Henneaux, I. S. Krasil’shchik, and A. M. Vinogradov (eds.), Secondary calculus and cohomological physics, Contemporary Mathematics, vol. 219, Amer. Math. Soc., Providence, RI, 1998, 562–568.

Moscow Institute for Municipal Economy and Independent University of Moscow Correspondence to: 1st Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia E-mail address: josephk@glasnet.ru

Received July 27, 1999