The Diffiety Institute Preprint Series

Preprint DIPS–9/98 December 4, 1995

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The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

On the Vinogradov C-spectral sequence for determined systems

of differential equations

by

Dmitri GESSLER

On the Vinogradov C-spectral sequence for determined systems of differential equations

Dmitri GESSLER

Published in Differential Geom. Appl. 7(1997), 303–324

Abstract. In this paper a description of the term E_{1} of the Vinogradov C-spectral sequence is given. Vanishing theorem for the group E_{1}^{p,n−1}, where n is a number of independent variables and p ≥ 3, is proved for a system of evolution equations with a nondegenerate symbol. The group E_{1}^{2,n−1} for a scalar linear evolution equation with constant coefficients is computed.

1. Introduction

Homological methods play an important role in the study of systems of differential equations. In [4] it is shown that infinitesimal symmetries of differential equations and recursion operators are of cohomological nature. The C-spectral sequence (variational bicomplex) introduced by Vinogradov [8] contains important invariants of differential equations such as conservation laws and characteristic classes of families of solutions. It provides means for studying different aspects of Lagrangian formalism, the inverse problem of the calculus of variations, etc. The term E_{1} of the C-spectral sequence is the analog of the de Rham complex in the category of nonlinear partial differential equations (for a very enlightening discussion see [12]).

There exists, however, no general method of computing this important spectral sequence. In [10] a powerful technique based on the Spencer type cohomology systems is developed, the “two line” theorem estimating the term E_{1} for determined systems of differential equations is proved, a concrete method of calculating for the term E_{1}^{1,n−1}, which is related to the theory of conservation laws, is given, where n is a number of independent variables. Due to this method a complete description of the set of conservation laws is possible for determined differential equations (see, for example, [11]). Further development of results of Vinogradov is done in [7, 6], where the Janet sequence for involutive differential equations is used, and a general approach to the calculation of the horizontal cohomology is proposed. In [2] this approach is applied to the C-spectral sequence for overdetermined equations.

1991 Mathematics Subject Classification. Primary 35A30; Secondary 58A20, 58G05.

Key words and phrases. Nonlinear differential equations, Spectral sequences, Spencer cohomology, C-differential operators.

Some results concerning the computation of the term E_{1}^{p,n−1}, p ≥ 2, are obtained in [3], where the C-spectral sequence is considered for scalar evolution equations with a single space variable. For a large class of such equations, in particular KdV, MKdV, PKdV, Burgers equation, the term E_{1}^{2,1} is computed and vanishing theorems are proved for the groups E_{1}^{p,n−1}, p ≥ 3, under a very restrictive assumption that an equation possesses an infinite dimensional space of higher infinitesimal symmetries. Some examples of computation of the C-spectral sequence also can be found in [1].

In this paper we develop a method of calculating the terms E_{1}^{p,n−1}, p ≥ 1, for determined differential equations and apply this method to evolution equations. This paper is organized as follows. Section 2 is a summary of the geometrical theory of nonlinear differential equations and the C-spectral sequence. In Section 3 we give a description of the term E_{1} of the C-spectral sequence for determined equations. In Section 4 this general description is applied to evolution equations, vanishing theorem for the group E_{1}^{p,n−1}, p ≥ 3, is proved for evolution equation with a nondegenerate symbol, the group E_{1}^{2,n−1} is computed for a scalar linear evolution equation with constant coefficients.

2. Jet manifolds and infinitely prolonged differential equations

In this section we define the basic concepts of the geometrical theory of differential equations and the theory of the C-spectral sequence ([5, 10, 9, 1]).

2.1. Jets. Let M be a smooth manifold and π : E → M be a smooth fiber bundle over M, dim M = n, dim E = m + n. Denote by Γ(π) the set of all (local) sections of π.

Let π_{k} : J^{k} → M be the bundle of k-jets for π,

J^{k}(π) = {[f]_{x}^{k} | f ∈ Γ(π), x ∈ M} ,

where [f]_{x}^{k} denotes the k-jet of a local section f at x. Denote by J^{∞}(π) the manifold of infinite jets for π. J^{∞}(π) is the inverse limit with respect to the following system of mappings

π_{k,l} : J^{k}(π) → J^{l}(π), π_{k,l} ([f]_{x}^{k}) = [f]_{x}^{l}, k ≥ l,
π_{k} : J^{k}(π) → M, π_{k} ([f]_{x}^{k}) = x.

By definition, one has natural projections

π_{∞,k} : J^{∞}(π) → J^{k}(π),
π_{∞} : J^{∞}(π) → M.

C-SPECTRAL SEQUENCE 3

Choose a coordinate neighborhood U in M such that the restriction of the bundle E is trivial. Let x_{1}, …, x_{n} be local coordinates in U and u^{1}, …, u^{m} be coordinates along the fiber of π in π^{−1}(U). Each local section f ∈ Γ(π) is of the form f = (u^{1}(x_{1}, …, x_{n}), …, u^{m}(x_{1}, …, x_{n})). Define functions p_{σ}^{j} by following

p_{σ}^{j} ([f]_{x_{0}}^{k}) = \frac{∂^{|σ|}u^{j}}{∂x_{1}^{σ_{1}} … ∂x_{n}^{σ_{n}}} |_{x=x_{0}} ,

where σ = (σ_{1}, …, σ_{n}), |σ| = σ_{1} + ⋯ + σ_{n}. Then smooth functions (x^{i}, p_{σ}^{j}), 1 ≤ i ≤ n, 1 ≤ j ≤ m, 0 ≤ |σ| ≤ k form local coordinates in J^{k}(π), 0 ≤ k ≤ ∞.

Let F_{k}(π) denote the algebra C^{∞} (J^{k}(π)). Then one has a system of embeddings

π_{k,l}^{∗} : F_{l}(π) → F_{k}(π), l ≤ k,
π_{k}^{∗} : C^{∞}(M) → F_{k}(π).

The direct limit F(π) with respect to the system {π_{k,l}^{∗}} is called the algebra of smooth functions on J^{∞}(π). We identify F_{k}(π) and C^{∞}(M) with their limits in F(π),

F(π) = ⋃_{k≥0} F_{k}(π).

In the same manner one can define the module of i-forms Λ^{i}(π), i ≥ 0, on J^{∞}(π)

Λ^{i}(π) = ⋃_{k≥0} Λ^{i} (J^{k}(π))

and consider a graded algebra

Λ^{∗}(π) = ∑_{i=0}^{∞} Λ^{i}(π)

of differential forms on J^{∞}(π).

A vector field X on J^{∞}(π) is an ℝ-linear mapping X : F(π) → F(π) such that for any φ, ψ ∈ F(π)

X(φψ) = φX(ψ) + ψX(φ),

and there exists r ≥ 0 such that for each k ≥ 0

X (F_{k}(π)) ⊂ F_{k+r}(π).

Denote by D(π) the set of all vector fields on J^{∞}(π). Obviously, D(π) is a F(π)-module and Lie algebra over ℝ. Locally each X ∈ D(π) can be represented as

X = ∑_{i} a_{i} \frac{∂}{∂x_{i}} + ∑_{j,σ} b_{σ}^{j} \frac{∂}{∂p_{σ}^{j}} ,

where b_{σ}^{j} ∈ F_{|σ|+r}(π) for some r.

Let ξ_{i} : F_{i} → J^{∞}(π), i = 1, 2, be vector bundles over J^{∞}(π), P_{i} = Γ(F_{i}) be F(π)-modules of sections. An ℝ-linear differential operator Δ : P_{1} → P_{2} is called C-differential if it can be restricted to the manifolds of the form [f]^{∞}, f ∈ Γ(π). That is

Δ(φ)|_{[f]^{∞}} = 0 as φ|_{[f]^{∞}} = 0, φ ∈ P_{1}.

If vector bundles ξ_{1}, ξ_{2} are finite dimensional, m_{i} = dim ξ_{i}, then in local coordinates each C-differential operator Δ can be represented as a m_{2} × m_{1} matrix

Δ = (\begin{matrix}∑_{σ} a_{11}^{σ}D_{σ} & & & … & & ∑_{σ} a_{1m_{1}}^{σ}D_{σ} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{σ} a_{m_{2}1}^{σ}D_{σ} & & & … & & ∑_{σ} a_{m_{2}m_{1}}^{σ}D_{σ} \end{matrix}) ,

where σ = (σ_{1}, …, σ_{n}), D_{σ} = (D_{1})^{σ_{1}} ∘ ⋯ ∘ (D_{n})^{σ_{n}}, and

D_{i} = \frac{∂}{∂x_{i}} + ∑_{j,σ} p_{σ+1_{i}}^{j} \frac{∂}{∂p_{σ}^{j}} , i = 1, …, n,

is the i-th total derivative.

The set of C-differential operators from P_{1} to P_{2} is, clearly, a F(π)- module and is denoted by CDiff(P_{1}; P_{2}).

Let ξ_{i} : F_{i} → M, i = 1, 2, be vector bundles, P_{i} = Γ(ξ_{i}), and Δ : P_{1} → P_{2} be an ℝ-linear differential operator, Δ ∈ Diff(P_{1}; P_{2}). By definition, put \overline{P_{i}} = F(π) ⊗_{C^{∞}(M)} P_{i} = Γ (π_{∞}^{∗}(ξ_{i})), where π_{∞}^{∗}(ξ_{i}) is the induced vector bundle over J^{∞}(π). A unique C-differential operator \overline{Δ} : \overline{P_{1}} → \overline{P_{2}} such that \overline{Δ}(1 ⊗ p) = 1 ⊗ Δ(p) is called the lifting of Δ.

Consider a vector field X ∈ D(M) on M. It is an operator of the first order acting from C^{∞}(M) to C^{∞}(M). Then the lifting \overline{X} ∈ CDiff (F(π), F(π)) is a vector field on J^{∞}(π). Consider a submodule CD(π) ⊂ D(π) generated by vector fields of the form \overline{X}. Thus we have a distribution on J^{∞}(π) which is called the Cartan distribution. The Cartan distribution is completely integrable in the sense that it satisfies Frobenius integrability condition

[CD(π), CD(π)] ⊂ CD(π).  (1)

In local coordinates if X = ∑_{i} a_{i} \frac{∂}{∂x_{i}}, a_{i} ∈ C^{∞}(M), then \overline{X} = ∑_{i} a_{i}D_{i}, and

CD(π) = {∑_{i} φ_{i}D_{i} | φ_{i} ∈ F(π)} .

Vector field X ∈ D(π) is called vertical if X(φ) = 0 for each φ ∈ C^{∞}(M) ⊂ F(π). Denote by D^{V} (π) ⊂ D(π) the F(π)-module of vertical vector fields. D^{V} is also a subalgebra of the Lie algebra D(π),

[D^{V} (π), D^{V} (π)] ⊂ D^{V} (π),  (2)

and D(π) splits into a direct sum

D(π) = CD(π) ⊕ D^{V} (π).

Dually, the module of 1-form on J^{∞}(π) splits into a direct sum

Λ^{1}(π) = C^{1}Λ^{1}(π) ⊕ \overline{Λ}^{1}(π),  (3)

where

\overline{Λ}^{1}(π) = \{ω ∈ Λ(π) | ω(X) = 0 for any X ∈ D^{V} (π)\}

is the module of horizontal 1-forms, and

C^{1}Λ^{1}(π) = {ω ∈ Λ(π) | ω(X) = 0 for any X ∈ CD(π)}

is the module of Cartan forms.

Locally,

\overline{Λ}^{1}(π) = \{∑_{i} φ_{i}dx_{i} | φ_{i} ∈ F(π)\} ,

C^{1}Λ^{1}(π) = {∑_{j,σ} φ_{j}^{σ}ω_{σ}^{j} | φ_{j}^{σ} ∈ F(pi)} ,

where ω_{σ}^{j} = dp_{σ}^{j} − ∑_{i} p_{σ+1_{i}}^{j}dx_{i}.

From (3) it follows that each Λ^{i}(π), i > 0, splits into

Λ^{i}(π) = ∑_{α+β=i} \overline{Λ}^{α}(π) ⊗ C^{β}Λ^{1}(π),

where

\overline{Λ}^{α}(π) = \underbrace{\overline{Λ}^{1}(π) ∧ ⋯ ∧ \overline{Λ}^{1}(π)}_{α times},

C^{β}Λ^{1}(π) = \underbrace{C^{1}Λ^{1}(π) ∧ ⋯ ∧ C^{1}Λ^{1}(π)}_{β times} .

The graded algebra \overline{Λ}^{∗}(π) is the lifting of the graded algebra Λ^{∗}(M) of differential forms on M. The lifting of the de Rham differential d : Λ^{i}(M) → Λ^{i+1}(M) is called the horizontal differential and is denoted by \overline{d} : \overline{Λ}^{i}(π) → \overline{Λ}^{i+1}(π). The complex

0 ⟶^{F} (π) ⟶^{\overline{d}} \overline{Λ}^{1}(π) ⟶^{\overline{d}}… ⟶^{\overline{d}} \overline{Λ}^{n}(π) ⟶^{0}

is called the horizontal de Rham complex and its cohomology at the term \overline{Λ}^{i}(π) is denoted by \overline{H}^{i}(π). In local coordinates

\overline{d}(dx_{i}) = 0, \overline{d}(φ) = ∑_{i} D_{i}(φ)dx_{i}, φ ∈ F(π).

Let V (π) : V (E) → E be the vector bundle of vertical vector fields on E, V (E) = \{v ∈ T (E) | π_{∗}v = 0\}. By definition, put κ = \overline{Γ(V (π))} = Γ (π_{∞,0}^{∗} (V (π))). Then there exists a map

∋: κ → D^{V} (π), φ ↦∋_{φ},

where ∋_{φ} is called evolutionary derivation and is defined by following

∋_{φ} (ψ)|_{[f]^{∞}} = \frac{d}{dt}|_{t=0} (ψ|_{[f_{t}]^{∞}}) ,

where f ∈ Γ(π), φ ∈ κ, ψ ∈ F(π), and f_{t} is a 1-parameter family of sections of π such that \frac{d}{dt} |_{t=0} f_{t} = φ|_{[f]^{∞}}, f_{0} = f.

In local coordinates, if φ = (φ^{1}, …, φ^{m}), then

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

2.2. Differential operators and equations. The system of nonlinear differential equations of order k imposed on sections of π : E → M is a submanifold Y^{k} ⊂ J^{k}(π). Denote by Y^{k+s} ⊂ J^{k+s}(π) the s-th prolongation of Y^{k}. We will always suppose that Y^{k} is formally integrable. Then Y^{s}, s ≥ k, is a smooth manifold, and π_{s,t}, s ≥ t ≥ k, maps Y^{s} onto Y^{t} surjectively. The inverse limit of the system of maps

π_{s,t} : Y^{s} → Y^{t}, s ≥ t ≥ k,

is called an infinitely prolonged differential equation, or simply a differential equation, and is denoted by Y = Y^{∞}. Obviously, the infinite jets manifold J^{∞}(π) is a differential equation of zero order with Y^{k} = J^{k}(π), k ≥ 0.

One can construct a calculus on a differential equation Y ⊂ J^{∞}(π) in the same way as for the jets manifold J^{∞}(π). Let F, Λ^{∗}, D denote the algebra of smooth functions, the graded algebra of differential forms and the module of vector fields on Y respectively.

As for the jets manifold J^{∞}(π), there exists a splitting of the modules of vector fields D and 1-forms Λ^{1}

D = D^{V} ⊕ CD,
Λ^{1} = \overline{Λ}^{1} ⊕ C^{1}Λ^{1}.  (4)

Let ξ : E′ → M be a vector bundle, F : Γ(π) → Γ(ξ) be a nonlinear differential operator, F can be considered as a section of the induced vector bundle π_{∞}^{∗}(ξ) over J^{∞}(π). Define a smooth map

J(F) : J^{∞}(π) → J^{∞}(ξ), [f]_{x}^{∞} ↦ [F (f)]_{x}^{∞}.

Let Y = Y(F) be the differential equation defined by F,

Y(F) = ker J(F) = {θ ∈ J^{∞}(π) | J(F)(θ) = 0} .

In local coordinates, if F = (F^{1}, …, F^{s}), F^{l} = F^{l}(x_{i}, p_{σ}^{j}), then Y(F) is the infinite prolongation of the system

\begin{cases}F^{1} = 0, \\ . & . & . & . & . & . & . \\ F^{s} = 0. \end{cases}

Denote by P = Γ (π_{∞}^{∗}(ξ)) the F(π)-module of sections of the induced vector bundle over J^{∞}(π). The universal linearization of F is a Cdifferential operator ℓ_{F} ∈ CDiff(κ; P) such that

ℓ_{F} (φ)|_{[f]^{∞}} = \frac{d}{dt}|_{t=0} F (f_{t}), φ ∈ κ, f ∈ Γ(π),

where f_{t} is a 1-parameter family of sections of π such that \frac{d}{dt} |_{t=0} f_{t} = φ|_{[f]^{∞}}, f_{0} = f. In local coordinates

ℓ_{F} = (\begin{matrix}∑_{σ} \frac{∂F^{1}}{∂p_{σ}^{1}} D_{σ} & & & … & & ∑_{σ} \frac{∂F^{1}}{∂p_{σ}^{m}} D_{σ} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{σ} \frac{∂F^{s}}{∂p_{σ}^{1}} D_{σ} & & & … & & ∑_{σ} \frac{∂F^{s}}{∂p_{σ}^{m}} D_{σ} \end{matrix}) .

Denote by I_{Y} ⊂ F(π) the ideal of functions identically equal to 0 on Y. For each F(π)-module R denote by [R] the restriction of R to Y, [R] = R/(I_{Y} · R). If R is a projective F(π)-module, then [R] is a projective F-module.

A differential equation Y = Y(F) is called regular with respect to the nonlinear differential operator F = (F^{1}, …, F^{s}) if

1. ideal I_{Y} is generated by functions D_{σ}(F^{l}), 1 ≤ l ≤ s, 0 ≤ |σ| < ∞;

2. the module of 1-forms Λ^{1} is projective.

This definition does not depend on the choice of coordinate chart. In the geometrical theory of nonlinear differential equations a regular differential equation is an analog of a smooth (without singularities) manifold.

If the F-module Λ^{1} is projective, then from (4) it follows that modules \overline{Λ}^{1} and C^{1}Λ^{1} are also projective. From now onwards we assume that all the modules under consideration are projective.

Let i : Y → J^{∞}(π) be natural inclusion, i^{∗} : [C^{1}Λ^{1}(π)] → C^{1}Λ^{1} be induced homomorphism, [ℓ_{F}] : [κ] → [P] be the restriction of the universal linearization to Y.

Proposition 1 ([10]). 1. [C^{1}Λ^{1}(π)] = CDiff([κ]; F); 2. The following sequence is exact

CDiff([P]; F) ⟶^{[ℓ_{F}]} CDiff([κ]; F) ⟶^{i^{∗}} C^{1}Λ^{1} ⟶^{0},

[ℓ_{F}](∇) = ∇ ∘ [ℓ_{F}].

In local coordinates the isomorphism of statement (1) is given by following

ω_{σ}^{j} = dp_{σ}^{j} − ∑_{i} p_{σ+1_{i}}^{j}dx_{i} ↦ ∇_{σ}^{j}, ∇_{σ}^{j}(φ^{1}, …, φ^{m}) = D_{σ}(φ^{j}).

A differential equation Y = Y(F) is called determined if the homomorphism

[ℓ_{F}] : CDiff([P]; F) → CDiff([κ]; F)

is an injection. This condition always holds for evolution equations, or for differential equations Y(F) such that the symbol of the matrix differential operator ℓ_{F} : κ → P has the maximal rank almost everywhere.

If a differential equation Y(F) is regular and determined then the short exact sequence

0 ⟶^{CDiff} ([P]; F) ⟶^{[ℓ_{F}]} CDiff([κ]; F) ⟶^{i^{∗}} C^{1}Λ^{1} ⟶^{0}  (5)

splits and C^{1}Λ^{1} can be considered as a submodule of CDiff([κ]; F).

2.3. Adjoint operator. Let Y be a differential equation. For any F-module Q consider the following complex (S(Q), \overline{d}_{1}):

0 ⟶^{CDiff} (Q; F) ⟶^{\overline{d}_{1}} CDiff(Q; \overline{Λ}^{1}) ⟶^{\overline{d}_{1}} … ⟶^{\overline{d}_{1}} CDiff(Q; \overline{Λ}^{n}) ⟶^{0},

\overline{d}_{1}(Δ) = −\overline{d} ∘ Δ.  (6)

The cohomology of complex (6) is described by

Proposition 2 ([10]). 1. H^{i} (S(Q)) = 0 if i ≠ n; 2. H^{n} (S(Q)) = Hom_{F}(Q; \overline{Λ}^{n}).

By definition, put Q̂ = Hom_{F}(Q; \overline{Λ}^{n}) for any module Q.

Each C-differential operator ∇ : Q_{1} → Q_{2} induces a homomorphism of complexes S(∇) : S(Q_{2}) → S(Q_{1}), S(∇)(Δ) = Δ ∘ ∇, Δ ∈ CDiff(Q_{2}; \overline{Λ}^{i}), and an ℝ-linear map of the cohomology

∇^{∗} : Q̂_{2} → Q̂_{1}.

The operator ∇^{∗} is called the adjoint operator for ∇.

Proposition 3 ([10]). 1. ∇^{∗} ∈ CDiff(Q̂_{2}; Q̂_{1}).

2. For all ∇_{1} ∈ CDiff(Q_{1}; Q_{2}), ∇_{2} ∈ CDiff(Q_{2}; Q_{3})

(∇_{2} ∘ ∇_{1})^{∗} = ∇_{1}^{∗} ∘ ∇_{2}^{∗}.

3. If in local coordinates ∇ is a m_{1} × m_{2}-matrix

∇ = (∑_{σ} a_{ij}^{σ}D_{σ}),

then ∇^{∗} is the m_{2} × m_{1}-matrix

∇^{∗} = (∑_{σ}(−1)^{|σ|}D_{σ} ∘ a_{ji}^{σ}) .

2.4. The C-spectral sequence. Consider the C-filtration in the de Rham complex on an equation Y

Λ = C^{0}Λ ⊃ C^{1}Λ ⊃ C^{2}Λ ⊃ …,

where

C^{p}Λ = ∑_{\begin{matrix}α≥p \\ β≥0 \end{matrix}} C^{α}Λ^{1} ⊗ \overline{Λ}^{β}.

The spectral sequence {E_{r}^{p,q}, d_{r}^{p,q}} determined by this filtration is called the C-spectral sequence for the differential equation Y. As usual p denotes the filtration degree and p + q denotes the total degree.

Proposition 4 ([10]). 1. E_{0}^{p,q} = C^{p}Λ^{1} ⊗ \overline{Λ}^{q}; 2. E_{0}^{0,q} = \overline{Λ}^{q}, d_{0}^{0,q} = \overline{d};

3. The C-spectral sequence converges and its term E_{∞} = {E_{∞}^{p,q}} is attached to H_{de Rham}^{∗}(Y).

Keeping in mind statement (2) of Proposition 4 denote

d_{0}^{p,q} = \overline{d} : C^{p}Λ^{1} ⊗ \overline{Λ}^{q} → C^{p}Λ^{1} ⊗ \overline{Λ}^{q+1}.

If Y = J^{∞}(π), then in local coordinates \overline{d} is uniquely defined by following

\overline{d}(f) = ∑_{i} D_{i}(f)dx_{i}, f ∈ F;

\overline{d}(dx_{i}) = 0, dx_{i} ∈ \overline{Λ}^{1};

\overline{d}(ω_{σ}^{j}) = dω_{σ}^{j} = ∑_{i} dx_{i} ∧ ω_{σ+1_{i}}^{j}, ω_{σ}^{j} ∈ C^{1}Λ^{1}.

3. The term E_{1} of the C-spectral sequence

In this section we suppose that Y = Y(F) is a regular differential equation defined by a nonlinear differential operator F.

3.1. Multilinear C-differential operators. Let Q_{1}, …, Q_{p}, R be Fmodules. Consider the module of ℝ-multilinear C-differential operators Δ : Q_{1} × ⋯ × Q_{p} → R

CDiff(Q_{1}, …, Q_{p}; R) = CDiff (Q_{1}; CDiff (Q_{2}; … CDiff(Q_{p}; R) …))
= CDiff(Q_{1}; F) ⊗ ⋯ ⊗ CDiff(Q_{p}; F) ⊗ R.

For any F-modules Q, R, by definition, put

CDiff_{p}(Q; R) = CDiff(\underbrace{Q, …, Q}_{p times}; R) = ⨂^{p} CDiff(Q; F) ⊗ R,

and consider the submodules CDiff_{p}^{alt}(Q; R) ⊂ CDiff_{p}(Q; R)

CDiff_{p}^{alt}(Q; R) = ⋀^{p} CDiff(Q; F) ⊗ R.

Module CDiff_{p}^{alt}(Q; R) can be identified with the module of skew-symmetric C-differential operators Δ : \underbrace{Q × ⋯ × Q}_{p times} → R.

Let ∇ : R_{1} → R_{2} be a C-differential operator. For each F-module Q define an operator

∇_{p} : CDiff_{p}^{alt}(Q; R_{1}) → CDiff_{p}^{alt}(Q; R_{2}),
Δ ↦ (−1)^{p}∇ ∘ Δ.

Proposition 5 ([10]). Let {E_{0}^{p,q}(π), d_{0}^{p,q}(π)} be the zero term of the C-spectral sequence for J^{∞}(π). Then

E_{0}^{p,q}(π) = C^{p}Λ^{1}(π) ⊗ \overline{Λ}^{q}(π) = CDiff_{p}^{alt} (κ; \overline{Λ}^{q}(π));
d_{0}^{p,q}(π) = (\overline{d})_{p},

where \overline{d} : \overline{Λ}^{q}(π) → \overline{Λ}^{q+1}(π) is the horizontal differential on J^{∞}(π).

Lemma 1. 1. Let {[E_{0}^{p,q}(π)] , [d_{0}^{p,q}(π)]} be the restriction of the zero term of the C-spectral sequence for J^{∞}(π) to the equation Y. Then

[E_{0}^{p,q}(π)] = [C^{p}Λ^{1}(π)] ⊗ \overline{Λ}^{q} = CDiff_{p}^{alt} ([κ]; \overline{Λ}^{q});

[d_{0}^{p,q}(π)] = (\overline{d})_{p},

where \overline{d} : \overline{Λ}^{q} → \overline{Λ}^{q+1} is the horizontal differential.

2. For each F-module Q the following sequence is exact

CDiff ([P]; F) ⊗ CDiff_{p−1}^{alt} ([κ]; Q) ⟶^{ℓ}CDiff_{p}^{alt} ([κ]; Q) ⟶^{i^{∗}} C^{p}Λ^{1} ⊗ Q ⟶^{0},

where

ℓ(Δ_{1} ⊗ Δ_{2}) = (Δ_{1} ∘ [ℓ_{F}]) ∧ Δ_{2}, Δ_{1} ∈ CDiff([P]; F), Δ_{2} ∈ CDiff_{p−1}^{alt}([κ]; Q).

3. If ∇ : Q_{1} → Q_{2} is a C-differential operator, then there exists a unique C-differential operator ∇_{p} : C^{p}Λ^{1} ⊗ Q_{1} → C^{p}Λ^{1} ⊗ Q_{2} such that the following diagram is commutative

CDiff_{p}^{alt} ([κ]; Q_{1}) ⟶^{∇_{p}} CDiff_{p}^{alt} ([κ]; Q_{2})
i^{∗}↓  ^{i^{∗}}↓
C^{p}Λ^{1} ⊗ Q_{1}  ⟶^{∇_{p}} C^{p}Λ^{1} ⊗ Q_{2}

4. d_{0}^{p,q} = (\overline{d})_{p}, where d_{0}^{p,q} denotes the zero differential in the C-spectral sequence for Y.

Proof. Statement (1) follows from Proposition 5. Statement (2) follows from Proposition 1. Statement (3) follows from (2). Finally, (4) follows from (1), (3), and the fact that the following diagram is commutative

[E_{0}^{p,q}(π)] ⟶^{[d_{0}^{p,q}(π)]} [E_{0}^{p,q+1}(π)]
i^{∗}↓  ^{i^{∗}}↓
E_{0}^{p,q}  ⟶^{d_{0}^{p,q}} E_{0}^{p,q+1}

The action of C-differential operators on the module C^{p}Λ^{1} ⊗ Q_{1} of differential forms with coefficients in Q_{1} defined by Lemma 1 coincides with the one introduced in [4].

Let K = {K^{i}, δ^{i}} be a complex such that δ^{i} for any i is a C-differential operator. By Lemma 1 we have a complex CDiff(R_{1}, …, R_{p}; K) = {CDiff(R_{1}, …, R_{p}; K^{i}), δ_{1...1}^{i}} for any modules R_{1}, …, R_{p}, and a complex C^{p}Λ^{1} ⊗ K = {C^{p}Λ^{1} ⊗ K^{i}, δ_{p}^{i}} for any integer p ≥ 0.

Proposition 6 ([10]). 1. Let R_{1}, …, R_{p}, Q be F-modules. Then

H^{i} (CDiff(R_{1}, …, R_{p}; S(Q))) = \begin{cases}0 & , i ≠ n; \\ CDiff(R_{1}, …, R_{p}; Q̂) & , i = n. \end{cases}

2. Let ∇ : Q_{1} → Q_{2} be a C-differential operator. Then

(∇^{∗})_{1...1} : CDiff(R_{1}, …, R_{p}; Q̂_{2}) → CDiff(R_{1}, …, R_{p}; Q̂_{1})

is the induced map of the cohomology.

Let ∇ ∈ CDiff(R_{1}, …, R_{p}; Q). Take r_{i} ∈ R_{i}, i = 2, …, p, and consider a C-differential operator

∇_{r_{2},...,r_{p}}^{(1)} : R_{1} → Q, ∇_{r_{2},...,r_{p}}^{(1)}(r) = ∇(r, r_{2}, …, r_{p}),

where r ∈ R_{1}. A C-differential operator

∇^{∗_{1}} : Q̂ × R_{2} × ⋯ × R_{p} → R̂_{1}, ∇^{∗_{1}}(q̂, r_{2}, …, r_{p}) = (∇_{r_{2},...,r_{p}}^{(1)})^{∗} (q̂),

where q̂ ∈ Q̂, r_{i} ∈ R_{i}, is called adjoint to ∇ with respect to the 1-st argument. In the same manner one can define for any s = 1, …, p an operator ∇^{∗_{s}} adjoint to ∇ with respect to the s-th argument.

Denote by ⟨·, ·⟩ : Q × Q̂ → \overline{Λ}^{n} the natural pairing, ⟨q, φ⟩ = φ(q), where q ∈ Q, φ ∈ Q̂ = Hom_{F}(Q; \overline{Λ}^{n}).

Proposition 7 ([10], the Green formula). Let ∇ ∈ CDiff(R_{1}, …, R_{p}; Q̂). Then there exists a C-differential operator G ∈ CDiff(R_{1}, …, R_{p}, Q; \overline{Λ}^{n−1}) such that for any r_{i} ∈ R_{i}, 1 ≤ i ≤ p, and q ∈ Q

⟨∇(r_{1}, …, r_{p}), q⟩ − ⟨r_{1}, ∇^{∗_{1}}(q, r_{2}, …, r_{p})⟩ = (\overline{d}_{1...1}G)(r_{1}, …, r_{p}, q).(7)

The following Lemma is a main tool for calculating the C-spectral sequence. For more general discussion see [2].

Lemma 2. 1. Let Q be a F-module. Then

H^{i}(C^{p}Λ^{1} ⊗ S(Q)) = \begin{cases}0 & , i ≠ n; \\ C^{p}Λ^{1} ⊗ Q̂ & , i = n. \end{cases}

2. Let ∇ : Q_{1} → Q_{2} be a C-differential operator. Then (∇^{∗})_{p} : C^{p}Λ^{1} ⊗ Q̂_{2} → C^{p}Λ^{1} ⊗ Q̂_{1} is the induced map of the cohomology.

Proof. Consider a natural filtration in the complex \{S(Q), \overline{d}_{1}\} by the modules

F^{k} = \{Δ ∈ CDiff(Q; \overline{Λ}^{i}) | ord Δ ≤ k + i − n\},  (8)

where ord Δ denotes the order of a C-differential operator Δ. Clearly, F^{0} ⊂ F^{1} ⊂ ⋯ ⊂ F^{k} ⊂ …, and \overline{d}_{1}(F^{k}) ⊂ F^{k} for any k ≥ 0. Let {F_{r}^{k,q}, δ_{r}^{k,q}} be the spectral sequence for the complex S(Q) with respect to filtration (8). It is easy to see that

F_{0}^{k,q} = Q^{∗} ⊗ S^{2k+q−n}\overline{Λ}^{1∗} ⊗ \overline{Λ}^{k+q} = Q^{∗} ⊗ S^{2k+q−n}V^{∗} ⊗ Λ^{n−k−q}V^{∗} ⊗ \overline{Λ}^{n},

where V = \overline{Λ}^{1}, if 0 ≤ k +q ≤ n, 2k +q −n ≥ 0, and F_{0}^{k,q} = 0 otherwise. Further, the zero differential δ_{0}^{k,q} has a simple form δ_{0}^{k,q} = 1 ⊗ δ^{k} ⊗ 1, where

δ^{k} : S^{i}V^{∗} ⊗ Λ^{k−i}V^{∗} → S^{i+1}V^{∗} ⊗ Λ^{k−i−1}V^{∗}

is the Koszul differential. Hence, F_{1}^{k,q} = 0 if k ≠ 0 or q ≠ n, and F_{1}^{0,n} = Q^{∗} ⊗ \overline{Λ}^{n} = Q̂.

Filtration (8) of the complex S(Q) yields a filtration of the complex C^{p}Λ^{1} ⊗ S(Q) by modules F̃^{k} = C^{p}Λ^{1} ⊗ F^{k}. The corresponding spectral sequence {F̃_{r}^{k,q}, δ̃_{r}^{k,q}} has the zero term F̃_{0}^{k,q} = C^{p}Λ^{1} ⊗ F_{0}^{k,q} and δ̃_{0}^{k,q} = 1 ⊗ δ_{0}^{k,q}. Hence, F̃_{1}^{k,q} = 0 if k ≠ 0 or q ≠ n, and F̃_{1}^{0,n} = C^{p}Λ^{1} ⊗ Q̂. This concludes the proof of statement (1). Statement (2) follows now from Proposition 6 and Lemma 1.

3.2. The term E_{1} for determined equations. In this subsection we suppose that Y = Y(F) is a regular determined equation.

Lemma 3. Let H^{i} be the cohomology of the complex \{C^{p−1}Λ^{1} ⊗ C^{1}Λ^{1} ⊗ \overline{Λ}^{∗}, \overline{d}_{1,p−1}\}. Then

H^{i} = \begin{cases}0, & i ≠ n − 1, n, \\ ker [ℓ_{F}]_{p−1}^{∗}, & i = n − 1, \\ coker [ℓ_{F}]_{p−1}^{∗}, & i = n, \end{cases} [ℓ_{F}]_{p−1}^{∗} : C^{p−1}Λ^{1} ⊗ ˆ[P] → C^{p−1}Λ^{1} ⊗ ˆ[κ].

Proof. Since the equation Y is regular and determined, sequence (5) is exact. Consider the following short exact sequence of complexes

0 ⟶^{C}^{p−1} Λ^{1} ⊗ S([P]) ⟶^{1⊗S([ℓ_{F}])} C^{p−1}Λ^{1} ⊗ S([κ]) ⟶^{1⊗i^{∗}} C^{p−1}Λ^{1} ⊗ C^{1}Λ^{1} ⊗ \overline{Λ}^{∗} ⟶^{0}.

By Lemma 2 the long exact sequence of the cohomology has the form

0 ⟶^{H}^{n−1}⟶^{C}^{p−1} Λ^{1} ⊗ [P̂] ⟶^{[ℓ_{F}]_{p−1}^{∗}} C^{p−1}Λ^{1} ⊗ [κ̂] ⟶^{H}^{n}⟶^{0},

and the Lemma is obvious.

The permutation group S_{p} acts in the complex \{CDiff_{p}([κ]; Λ^{∗}), \overline{d}_{p}\}. For each τ ∈ S_{p} and ∇ ∈ CDiff_{p}([κ]; Λ^{q}) we have

τ(∇)(χ_{1}, …, χ_{p}) = ∇(χ_{τ(1)}, …, χ_{τ(p)}).  (9)

It is easy to see that action (9) projects to an action of the group S_{p} in the complex \{C^{p−1}Λ^{1} ⊗ C^{1}Λ^{1} ⊗ \overline{Λ}^{∗}, \overline{d}_{1,p−1}\} and, therefore, induces

an action in the cohomology. The complex C^{p}Λ^{1} ⊗ \overline{Λ}^{∗} = E_{0}^{p,∗} is the antisymmetric part of the complex C^{p−1}Λ^{1} ⊗ C^{1}Λ^{1} ⊗ \overline{Λ}^{∗} with respect to the action of S_{p}. Hence, the cohomology E_{1}^{p,q} = H^{q}(C^{p}Λ^{1} ⊗ \overline{Λ}^{∗}) is the antisymmetric part of H^{q} (C^{p−1}Λ^{1} ⊗ C^{1}Λ^{1} ⊗ \overline{Λ}^{∗}). Combining this with Lemma 3 one immediately obtain the following

Theorem 1 ([10],the two line theorem). Let Y be a regular determined differential equation. Then

1. E_{1}^{p,q} = 0, p ≥ 1, q ≠ n − 1, n;

2. E_{1}^{p,n−1} (resp. E_{1}^{p,n}) is the antisymmetric part of ker[ℓ_{F}]_{p−1}^{∗} (resp. coker[ℓ_{F}]_{p−1}^{∗}) with respect to the induced action of the permutation group S_{p}.

Let us now describe the action of the permutation group S_{p} in ker [ℓ_{F}]_{p−1}^{∗} and coker [ℓ_{F}]_{p−1}^{∗}. Denote ℓ = [ℓ_{F}], ℓ_{p−1} = [ℓ_{F}]_{p−1}.

Lemma 4. Let ω ∈ ker ℓ_{p−1}^{∗} ⊂ C^{p−1}Λ^{1}⊗[P̂] and ∇ = ∇_{ω} ∈ CDiff_{p−1}([κ]; [P̂]) be a C-differential operator such that i^{∗}(∇) = ω. Then the following is true.

1. There exist operators Δ_{s} ∈ CDiff(\underbrace{[κ], …}_{s−1}, [P], \underbrace{…, [κ]}_{p−s−1}; [κ̂]), 1 ≤

s ≤ p − 1, such that

ℓ^{∗} (∇(χ_{1}, …, χ_{p−1})) = ∑_{s=1}^{p−1} Δ_{s}(χ_{1}, …, ℓ(χ_{s}), …, χ_{p−1}).  (10)

2. If τ ∈ S_{p−1} ⊂ S_{p}, τ(p) = p, then

τ(ω) = i^{∗}(τ(∇)),

where τ(∇) is defined by (9).

3. If τ = (s, p) is a transposition that interchanges s and p then

τ(ω) = −i^{∗} (Δ_{s}^{∗_{s}}) ,

where Δ^{∗_{s}} denotes the adjoint operator for Δ with respect to the s-th argument.

Proof. Statement (1) follows from the definition of ℓ_{p−1}.

In order to prove statements (2) and (3) let us describe the isomorphism between ker ℓ_{p−1}^{∗} and H^{n−1}(C^{1}Λ^{1}⊗C^{p−1}Λ^{1}⊗\overline{Λ}^{∗}) given by Lemma

2. If ω ∈ ker ℓ_{p−1}^{∗} and ω = i^{∗}(∇), where ∇ ∈ CDiff_{p−1}([κ]; [P̂]), then by Green formula (7) applied to the operator ℓ there exists a C-differential operator A ∈ CDiff_{p}([κ]; \overline{Λ}^{n−1}) such that

⟨ℓ^{∗}∇(χ_{1}, …, χ_{p−1}), χ_{p}⟩ − ⟨∇(χ_{1}, …, χ_{p−1}), ℓχ_{p}⟩ = (\overline{d}_{p}A)(χ_{1}, …, χ_{p}).

The cohomology class corresponding to ω is defined by a differential form η = i^{∗}(A) ∈ C^{1}Λ^{1} ⊗ C^{p−1}Λ^{1} ⊗ \overline{Λ}^{n−1}.

Now statement 2 is obvious. Indeed, if τ ∈ S_{p−1} ⊂ S_{p}, then

⟨ℓ^{∗}τ(∇)(χ_{1}, …, χ_{p−1}), χ_{p}⟩ − ⟨τ(∇)(χ_{1}, …, χ_{p−1}), ℓχ_{p}⟩ = (\overline{d}_{p}τ(A))(χ_{1}, …, χ_{p}),

and the cohomology class corresponding to τ(∇) is defined by a differential form η′ = i^{∗}(τ(A)) = τ(i^{∗}(A)) = τ(η).

Let us prove statement (3). Without loss of generality assume that τ is a transposition (1, p). From (10) we have

ℓ^{∗}Δ_{1}^{∗_{1}}(χ_{1}, …, χ_{p−1}) = ∇^{∗_{1}}(ℓχ_{1}, …, χ_{p−1}) − ∑_{k=2}^{p−1} Δ_{k}^{∗_{1}}(χ_{1}, …, ℓχ_{k}, …, χ_{p−1}). (11)

A C-differential operator −Δ_{1}^{∗_{1}} defines a cohomology class η′ = i^{∗}(B), where B ∈ CDiff_{p}([κ]; \overline{Λ}^{n−1}) satisfies the following equation

(\overline{d}_{p}B)(χ_{1}, …, χ_{p}) = −⟨ℓ^{∗}Δ_{1}^{∗_{1}}(χ_{1}, …, χ_{p−1}), χ_{p}⟩ + ⟨Δ_{1}^{∗_{1}}(χ_{1}, …, χ_{p−1}), ℓχ_{p}⟩.

But from (11) we have

−⟨ℓ^{∗}Δ_{1}^{∗_{1}}(χ_{1}, …, χ_{p−1}), χ_{p}⟩ + ⟨Δ_{1}^{∗_{1}}(χ_{1}, …, χ_{p−1}), ℓχ_{p}⟩ =

− ⟨∇^{∗_{1}}(ℓχ_{1}, …, χ_{p−1}), χ_{p}⟩ +

∑_{k=2}^{p−1}⟨Δ_{k}^{∗_{1}}(χ_{1}, …, ℓχ_{k}, …, χ_{p−1}), χ_{p}⟩ + ⟨Δ_{1}^{∗_{1}}(χ_{1}, …, χ_{p−1}), ℓχ_{p}⟩ =

− ⟨ℓχ_{1}, ∇(χ_{p}, χ_{2}, …, χ_{p−1})⟩ + (\overline{d}_{p}C_{1})(ℓχ_{1}, …, χ_{p})

+∑_{k=2}^{p−1}⟨χ_{1}, Δ_{k}(χ_{p}, χ_{2}, …, ℓχ_{k}, …, χ_{p−1})⟩+∑_{k=2}^{p−1}(\overline{d}_{p}C_{k})(χ_{1}, …, ℓχ_{k}, …, χ_{p})

+ ⟨χ_{1}, Δ_{1}(ℓχ_{p}, χ_{2}, …, χ_{p−1})⟩ + (\overline{d}_{p}C_{p})(χ_{1}, …, ℓχ_{p}) =

− ⟨ℓχ_{1}, ∇(χ_{p}, χ_{2}, …, χ_{p−1})⟩

+ ⟨χ_{1}, (ℓ^{∗}∇)(χ_{p}, χ_{2}, …, χ_{p−1})⟩ + ∑_{k=1}^{p}(\overline{d}_{p}C_{k})(χ_{1}, …, ℓχ_{k}, …, χ_{p}) =

(\overline{d}_{p}τ(A))(χ_{1}, …, χ_{p}) + ∑_{k=1}^{p}(\overline{d}_{p}C_{k})(χ_{1}, …, ℓχ_{k}, …, χ_{p}),

where the C-differential operators C_{k}, k = 1, …, p, are defined by Green formula (7). Hence, one can choose the operator B such that

B(χ_{1}, …, χ_{p}) = τ(A)(χ_{1}, …, χ_{p}) + ∑_{k=1}^{p} C_{k}(χ_{1}, …, ℓχ_{k}, …, χ_{p}).

But, obviously, i^{∗}(B) = i^{∗}(τ(A)), and the operator −Δ_{1}^{∗_{1}} corresponds to the cohomology class τ(η)

Proposition 8 ([10]). The action of the permutation group S_{p} in the n-th cohomology group CDiff_{p−1}([κ]; [κ̂]) can be described as follows.

1. If τ ∈ S_{p−1} ⊂ S_{p}, τ(p) = p, then

τ(∇)(χ_{1}, …, χ_{p−1}) = ∇(χ_{τ(1)}, …, χ_{τ(p−1)}),

2. if τ = (s, p) is a transposition that interchanges s and p, then

τ(∇) = ∇^{∗_{s}},

where ∇ ∈ CDiff_{p−1}([κ]; [κ̂]).

Corollary 1. Let ω ∈ coker ℓ_{p−1}^{∗} and ∇ = ∇_{ω} ∈ CDiff_{p−1}^{alt}([κ]; [κ̂]) be a C-differential operator such that ω ≡ i^{∗}(∇) mod im ℓ^{∗}. Then the following is true.

1. If τ ∈ S_{p−1} ⊂ S_{p}, τ(p) = p, then

τ(ω) ≡ i^{∗}(τ(∇)) mod im ℓ^{∗},

where τ(∇) is defined by (9).

2. If τ = (s, p) is a transposition that interchanges s and p then

τ(ω) ≡ i^{∗} (∇^{∗_{s}}) mod im ℓ^{∗}.

4. The C-spectral sequence for evolution equations

4.1. Let Y be an evolution equation represented locally as

Y = {u_{t} − f(x_{i}, t, u_{σ}^{j}) = 0} , k = 1, …, m,

where x_{1}, …, x_{n−1}, t are independent variables, u = (u^{1}, …, u^{m}) are dependent variables and

u_{σ}^{j} = \frac{∂^{|σ|}u^{j}}{∂x_{1}^{σ_{1}} … ∂x_{n−1}^{σ_{n−1}}} , σ = (σ_{1}, …, σ_{n−1}).

Functions x_{i}, t, p_{σ}^{j} form a system of local coordinates on Y. Total derivatives D_{t}, D_{i} have the following form

D_{t} = \frac{∂}{∂t} + ∑_{j,σ} D_{σ}(f^{j})\frac{∂}{∂p_{σ}^{j}} , D_{i} = \frac{∂}{∂x_{i}} + ∑_{j,σ} p_{σ+1_{i}}^{j} \frac{∂}{∂p_{σ}^{j}} , i = 1, …, n − 1,

and the universal linearization is a m × m matrix

ℓ = [ℓ_{F}] = D_{t} − L_{f} = (\begin{matrix}D_{t} & 0 & … & 0 \\ 0 & D_{t} & … & 0 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & . . . . . . . & & D_{t} \end{matrix}) − (\begin{matrix}∑_{σ} \frac{∂f^{1}}{∂p_{σ}^{1}} D_{σ} & & … & & & ∑_{σ} \frac{∂f^{1}}{∂p_{σ}^{m}} D_{σ} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{σ} \frac{∂f^{m}}{∂p_{σ}^{1}} D_{σ} & & … & & & ∑_{σ} \frac{∂f^{m}}{∂p_{σ}^{m}} D_{σ} \end{matrix}) .

By definition, put F^{m} = \underbrace{F ⊕ ⋯ ⊕ F}_{m times}. Then in local coordinates one

can identify [κ] = [P] = [κ̂] = [P̂] = F^{m}. Denote by CDiff_{x}(F; F) the submodule of CDiff(F; F) generated by operators D_{σ}, σ = (σ_{1}, …, σ_{n−1}). For any F-modules Q_{1}, Q_{2} by definition put CDiff_{x}(Q_{1}; Q_{2}) = Hom_{F}(Q_{1}; Q_{2})⊗ CDiff_{x}(F; F) ⊂ CDiff(Q_{1}; Q_{2}) = Hom_{F}(Q_{1}; Q_{2})⊗CDiff(F; F), and define a linear projection i^{∗} : CDiff(F^{m}; F) → CDiff_{x}(F^{m}; F) by following

i^{∗}(∇) = ∇, if ∇ ∈ CDiff_{x}(F^{m}; F),
i^{∗}(∇ ∘ D_{t}) = i^{∗}(∇ ∘ L_{f}).

Lemma 5. The sequence

0 ⟶^{CDiff} (F^{m}; F) ⟶^{ℓ}CDiff(F^{m}; F) ⟶^{i^{∗}} CDiff_{x}(F^{m}; F) ⟶^{0},

where ℓ(Δ) = Δ ∘ ℓ is exact.

Proof. Obvious.

Corollary 2. If Y is an evolution equation,then 1. Y is regular and determined;

2. C^{1}Λ^{1} = CDiff_{x}(F^{m}; F);

Each multilinear C-differential operator Δ ∈ CDiff_{x,p}(F^{m}; F^{m}) can be written in the form

Δ = (Δ_{ij_{1}...j_{p}}) = ( ∑_{σ^{1},...,σ^{p}} a_{ij_{1}...j_{p}}^{σ^{1}...σ^{p}}D_{σ^{1}} ⊗ ⋯ ⊗ D_{σ^{p}} ) ,

(Δ(χ_{1}, …, χ_{p}))^{i} = ∑_{\begin{matrix}σ^{1},...,σ^{p} \\ i,j_{1},...,j_{p} \end{matrix}} a_{ij_{1}...j_{p}}^{σ^{1}...σ^{p}}D_{σ^{1}}(χ_{1}^{j_{1}}) · · · · · D_{σ^{p}}(χ_{p}^{j_{p}}),

where 1 ≤ i, j_{1}, …, j_{p} ≤ m, σ^{s} = (σ_{1}^{s}, …, σ_{n−1}^{s}), χ_{s} = (χ_{s}^{1}, …, χ_{s}^{m}) ∈ F^{m}.

For each ∇ ∈ CDiff(F; F) and Δ ∈ CDiff_{x,p}(F^{m}; F^{m}) define a Cdifferential operator ∇(Δ) ∈ CDiff_{x,p}(F^{m}; F^{m}) such that

(∇(Δ))_{ij_{1}...j_{p}} = ∑_{σ^{1},...,σ^{p}} ∇(a_{ij_{1}...j_{p}}^{σ^{1}...σ^{p}})D_{σ^{1}} ⊗ ⋯ ⊗ D_{σ^{p}}.

Theorem 2. The term E_{1}^{p,n−1}, p ≥ 2, for an evolution equation Y consists of C-differential operators ∇ ∈ CDiff_{x,p−1}^{alt}(F^{m}; F^{m}) such that

1. ∇^{∗_{s}} = −∇, 1 ≤ s ≤ p − 1;

2. for any χ_{1}, …, χ_{p−1} ∈ F^{m}

D_{t}(∇)(χ_{1}, …, χ_{p−1}) + L_{f}^{∗} (∇(χ_{1}, …, χ_{p−1})) + ∑_{s=1}^{p−1} ∇ (χ_{1}, …, L_{f}(χ_{s}), …, χ_{p−1}) = 0. (12)

Proof. By Theorem 1 if ∇ ∈ E_{1}^{p,n−1}, then there exist operators Δ_{1}, …, Δ_{p−1} such that

ℓ^{∗} (∇(χ_{1}, …, χ_{p−1})) = ∑_{s=1}^{p−1} Δ_{s}(χ_{1}, …, ℓ(χ_{s}), …, χ_{p−1}).

But we have

−ℓ^{∗} (∇(χ_{1}, …, χ_{p−1})) = (D_{t} + L_{f}^{∗}) (∇(χ_{1}, …, χ_{p−1})) =

∑_{s=1}^{p−1} ∇(χ_{1}, …, D_{t}(χ_{s}), …, χ_{p−1}) +

D_{t}(∇)(χ_{1}, …, χ_{p−1}) + L_{f}^{∗} (∇(χ_{1}, …, χ_{p−1})) =

∑_{s=1}^{p−1} ∇(χ_{1}, …, ℓ(χ_{s}), …, χ_{p−1}) + D_{t}(∇)(χ_{1}, …, χ_{p−1}) +

L_{f}^{∗} (∇(χ_{1}, …, χ_{p−1})) + ∑_{s=1}^{p−1} ∇(χ_{1}, …, L_{f}(χ_{s}), …, χ_{p−1}).

Therefore,

D_{t}(∇)(χ_{1}, …, χ_{p−1}) + L_{f}^{∗} (∇(χ_{1}, …, χ_{p−1})) + ∑_{s=1}^{p−1} ∇ (χ_{1}, …, L_{f}(χ_{s}), …, χ_{p−1}) = 0,

and Δ_{s} = −∇. Hence, if τ = (s, p) ∈ S_{p}, then τ(∇) = −Δ_{s}^{∗_{s}}, and τ(∇) = −∇ if and only if ∇ = −∇^{∗_{s}}.

In any module CDiff(Q; R) there exists a filtration by the modules CDiff^{(k)}(Q; R) consisting of C-differential operators of order ≤ k. Consider the module of C-symbols

Csmbl(Q; R) = ∑_{k=0}^{∞} Csmbl^{(k)}(Q; R),

Csmbl^{(k)}(Q; R) = CDiff^{(k)}(Q; R)/ CDiff^{(k−1)}(Q; R).

By definition, one has projections

Csmbl^{(k)} : CDiff^{(k)}(Q; R) → Csmbl^{(k)}(Q; R).

For each Δ ∈ CDiff(Q; R), Δ ≠ 0, define the order of Δ

ord Δ = min{k | Δ ∈ CDiff^{(k)}(Q; R)},

and the C-symbol s(Δ) = Csmbl^{(ord Δ)}(Δ). Symbols of C-differential operators from CDiff_{x}(Q; R) generate a submodule Csmbl_{x}(Q; R) ∈ Csmbl(Q; R).

By definition, put s(D_{i}) = θ_{i}, s(D_{t}) = θ_{t}, then θ_{i}, θ_{t} ∈ Csmbl^{(1)}(F; F). The module Csmbl(F; F) can be identified with the module of polynomials with coefficients in F, Csmbl(F; F) = F[θ_{1}, …, θ_{n−1}, θ_{t}] and Csmbl_{x}(F; F) = F[θ_{1}, …, θ_{n−1}].

The composition of differential operators in CDiff(F; F) induces a multiplication in Csmbl(F; F), which can be identified with the multiplication of polynomials in the algebra F[θ_{1}, …, θ_{n−1}, θ_{t}].

In the same way define a module Csmbl(Q_{1}, …, Q_{p}; R) = Csmbl(Q_{1}; F)⊗ ⋯ ⊗ Csmbl(Q_{p}; F) ⊗ R and for each multilinear C-differential operator the order and the C-symbol by following

ord(Δ_{1} ⊗ ⋯ ⊗ Δ_{p} ⊗ r) = ord Δ_{1} + ⋯ + ord Δ_{p},
s(Δ_{1} ⊗ ⋯ ⊗ Δ_{p} ⊗ r) = s(Δ_{1}) ⊗ ⋯ ⊗ s(Δ_{p}) ⊗ r,

where Δ_{i} ∈ CDiff(Q_{i}; F), 1 ≤ i ≤ p, r ∈ R.

We identify

Csmbl(Q_{1}, …, Q_{p}; R) = Hom_{F}(Q_{1}, …, Q_{p}; R) ⊗ Csmbl(F; F) ⊗ ⋯ ⊗ Csmbl(F; F),

Csmbl_{x}(Q_{1}, …, Q_{p}; R) =

Hom_{F}(Q_{1}, …, Q_{p}; R) ⊗ Csmbl_{x}(F; F) ⊗ ⋯ ⊗ Csmbl_{x}(F; F) =

Hom_{F}(Q_{1}, …, Q_{p}; R) ⊗ F[θ_{i}^{j}], 1 ≤ i ≤ n − 1, 1 ≤ j ≤ p.

Let us simplify notations by introducing θ = (θ_{1}, …, θ_{n−1}) and θ^{j} = (θ_{1}^{j}, …, θ_{n−1}^{j}). The composition of differential operators induces a left F[θ]-module structure in Csmbl(F, …, F; F) = F[θ_{i}^{j}] as follows

Δ_{1}(θ) ∘ Δ_{2}(θ^{1}, …, θ^{p}) = Δ_{1}(θ^{1} + ⋯ + θ^{p}) · Δ_{2}(θ^{1}, …, θ^{p}),

where Δ_{1} · Δ_{2} denotes the multiplication of polynomials.

Theorem 3. Let Y be an evolution equation such that the C-symbol of the C-differential operator L_{f} is nondegenerate on the dense open subset of Y and ord L_{f} ≥ 2. Then

E_{1}^{p,n−1} = 0, p ≥ 3.

Proof. Let Δ ∈ CDiff_{x,p−1}(F^{m}; F^{m}) be a solution of (12) from Theorem

2. Choose a point π ∈ Y such that the C-symbol s(L_{f}) is nondegenerate at this point. Denote the C-symbols of Δ and L_{f} at the point π by δ = s(Δ)(π) and λ = s(L_{f})(π) respectively. Then λ ∈ Hom_{ℝ}(V; V) ⊗ F[θ], δ ∈ Hom_{ℝ}(⋀^{p−1} V; V) ⊗ F[θ^{1}, …, θ^{p−1}], where V = ℝ^{m}.

Fix a basis of V and identify λ with an m × m matrix λ = (λ_{j}^{i}), i, j = 1, …, m, where λ_{j}^{i} = λ_{j}^{i}(θ) ∈ ℝ[θ] is a homogeneous polynomial of degree l = ord L_{f}. Then (λ^{∗})_{j}^{i} = (−1)^{l}λ_{i}^{j}. Let δ_{i_{1}...i_{p−1}}^{k} = δ_{i_{1}...i_{p−1}}^{k}(θ^{1}, …, θ^{p−1}) be the components of δ in the chosen basis. Now one can rewrite the C-symbol of equation (12) from Theorem 2 at point π in the form

(−1)^{l} ∑_{i=1}^{m} λ_{i}^{k}(θ^{1} + ⋯ + θ^{p−1})δ_{i_{1}...i_{p−1}}^{i} + ∑_{s=1}^{p−1} ∑_{i=1}^{m} δ_{i_{1}...i_{s−1}i...i_{p−1}}^{k}λ_{i}^{i_{s}}(θ^{s}) = 0, (13)

where 1 ≤ i_{1}, …, i_{s}, k ≤ m.

System (13) 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 equal to 0. Since λ = λ(θ) is not degenerate, there exists v ∈ ℂ^{m} such that detλ(v) ≠ 0. One can suppose that λ(v) has an uppertriangular (Jordan) form, λ_{j}^{i}(v) = 0 if i ≥ j and λ_{i}^{i}(v) ≠ 0. Then for any α ∈ ℂ the matrix λ(αv) has also an uppertriangular form λ_{j}^{i}(αv) = α^{l}λ_{j}^{i}(v). Since l = ord L_{f} ≥ 2 and p ≥ 3, there exist complex numbers α_{s} ∈ ℂ, s = 1, …, p − 1, such that for any i_{1}, …, i_{p−1}, k, 1 ≤ i_{1}, …, i_{p−1}, k ≤ m

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

Put θ^{i} = α_{i}v. Then system (13) can be rewritten in an uppertriangular form with diagonal entries A_{k,i_{1},...,i_{p−1}} ≠ 0. Hence, the determinant of system (13) does not equal to 0, and the solution of this system δ = 0. Therefore, C-symbol of Δ vanishes on a dense subset of Y. But this means that Δ = 0.

4.2. Scalar evolution equations. In this subsection we consider the case when Y is a scalar evolution equation. If the order of an equation Y is greater then 1, then from Theorem 3 it follows that E_{1}^{p,n−1} = 0 if p ≥ 3. If the order of Y is equal to 1 then the following Theorem is true.

Theorem 4. Let Y be a scalar evolution equation of the 1-st order. Then E_{1}^{p,n−1}, p ≥ 1, is an infinite dimensional vector space.

Proof. Locally each scalar evolution equation of the 1-st order by a contact transformation can be reduced to the equation u_{t} = 0. In this case equation (12) has a simple form D_{t}(Δ) = \frac{∂Δ}{∂t} = 0. Denote by F_{x} the algebra of functions on Y that do not depend on the variable t. Then E_{1}^{1,n−1} = F_{x} and E_{1}^{p,n−1}, p ≥ 2, consists of operators Δ ∈ CDiff_{x,p−1}^{alt}(F; F) with coefficients in F_{x} such that Δ^{∗_{s}} = −Δ, 1 ≤ s ≤ p − 1.

The term E_{1}^{p,n−1}, p = 1, 2 need not be trivial. E_{1}^{1,n−1} contains generating functions of conservation laws. In [3] the term E_{1}^{2,1} is computed for some evolution equations with single space variable, n = 2. Here we compute the term E_{1}^{2,n−1} for a scalar linear evolution equation with constant coefficients.

Let Y be a scalar linear evolution equation with constant coefficients u_{t} = Lu, L ∈ ℝ[θ]. In this case L_{f} = \overline{L} is the lifting of the linear differential operator L. We identify L and L_{f} in the polynomial ring ℝ[θ].

The equation defining E_{1}^{2,n−1} has the following form:

D_{t}(Δ) + L^{∗} ∘ Δ + Δ ∘ L = 0  (14)

and the composition of differential operators ∘ : F[θ] × F[θ] → F[θ] can be described as follows. Let Δ_{i} = Δ_{i}(θ_{1}, …, θ_{n−1}) ∈ F[θ], i = 1, 2, be polynomials with coefficients in F. Then

Δ_{1} ∘ Δ_{2} = Δ_{1}(θ_{1} + D_{1}, …, θ_{n−1} + D_{n−1}) · Δ_{2}

= Δ_{1} · Δ_{2} + ∑_{i} \frac{∂Δ_{1}}{∂θ_{i}} · D_{i}(Δ_{2}) + \frac{1}{2} ∑_{i≤j} \frac{∂^{2}Δ_{1}}{∂θ_{i}∂θ_{j}} · D_{i,j}(Δ_{2}) + …,

where Δ_{1} · Δ_{2} denotes the multiplication in the polynomial ring F[θ] and D_{i} · Δ = D_{i}(Δ).

If L is a linear C-differential operator with constant coefficients, L ∈ ℝ[θ], then L = ∑_{i=0}^{k} L_{i}, where L_{i} is a homogeneous polynomial, ord L_{i} = i, and

Δ ∘ L = Δ · L = L · Δ, L^{∗} = L(−θ) = ∑_{i=0}^{k}(−1)^{i}L_{i},

where Δ ∈ F[θ].

Each operator Δ ∈ F[θ] is a sum of homogeneous operators, Δ = ∑_{i=0}^{l} Δ_{i}, ord Δ_{i} = i, Δ_{l} ≠ 0. The left-hand side of equation (14) is also a sum of homogeneous operators. Hence one can solve (14) by equating homogeneous terms to 0. Let us wright out 2 terms of maximal order k + l and k + l − 1 if k ≥ 2

(1 + (−1)^{k})L_{k} · Δ_{l} = 0;  (15)

(1 + (−1)^{k})L_{k} · Δ_{l−1} + (1 − (−1)^{k})L_{k−1} · Δ_{l} + ∑_{i=1}^{n−1} \frac{∂L_{k}}{∂θ_{i}} · D_{i}(Δ_{l}) = 0. (16)

Let L ∈ ℝ[θ] ⊂ F[θ] be a linear C-differential operator with constant coefficients. Consider a transformation

L ↦ L′ = e^{−(λ,x)} ∘ L ∘ e^{(λ,x)} + A, L′ = L(θ + λ) + A,  (17)

where λ ∈ ℝ^{n−1}, (λ, x) = λ_{1}x_{1} + ⋯ + λ_{n−1}x_{n−1}, A ∈ ℝ[θ] is a linear C-differential operator with constant coefficients, ord A ≤ 1. Obviously, the operator L′ is also a C-differential operator with constant coefficients. We say that C-differential operators L_{1} and L_{2} are equivalent if by applying transformation (17) and a linear change of variables θ = (θ_{1}, …, θ_{n−1}) one can obtain L_{2} from L_{1}. If operators L_{1} and L_{2} are equivalent then the corresponding evolution equations u_{t} = L_{1}u and v_{t} = L_{2}v can be obtained one from another by a change of variables.

An operator L ∈ ℝ[θ], L = ∑_{i=0}^{k} L_{i} is said to be in a normal form if there exist integer numbers s_{0}, …, s_{k}, 1 ≤ s_{k} ≤ ⋯ ≤ s_{0} ≤ n − 1, such that for any i, 0 ≤ i ≤ k the following conditions hold:

1. \frac{∂L_{i}}{∂θ_{j}} = 0, if j > s_{i};

2. polynomials \frac{∂L_{i}}{∂θ_{j}}, s_{i+1} ≤ j ≤ s_{i} are linearly independent.

The integer number s_{0} is called the rank of the operator L and is denoted by rk L.

Lemma 6. Let L = ∑_{i=0}^{k} L_{i} be an operator, L ∈ ℝ[θ_{1}, …, θ_{n−1}]. There exists a linear change of variables (θ_{1}, …, θ_{n−1}) that transforms L to an operator L′ in a normal form.

Proof. Straightforward.

Theorem 5. Let Y be a scalar linear evolution equation with constant coefficients u_{t} = Lu, L ∈ ℝ[θ]. If E_{1}^{2,n−1} ≠ 0, then L is equivalent to a skew-selfadjoint operator L′, L′^{∗} = −L′.

Proof. Let Δ be a C-differential operator satisfying equation (14), Δ = ∑_{i=0}^{l} Δ_{i}, ord Δ_{i} = i, Δ_{r} ≠ 0. Consider a homogeneous part (15) of equation (14) of the order r+k. Since there exists a non trivial solution Δ_{l}, the order k of L is odd.

By induction, suppose that there exists r ≥ 0 such that L_{k−2i+1} = 0, 0 ≤ i ≤ r. Prove that the operator L is equivalent to an operator L′ = ∑_{i=0}^{k} L′_{i} such that L′_{k−2i+1} = 0, 0 ≤ i ≤ r + 1.

If k − 2r − 1 = 0 then put L′ = L − L_{0}. Suppose that k − 2r − 1 ≥ 2. By Lemma 6 we can assume that there exist integer numbers s_{1}, …, s_{r}, 1 ≤ s_{1} ≤ ⋯ ≤ s_{r} ≤ n − 1, such that for any i, 0 ≤ i ≤ r,

1. \frac{∂L_{k−2i}}{∂θ_{j}} = 0, if j > s_{i+1};

2. polynomials \frac{∂L_{k−2i}}{∂θ_{j}}, s_{i} ≤ j ≤ s_{i+1} are linearly independent.

Hence, the homogeneous part of equation (14) of the order k + l − 1 (16) simplifies as follows

∑_{i=1}^{s_{1}} \frac{∂L_{k}}{∂θ_{i}} D_{i}(Δ_{l}) = 0.

From condition (2) we have that D_{i}(Δ_{l}) = 0, 1 ≤ i ≤ s_{1}. Hence, we proved that if Δ is a solution of equation (14), then ord D_{i}(Δ) ≤ ord Δ−1 = l−1, 1 ≤ i ≤ s_{1}. But D_{i}(Δ) is also a solution of (14) for any i. Therefore, ord D_{ij}(Δ) ≤ l−2, 1 ≤ i, j ≤ s_{1}, and ord D_{i_{1}...i_{l+1}}(Δ) = 0, 1 ≤ i_{1}, …, i_{l+1} ≤ s_{1}. Now it is easy to see that one can find a non zero solution of (14) Δ′ = D_{i_{1}...i_{j}}(Δ) such that Δ′ ≠ 0 and D_{i}(Δ′) = 0 for any i = 1, …, s_{1}. Then

L_{k}^{∗} ∘ Δ′ + Δ′ ∘ L_{k} = −L_{k}(θ_{1} + D_{1}, …, θ_{s_{1}} + D_{s_{1}}) · Δ′ + L_{k} · Δ′ = −L_{k} · Δ′ + L_{k} · Δ′ = 0,

and one can continue solving equation (14) with L = L_{k−2} + . . .. Repeating arguments above we find a non zero solution of (14) Δ′′ such that D_{i}(Δ′′) = 0, 1 ≤ i ≤ s_{r}. Consider than the homogeneous part of (14) of the order k + l − 1 − 2r, which can be written as follows

2L_{k−2r−1} · Δ′′_{l} + ∑_{i=s_{r}+1}^{n−1} \frac{∂L_{k−2r}}{∂θ_{i}} · D_{i}(Δ′′_{l}) = 0.

Since Δ′′_{l} ≠ 0 ,one can find λ_{i} ∈ ℝ, s_{r} + 1 ≤ i ≤ n − 1, such that

L_{k−2r−1} + ∑_{i=s_{r}+1}^{n−1} \frac{∂L_{k−2r}}{∂θ_{i}} λ_{i} = 0.

Put L′ = e^{−(λ,x)} ∘ L′ ∘ e^{(λ,x)} = L(θ + λ), where (λ, x) = ∑_{i=s_{r}+1}^{n−1} λ_{i}x_{i}. Then L′_{i} = L_{i} if k − 2r ≤ i ≤ k and L′_{k−2r−1} = 0.

Therefore, one can find an operator L′ equivalent to L such that L′_{i} = 0 for even i. But then we have

L′^{∗}(θ) = L′(−θ) = −L′

and L′ is a skew-selfadjoint operator. The Theorem is proved.

Theorem 6. Let L ∈ ℝ[θ], L = ∑_{i=0}^{k} L_{i}, be a skew-selfadjoint operator in a normal form and L_{0} = L_{1} = 0. Then

1. solutions of equation (14) form an algebra A with respect to the operation of composition;

2. if Δ ∈ A then Δ^{∗} ∈ A;

3. the algebra A is isomorphic to the algebra of differential operators A_{0} = ℝ[X_{1}, …, X_{s}]⊗C^{∞}(x_{s+1}, …, x_{n−1})⊗ℝ[θ_{1}, …, θ_{n−1}], where s = rk L, and C^{∞}(x_{s+1}, …, x_{n−1}) denotes the algebra of smooth functions depending on variables x_{s+1}, …, x_{n−1}.

Proof. Since L^{∗} = −L, equation (14) has the following form

D_{t}(Δ) + [Δ, L] = 0.  (18)

If Δ_{1}, Δ_{2} are solutions of (18), then

D_{t}(Δ_{1} ∘ Δ_{2}) + [Δ_{1} ∘ Δ_{2}, L] = (D_{t}(Δ_{1}) + [Δ_{1}, L]) ∘ Δ_{2} + Δ_{1} ∘ (D_{t}(Δ_{2}) + [Δ_{2}, L]) = 0,

and this proves statement (1).

Further, if Δ is a solution of (18), then

0 = (D_{t}(Δ) + [Δ, L])^{∗} = D_{t}(Δ^{∗}) + [L^{∗}, Δ^{∗}] = D_{t}(Δ^{∗}) + [Δ^{∗}, L],

and (2) is proved.

Now we show that the algebra of solutions of equation (18) A contains as a subalgebra A_{0} = ℝ[X_{1}, …, X_{s}] ⊗ C^{∞}(x_{s+1}, …, x_{n−1}) ⊗ ℝ[θ]. Obviously, each Δ ∈ C^{∞}(x_{s+1}, …, x_{n−1}) ⊗ ℝ[θ] satisfies equation (18). Let X_{i} = x_{i} + t\frac{∂L}{∂θ_{i}}, 1 ≤ i ≤ s. Then

D_{t}(X_{i}) + [X_{i}, L] = \frac{∂L}{∂θ_{i}} + [x_{i}, L] = \frac{∂L}{∂θ_{i}} − \frac{∂L}{∂θ_{i}} = 0,

and X_{i} ∈ A.

Further,

[X_{i}, X_{j}] = t[\frac{∂L}{∂θ_{i}} , x_{j}] + t[x_{i}, \frac{∂L}{∂θ_{j}}] = t\frac{∂^{2}L}{∂θ_{i}∂θ_{j}} − t\frac{∂^{2}L}{∂θ_{i}∂θ_{j}} = 0,

[X_{i}, θ_{j}] = δ_{ij}, X_{i}^{∗} = X_{i}.

Hence, the algebra generated by X_{i}, 1 ≤ i ≤ s, and Δ ∈ C^{∞}(x_{s+1}, …, x_{n−1})⊗ ℝ[θ] is the algebra of differential operators A_{0}.

Let us prove that A ⊂ A_{0}. By induction, suppose that if Δ ∈ A and D_{i}(Δ) = 0 for all i, 1 ≤ i ≤ r ≤ s, then Δ ∈ A_{0}. Let Δ ∈ A and D_{i}(Δ) = 0 for all i, 1 ≤ i ≤ r − 1. In the same way as in the proof of Theorem 5 one can show that D_{r−1}^{l+1}(Δ) = 0. Hence, Δ is a polynomial with respect to the variable x_{r−1}, Δ = ∑_{i=0}^{d} x_{r−1}^{i}Δ^{(i)}, D_{j}(Δ^{(i)}) = 0, 1 ≤ j ≤ r. But Δ^{(d)} = \frac{1}{d!}D_{r−1}^{d}(Δ) is a solution of (18) and by the inductive hypothesis Δ^{(d)} ∈ A_{0}. We can apply the same argument to the operator Δ′ = Δ − X_{r−1}Δ^{(d)} and show that Δ = ∑_{i=0}^{d} X_{r−1}^{i}∇^{(i)}, where ∇^{(i)} ∈ A_{0}. Therefore, Δ ∈ A_{0} and A = A_{0}.

Corollary 3. Let Y = {u_{t} = Lu} be a scalar evolution equation, where L is a linear differential operator with constant coefficients. Then term E_{1}^{2,n−1} of the C-spectral sequence for Y is isomorphic to a linear space of differential operators Δ ∈ ℝ[X_{1}, …, X_{s}] ⊗ C^{∞}(x_{s+1}, …, x_{n−1}) ⊗ ℝ[θ_{1}, …, θ_{n−1}], s = rk L, such that Δ^{∗} = −Δ.

Acknowledgments

The author is grateful to Professor A. M. Vinogradov for very stimulating attention to this work and to A. M. Verbovetsky for useful discussions.

He also wishes to thank the SISSA for warm hospitality and the Ministero degli Affari Esteri for a fellowship.

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