The Diffiety Institute Preprint Series

Preprint DIPS–2/2001 March 30, 2001

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

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

On the horizontal gauge cohomology and

non-removability of the spectral parameter

by

Michal MARVAN

On the horizontal gauge cohomology and non-removability of the spectral parameter

MICHAL MARVAN

Abstract. Given a zero-curvature representation α, the horizontal gauge cohomology group H̄_{α}^{1} is shown to contain obstructions to removability of the “spectral parameter” of α. A method to compute H̄_{α}^{1} is suggested.

Introduction

Important integration techniques ([1, 6, 8, 9]) for soliton equations, especially in dimension two, involve a zero-curvature representation (ZCR) taking values in a non-solvable Lie algebra 𝔤 and depending on a non-removable “spectral” parameter ([2, 26]). This is why oneparametric families of ZCR’s, also referred to as “linear problems”, are often considered as attributes of “complete integrability” of nonlinear partial differential equations (PDE). From another perspective, ZCR’s are linear, finite-dimensional coverings [15], and often realizations of the Wahlquist and Estabrook [25] prolongation structures. Yet another interpretation has been given in terms of pseudospherical surfaces [19, 10]. Relations to the geometry of surfaces are also the subject of [5, 20, 21].

In [16], using the so-called first gauge cohomology, a characteristic element χ_{α} was associated to any ZCR α of a formally integrable system of nonlinear PDE’s; the case of 𝔤 = 𝔰𝔩_{2} being further developed in [17]. Independently, in his treatment of “continual classes” of evolution equations possessing a ZCR, Sakovich [18] introduced essentially the same concept, although without cohomological interpretation.

In this paper we deal with zeroth (horizontal) gauge cohomology H̄_{α}^{q} associated with any ZCR α (Section 1). For ZCR’s depending on a parameter we introduce a horizontal cohomology class [α̇] ∈ H̄_{α}^{1} that is an obstruction to removability of the parameter (Section 2). This paper is to a great extent parallel to the work by Krasil’shchik and Igonin [13], who study general coverings depending on a parameter, with the socalled C-cohomology in the background (Krasil’shchik [11, 12]).

2000 Mathematics Subject Classification. 35A30, 58G05.

Key words and phrases. Zero-curvature representation, horizontal cohomology, spectral parameter, pseudospherical surface.

ZCR’s related to pseudospherical surfaces need not depend on a parameter, but when they do, infinite-dimensional sequences of conservation laws are generated ([19, 5]), provided the parameter is not removable. We discuss in detail an example [21, Ch. I, Sect. 7.10], which features a removable parameter.

In Section 4 we show a simple way to compute H̄_{α}^{1}, applicable to typical equations of mathematical physics. The idea is to convert the problem into one already solved—finding a ZCR, although with coefficients in a larger algebra.

It should be noted that when the Lie algebra 𝔤 is one-dimensional, then the horizontal gauge cohomology becomes the horizontal cohomology H̄^{q} = E_{1}^{0,q} of Vinogradov [23] and is effectively computable by means of the C-spectral sequence [23, 22]. Also, according to Verbovetsky [24], for general 𝔤 we still have H̄_{α}^{q} = E_{1}^{0,q} for a generalized C-spectral sequence with coefficients in a C-module, which leads to a computation method for the terms E_{1}^{p,q} with p ≥ 1, but not the terms E_{1}^{0,q} = H̄_{α}^{q} we need here.

1. Preliminaries

We recall briefly the basics of geometry of PDE’s from [14, 23], which are the main references for all concepts unexplained below. All manifolds, functions, vector fields etc. are assumed to be smooth of class C^{∞}. Let X be an m-dimensional manifold, 1 < m < ∞, let Y → X be a fibered manifold over X of fibre dimension h ≤ ∞. Consider the infinite jet prolongation j^{∞}Y → X of Y → X. Local coordinates x^{i}, i = 1, …, m, on X and u^{k}, k = 1, …, h, along the fibres of Y → X induce natural local coordinates u_{I}^{k} along the fibres of j^{∞}Y → Y; here and in what follows I stands for a symmetric multi-index in the i’s. The commuting vector fields D_{i} = ∂/∂x^{i} + ∑_{r,I} u_{Ii}^{k} ∂/∂u_{I}^{k} on j^{∞}Y are called total derivatives. Differential operators D_{I} acting on functions on j^{∞}Y are defined recursively by D_{Ii} = D_{I} ∘ D_{i}.

In this framework we consider a finite system of finite-order PDE’s

(1)  F^{l}(x^{i}, u, …, u_{I}^{k}, …) = 0,

l = 1, …, n, on unknowns u^{k} and their derivatives u_{I}^{k}. We assume that system (1), along with all its differential consequences D_{I}F^{l} = 0, determines a submanifold E ⊆ j^{∞}Y (typically infinite-dimensional). The total derivatives are tangent to E, hence they have a well-defined action on smooth functions on E. The restricted fields D_{i} = D_{i}|_{E} span the so-called Cartan distribution on E, denoted C. The structure (E, C) is a particular instance of a diffiety (see loc. cit.).

On E, we have the direct decomposition of the tangent bundle T E as C ⊕V E, the Cartan distribution and the vertical vector bundle with respect to the projection on X. Let C^{∞}E denote the ring of C^{∞} functions on E, let Λ^{1,0}E = Ann C and Λ^{0,1}E = Ann V E denote the C^{∞}E-modules of contact 1-forms and horizontal 1-forms, respectively. Denoting by Λ^{r}E the C^{∞}E-modules of antisymmetric r-forms on E, we have the induced splittings Λ^{r}E = ⨁_{p+q=r} Λ^{p,q}E into r + 1 direct summands Λ^{p,q}E = ⋀^{p} Λ^{1,0}E ∧ ⋀^{q} Λ^{0,1}E. Accordingly, the exterior differentials split into the sum d = d̄+ ℓ of the horizontal differential d̄: Λ^{p,q}E → Λ^{p,q+1}E and the vertical differential ℓ: Λ^{p,q}E → Λ^{p+1,q}E. With respect to d̄ and ℓ, the collection of the C^{∞}E-modules Λ^{p,q}E, p, q ≥ 0, is the variational bicomplex, whose associated spectral sequence (with E_{0}^{pq} = Λ^{p,q}) is the C-spectral sequence. Coordinate formulas for the differentials d̄, ℓ are derived from their action on Λ^{0,0}E = C^{∞}E, which is

d̄f = ∑_{i} D_{i}f dx^{i},

ℓf = ∑_{k,I} \frac{∂f}{∂u_{I}^{k}} ω_{I}^{k},  ω_{I}^{k} = du_{I}^{k} − ∑_{i} u_{iI}^{k} dx^{i}.

Turning to the material of [16], we consider a finite-dimensional real or complex Lie algebra 𝔤. Its tensor product with the graded exterior algebra Λ̄E = ⨁_{q} Λ̄^{q}E is a graded nonassociative algebra under the bracket [Aμ, Bν] = [A, B]μ ∧ ν for A, B ∈ 𝔤 and μ, ν ∈ Λ̄E. Then

[ρ, σ] = −(−1)^{rs}[σ, ρ],  d̄[ρ, σ] = [d̄ρ, σ] + (−1)^{r}[ρ, d̄σ]

for ρ ∈ Λ̄^{r}E ⊗𝔤, σ ∈ Λ̄^{s}E ⊗𝔤. It is technically convenient to assume that 𝔤 is a matrix algebra (the assumption being irrestrictive by the Ado theorem [3]), i.e., that 𝔤 is a subalgebra in some 𝔤𝔩_{n}. Then Λ̄E ⊗𝔤𝔩_{n} is a graded associative algebra with respect to the multiplication Aμ·Bν = (A · B)μ ∧ ν induced by the ordinary matrix multiplication, while

[ρ, σ] = ρ · σ − (−1)^{rs}σ · ρ,  d̄(ρ · σ) = d̄ρ · σ + (−1)^{r}ρ · d̄σ

for ρ ∈ Λ̄^{r}E ⊗ 𝔤𝔩_{n}, σ ∈ Λ̄^{s}E ⊗ 𝔤𝔩_{n}. Elements of C^{∞}E ⊗ 𝔤 will be called 𝔤-matrices.

A 𝔤-valued zero-curvature representation (ZCR) for E is a horizontal 1-form α ∈ Λ̄^{1}E ⊗ 𝔤 satisfying

(2)  d̄α = \frac{1}{2}[α, α].

If α = ∑_{i} A_{i} dx^{i}, A_{i} ∈ 𝔤, then eq. (2) becomes D_{j}A_{i}−D_{i}A_{j} +[A_{i}, A_{j}] = 0, for pairs {i, j} such that i ≠ j.

Given a ZCR α, we consider operators

¯∂_{α} = d̄− ad_{α}: Λ̄^{q}E ⊗ 𝔤 → Λ̄^{q+1}E ⊗ 𝔤,

where ad_{α} ρ = [α, ρ] for any ρ ∈ Λ̄E ⊗ 𝔤. We have ¯∂_{α} ∘ ¯∂_{α} = 0 as a consequence of (2), which gives the horizontal gauge complex (or 0-th linear gauge complex [16])

0 → Λ̄^{0}E ⊗ 𝔤 ⟶^{¯∂_{α}} Λ̄^{1}E ⊗ 𝔤 ⟶^{¯∂_{α}} Λ̄^{2}E ⊗ 𝔤 → ⋯ → Λ̄^{m}E ⊗ 𝔤 → 0.

The groups

H̄_{α}^{q}(E, 𝔤) = \frac{Ker(Λ̄^{q}E ⊗ 𝔤 ⟶^{¯∂_{α}} Λ̄^{q+1}E ⊗ 𝔤)}{Im(Λ̄^{q−1}E ⊗ 𝔤 ⟶^{¯∂_{α}} Λ̄^{q}E ⊗ 𝔤)}

are called the horizontal gauge cohomology groups with respect to the ZCR α.

If α = ∑_{i} A_{i} dx^{i}, A_{i} ∈ 𝔤, then for an arbitrary 𝔤-matrix C ∈ C^{∞}E ⊗𝔤 we have

(3)  ¯∂_{α}C = ∑_{i} D̂_{i}C dx^{i},  D̂_{i}C = D_{i}C − [A_{i}, C].

Operators D̂_{i} commute whenever α is a ZCR. We define recursively D̂_{Ii} = D̂_{I} ∘ D̂_{i}.

Likewise, for p = 1 the corresponding modules Λ^{1,q} ⊗ 𝔤 form the so-called first linear gauge complex

0 → Λ^{1,0}E ⊗ 𝔤 ⟶^{¯∂_{α}} Λ^{1,1}E ⊗ 𝔤 → ⋯ → Λ^{1,m}E ⊗ 𝔤 → 0.

The groups

H_{α}^{1,q}(E, 𝔤) = \frac{Ker(Λ^{1,q}E ⊗ 𝔤 ⟶^{¯∂_{α}} Λ̄^{1,q+1}E ⊗ 𝔤)}{Im(Λ̄^{1,q−1}E ⊗ 𝔤 ⟶^{¯∂_{α}} Λ̄^{1,q}E ⊗ 𝔤)}

are computable in principle. We refer to [16] or [24] for details of the isomorphism H_{α}^{1,q}(E, 𝔤) ≅ Ker(𝔤⊗P̂_{m−q} → 𝔤⊗P̂_{m−q−1})/ Im(𝔤⊗P̂_{m−q+1} → 𝔤 ⊗ P̂_{m−q}). Here the starting point is the “compatibility complex” P_{0} → P_{1} → P_{2} → ⋯, where P_{i} are modules of sections of appropriate vector bundles over E (see [24]). We only remark here that dim P_{0} = h = the number of unknowns, dim P_{1} = n = the number of equations in system (1), and the first arrow P_{0} → P_{1} can be identified with the operator of universal linearization, while each of the subsequent arrows expresses the integrability conditions for the previous one. Always P_{j} = 0 for all j > m, while for non-overdetermined systems we have even P_{j} = 0 for all j > 1. The dual complex ⋯ → P̂_{2} → P̂_{1} → P̂_{0} is composed of formally adjoint operators between dual modules P̂ := Hom(P, Λ̄^{m}E). Finally,

⋯ → 𝔤 ⊗ P̂_{2} → 𝔤 ⊗ P̂_{1} → 𝔤 ⊗ P̂_{0}

is obtained by replacing each operator D_{i} with its covariant counterpart D̂_{i} = D_{i} − ad_{A_{i}}.

The element ℓ(α) ∈ Λ^{1,1}E ⊗ 𝔤 is a cocycle by (2); the corresponding 1-st cohomology class ch(α) = [ℓ(α)] ∈ H_{α}^{1,1}(E, 𝔤) will be called the characteristic class of the ZCR α. The characteristic element χ_{α} we introduced in [16] is the image of ch(α) under the above isomorphism

H_{α}^{1,1}(E, 𝔤) ≅ \frac{Ker(𝔤 ⊗ P̂_{m−1} → 𝔤 ⊗ P̂_{m−2})}{Im(𝔤 ⊗ P̂_{m} → 𝔤 ⊗ P̂_{m−1})}.

In [16, Prop. 4.2] we proved that if H_{α}^{2,0} = 0 and ch(α) = 0, then α = d̄θ · θ^{−1} for an appropriate G-matrix θ (such α’s are called trivial).

Assuming system (1) non-overdetermined, P_{2} is zero. Then H_{α}^{1,1}(E, 𝔤) vanishes for m > 2 and every ZCR is trivial, while for m = 2 we have an isomorphism H_{α}^{1,1}(E, 𝔤) ≅ Ker(𝔤 ⊗ P̂_{1} → 𝔤 ⊗ P̂_{0}) and characteristic elements are n-tuples of 𝔤-matrices χ_{l} defined on E and satisfying

(4)  ∑_{l,J}(−1)^{|J|}D̂_{J} (\frac{∂F^{l}}{∂u_{J}^{k}} χ_{l}) = 0,  D̂_{i} = D_{i} − ad_{A_{i}} .

We also have an explicit formula for the characteristic element for α, first written by Sakovich [18] for evolution equations: if 𝔤-matrices C_{l}^{J} satisfy

d̄α − \frac{1}{2}[α, α] = ∑_{l,J} D_{J}F^{l} · C_{l}^{J},

then

(5)  χ_{l} = ∑_{J}(−D̂)_{J}C_{l}^{J} |_{E} .

For a matrix function S: E → G, we have the conjugation Ad_{S}: Λ̄^{q}E ⊗ 𝔤 → Λ̄^{q}E ⊗ 𝔤 defined by γ ↦ S · γ · S^{−1}. For any ZCR α, the form

α^{S} = d̄S · S^{−1} + S · α · S^{−1}

is another ZCR; we call it gauge equivalent to α. One easily checks that

(6)  ¯∂_{α^{S}} ∘ Ad_{S} = Ad_{S} ∘ ¯∂_{α},

so that Ad_{S} is a morphism of the horizontal gauge complexes. Since Ad_{S} is invertible (with inverse Ad_{S^{−1}}), we have

(7)  H̄_{α}^{q}(E, 𝔤) ≅ H̄_{α^{S}}^{q}(E, 𝔤).

Similarly to (7), we have the isomorphism H_{α}^{1,q}(E, 𝔤) ≅ H_{α^{S}}^{1,q}(E, 𝔤) induced by the conjugation Ad_{S}. Hence, gauge equivalent ZCR’s have conjugate characteristic elements. The converse is not true in general and two ZCR’s with conjugate characteristic elements may still be gauge inequivalent. Examples are provided by numerous completely integrable equations of mathematical physics (e.g., KdV, mKdV, sine-Gordon, etc.) that have their characteristic elements independent of the spectral parameter.

We close this section a result easy to prove:

Proposition 1. Let α ∈ Λ̄^{1}(E, 𝔤) be a 𝔤-valued ZCR on E, with 𝔤 = 𝔤′ ⊕ 𝔤′′, let α = α′ + α′′ with α′ ∈ Λ̄^{1}(E, 𝔤′), α′′ ∈ Λ̄^{1}(E, 𝔤′′). Then H̄_{α}^{q} = H̄_{α′}^{q} ⊕ H̄_{α′′}^{q}.

2. Removability

In this section we consider a ZCR depending on a parameter λ and show that the first horizontal gauge cohomology group H̄^{1} contains obstructions to removability of λ. Let α_{λ_{0}} be a member of a smooth family of ZCR’s

(8)  α_{λ} ∈ Λ̄^{1}E ⊗ 𝔤,

with λ taking values in an open interval containing λ_{0}. Denoting

α̇_{λ_{0}} = \frac{∂}{∂λ}|_{λ=λ_{0}}α_{λ} ∈ 𝔤,

we have

d̄α̇_{λ_{0}} = \frac{∂}{∂λ}|_{λ=λ_{0}} d̄α_{λ} = \frac{∂}{∂λ}|_{λ=λ_{0}}(α_{λ} · α_{λ}) = α̇_{λ_{0}} · α_{λ_{0}} + α_{λ_{0}} · α̇_{λ_{0}},

whence ¯∂_{α_{λ_{0}}} α̇_{λ_{0}} = 0. Thus,

Proposition 2. The form α̇_{λ} ∈ Λ̄^{1}E ⊗ 𝔤 is a 1-cocycle with respect to ∂_{α_{λ}}.

Recall that the parameter λ is removable if the forms α_{λ} are gauge equivalent for different values of λ, otherwise it is called non-removable. A parameter λ created by gauge action with respect to a matrix S_{λ} will be removed by the gauge action with respect to S_{λ}^{−1}.

Proposition 3. The following statements about the family (8) are equivalent:

(i) the parameter λ is removable; (ii) each of the forms α̇_{λ} ∈ Λ̄^{1}E ⊗ 𝔤 is a coboundary, α̇_{λ} = ¯∂_{α_{λ}}Q_{λ},

with 𝔤-matrices Q_{λ} depending smoothly on λ.

Proof. Let the matrix Lie algebra 𝔤 be associated to a connected and simply connected matrix Lie group G.

Assume (i) and fix λ_{0}. Then there exist G-matrices S_{λ} such that α_{λ_{0}}^{S_{λ}} = α_{λ} and S_{λ_{0}} = E = the identity matrix. Denote Ṡ_{λ_{0}} = ∂/∂λ|_{λ=λ_{0}} S_{λ}, which takes values in 𝔤, the tangent space to G at S_{λ_{0}} = E, the unit element. Then we have

0 = \frac{∂}{∂λ}|_{λ=λ_{0}}α_{λ_{0}} = \frac{∂}{∂λ}|_{λ=λ_{0}}α_{λ}^{S_{λ}^{−1}} = \frac{∂}{∂λ}|_{λ=λ_{0}}(−S_{λ}^{−1}d̄S_{λ} + S_{λ}^{−1}α_{λ}S_{λ})
= −d̄Ṡ_{λ_{0}} − Ṡ_{λ_{0}}α_{λ_{0}} + α_{λ_{0}} Ṡ_{λ_{0}} + α̇_{λ_{0}},

i.e., α̇_{λ_{0}} = ¯∂_{α_{λ_{0}}} Ṡ_{λ_{0}} for Ṡ_{λ_{0}} ∈ Λ̄^{0}E ⊗ 𝔤.

Conversely, let α̇_{λ} = ¯∂_{α_{λ}}Q_{λ} for some Q_{λ} ∈ C^{∞}E ⊗ 𝔤. Let S_{λ} be a G-matrix solution of the equation ∂S_{λ}/∂λ = Q_{λ}S_{λ} satisfying the initial condition S_{λ_{0}} = E. Considering the expression Z_{λ} = d̄S_{λ} + S_{λ}α_{λ_{0}} − α_{λ}S_{λ} = (α_{λ_{0}}^{S_{λ}} − α_{λ})S_{λ}, we have

\frac{∂}{∂λ}Z_{λ} = \frac{∂}{∂λ}(d̄S_{λ} + S_{λ}α_{λ_{0}} − α_{λ}S_{λ})
= (d̄Q_{λ} − [α_{λ}, Q_{λ}] − α̇_{λ})S_{λ} + Q_{λ}(d̄S_{λ} + S_{λ}α_{λ_{0}} − α_{λ}S_{λ})
= Q_{λ}Z_{λ},

while obviously Z_{λ_{0}} = 0. Therefore, Z_{λ} = 0 for all λ, which proves the assertion. □

Given a family α_{λ} of ZCR’s, the following procedure yields the parameter-removing matrix S_{λ}^{−1} whenever it exists:

(1) Compute the elements α̇_{λ} = ∂α_{λ}/∂λ.

(2) If α̇_{λ} are coboundaries, find 𝔤-matrices Q_{λ} such that α̇_{λ} = ¯∂_{α_{λ}}Q_{λ}.

(3) Solve the equation ∂S_{λ}/∂λ = Q_{λ}S_{λ} for a G-matrix S_{λ} under the initial condition S_{λ} = E.

For methods to perform the third step see, e.g., [7]. Concerning the second step, we shall show in the next section that H_{α}^{1} is effectively computable in principle. But before that we present an example.

Example 4. Sasaki [19] was probably the first to observe that equations possessing an 𝔰𝔩_{2}-valued ZCR also describe pseudospherical surfaces, and suggested a procedure to derive infinite families of conservation laws thereof, assuming dependence of the ZCR on a parameter. A similar study has been independently initiated by Chern and Tenenblat [5] and continued in a series of papers, see the references in [21]. Needless to say, no infinite series of conservation laws will be generated in the case of removable parameter. Here and in the appendix below we complete the results of [4, 5] by showing that exactly this happens in the example of the Burgers equation.

Writing the Burgers equation as u_{t} = u_{xx} + uu_{x}, authors of [4] consider the ZCR

(9)  \begin{matrix}α_{η} = A_{η} dx + B_{η} dt = (\begin{matrix}\frac{1}{2}η & \frac{1}{4}u + \frac{1}{2}η \\ \frac{1}{4}u − \frac{1}{2}η & −\frac{1}{2}η \end{matrix}) dx \\ +(\begin{matrix}\frac{1}{4}ηu & \frac{1}{4}u_{x} + \frac{1}{8}u^{2} + \frac{1}{4}ηu \\ \frac{1}{4}u_{x} + \frac{1}{8}u^{2} − \frac{1}{4}ηu & −\frac{1}{4}ηu \end{matrix}) dt. \end{matrix}

For η = 0 the ZCR reduces to the conservation law u dx+(u_{x} + \frac{1}{2}u^{2}) dt. We shall perform the above three-step procedure to show that for all η ≠ 0 the forms α_{η} are mutually gauge equivalent.

By differentiation we obtain

α̇_{η} = ( −\frac{1}{2} −\frac{1}{2}^{\frac{1}{2} \frac{1}{2}}) dx + ( \begin{matrix}\frac{1}{4}u & \frac{1}{4}u \\ −\frac{1}{4}u & −\frac{1}{4}u \end{matrix}) dt.

For all η ≠ 0 we solve α̇ = ¯∂_{α}Q = d̄Q − αQ + Qα as

Q_{η} = ( \begin{matrix}0 & −\frac{1}{2}η^{−1} \\ −\frac{1}{2}η^{−1} & 0 \end{matrix})

to show that each α̇_{η} is a coboundary. For η > 0, the equation Ṡ_{η} = Q_{η}S_{η} has a solution

S_{η} = (\begin{matrix}\frac{1}{2}η^{−1/2} + \frac{1}{2}η^{1/2} & \frac{1}{2}η^{−1/2} − \frac{1}{2}η^{1/2} \\ \frac{1}{2}η^{−1/2} − \frac{1}{2}η^{1/2} & \frac{1}{2}η^{−1/2} + \frac{1}{2}η^{1/2} \end{matrix})

in SL_{2} satisfying the initial condition S_{1} = E. Using S_{η}^{−1} as the gauge matrix, one transforms α_{η} to α_{1}, thus removing the parameter η. For η < 0 we similarly obtain a gauge equivalence between α_{η} and α_{−1}. But α_{−1} is transformable to α_{1} via the gauge matrix ({0 \atop 1} {1 \atop 0}).

This example is continued in the Appendix.

3. Computing the horizontal gauge cohomology

In the sequel we assume that the coefficient field is 𝔨 = ℝ or ℂ. Special attention is paid to Lie algebras 𝔤𝔩_{n} and 𝔰𝔩_{n}. Since 𝔤𝔩_{n} = 𝔰𝔩_{n}⊕𝔨, we have

H̄_{α}^{q}(E, 𝔤𝔩_{n}) = H̄_{α}^{q}(E, 𝔰𝔩_{n}) ⊕ H̄^{q}E

by Proposition 1. Here H̄^{q} stands for the usual (𝔨-valued) horizontal cohomology.

In the case of q = 0 we have simply H̄_{α}^{0}(E, 𝔤) = Ker ¯∂_{α}. However, ¯∂_{α}Q = d̄Q−[α, Q] = ∑_{i}(D_{i}Q−[A_{i}, Q]) dx^{i} = D̂_{i}Q dx^{i} for any 𝔤-matrix Q, and therefore the group H̄_{α}^{0}(E, 𝔤) may be easily computed from the overdetermined system of linear equations in total derivatives

(10)  D̂_{i}Q = 0.

E.g., for trivial α we have H̄_{α}^{0}(E, 𝔤) ≅ H̄_{0}^{0}(E, 𝔤) = 𝔤 ⊗ Ker d̄, where Ker d̄≅ 𝔨 unless the diffiety E has a trivial factor (see [15]).

Since each D̂_{i} is a 𝔨-linear differentiation, H̄_{α}^{0}(E, 𝔤𝔩_{n}) is a 𝔨-algebra. We have the following interpretation for its invertible elements:

Proposition 5. The set of all invertible elements in the 𝔨-algebra H̄_{α}^{0}(E, 𝔤𝔩_{n}) coincides with the stabilizer of α with respect to the gauge action.

Proof. Let Q ∈ C^{∞}E⊗𝔤𝔩_{n} be invertible. Then d̄Q = [α, Q] = α·Q−Q·α if and only if α^{Q} = d̄Q · Q^{−1} + Q · α · Q^{−1} = α. □

Remark 6. Nontrivial examples of ZCR’s taken from the literature typically satisfy

(11)  H̄_{α}^{0}(E, 𝔤𝔩_{n}) ≅ 𝔨,  H̄_{α}^{0}(E, 𝔰𝔩_{n}) = 0.

Assuming that (11) holds, we propose the following way to determine the cohomology group H̄_{α}^{1}(E, 𝔰𝔩_{n}). In what follows G denotes the connected and simply connected Lie group associated to 𝔤. Denote by 𝔤^{Δ} the product 𝔤 × 𝔤 endowed with the bracket

[(A, A′), (B, B′)] = ([A, B], [A′, B] + [A, B′]).

Then 𝔤^{Δ} is a Lie algebra—in fact, a semidirect product of the subalgebra {A′ = 0} and the ideal {A = 0}. Denote by G^{Δ} the product G × 𝔤 endowed with the group multiplication

(A, A′) · (B, B′) = (A · B, A′ + Ad_{A} B′),

unit element being (1, 0), and the inverse

(A, A′)^{−1} = (A^{−1}, − Ad_{A^{−1}} A′).

Then 𝔤^{Δ} is the Lie algebra of G^{Δ}. Given matrix realizations 𝔤 ⊆ 𝔤𝔩_{n} and G ⊆ GL_{n}, we henceforth use the matrix realizations

𝔤^{Δ} = \{(\begin{matrix}B & 0 \\ B′ & B \end{matrix}) | B, B′ ∈ 𝔤 \} ⊂ 𝔤𝔩_{2n}

and

G^{Δ} = \{(\begin{matrix}A & 0 \\ A′A & A \end{matrix}) | A ∈ G, A′ ∈ 𝔤 \} ⊂ GL_{2m}

for 𝔤^{Δ} and G^{Δ}.

The gauge action of a G^{Δ}-matrix Ã = (A, A′) on a 𝔤^{Δ}-valued horizontal q-form β̃ = (β, β′) ∈ Λ̄^{q} ⊗ 𝔤^{Δ} is given by

(12)  (β, β′)^{(A,A′)} = (β^{A}, d̄A′ + [A′, β^{A}] + Ad_{A} β′).

Finally, we introduce the mapping

L_{α}: Λ̄^{1}E ⊗ 𝔤 → Λ̄^{1}E ⊗ 𝔤^{Δ}, L_{α}β = (α, β).

Proposition 7. (i) A form β ∈ Λ̄^{1}E ⊗𝔤 is a ¯∂_{α}-cocycle if and only

if L_{α}β is a 𝔤^{Δ}-valued ZCR. (ii) Assuming (11), cocycles β_{1}, β_{2} are cohomological if and only if

L_{α}β_{1}, L_{α}β_{2} are gauge equivalent with respect to the group G^{Δ}.

Proof. Statement (i) immediately follows from the equality

d̄L_{α}β − L_{α}β · L_{α}β = (d̄α − α · α, ¯∂_{α}β).

(ii) Let β_{2} = β_{1} +d̄Q−[α, Q] for some Q ∈ C^{∞}E ⊗𝔤. Upon choosing S = (E, Q) ∈ G^{Δ}, formula (12) yields (L_{α}β_{1})^{S} = L_{α}β_{2}.

Conversely, let S = (P, Q) ∈ G^{Δ} exists such that (L_{α}β_{1})^{S} = L_{α}β_{2}, i.e.,

(13)  (α, β_{2}) = (α^{P} , ¯∂_{α^{P}} Q + Ad_{P} β_{1})

by eq. (12). We see that P stabilizes α, hence P ∈ H̄_{α}^{0}(E, 𝔤) by Proposition 5. According to assumption (11) then P = pE for some constant p ∈ 𝔨, so that eq. (13) simplifies to (α, β_{2}) = (α, ¯∂_{α}Q + β_{1}) and β_{1}, β_{2} are cohomological. □

4. Normal forms

To convert Proposition 7 into an effective computational procedure, one further needs to develop algebraic classification of normal forms of characteristic elements in 𝔤^{Δ} in the spirit of [17, Prop. 3]. For 𝔤 = 𝔰𝔩_{2} this is done below.

We assume that m = 2 and α = A dx + B dy. The procedure to compute ZCR’s of non-overdetermined systems (1) is as follows. Given eq. (1) and a Lie algebra 𝔤, solve the determining system

(14)  0 = D_{y}Ā − D_{x}B̄ + [Ā, B̄],
(15)  0 = ∑_{J,l}(−1)^{|J|}D̂_{J} (\frac{∂F^{l}}{∂u_{J}^{k}} χ̄_{l})

for unknowns χ̄, Ā, B̄ for χ̄, Ā, B̄ running through possible normal forms for triples (χ, A, B), χ ≠ 0, with respect to the group action

(χ, A, B) ↦ (SχS^{−1}, D_{x}SS^{−1} + SAS^{−1}, D_{y}SS^{−1} + SBS^{−1}).

The unknowns Ā, B̄ enter eqs. (14)–(15) via the coefficients of D̂_{x}, D̂_{y}; components of χ̄ are auxiliary unknowns. In general position we have (n + 1) · dim 𝔤 equations for the same number of unknowns. Possible normal forms χ̄, Ā, B̄ depend only on the algebra 𝔤.

We consider the simplest case of 𝔰𝔩_{2} here, with

A = (\begin{matrix}a_{11} & a_{12} \\ a_{21} & −a_{11} \end{matrix}), B = (\begin{matrix}b_{11} & b_{12} \\ b_{21} & −b_{11} \end{matrix}).

A ZCR may degenerate to a nonlocal conservation law. E.g., if a_{12}b_{21} = a_{21}b_{12}, then a_{11} dx + b_{11} dy is a conservation law; denoting h its potential, one easily checks that e^{−2h}(a_{12} dx + b_{12} dy) and e^{2h}(a_{21} dx + b_{21} dy) are two linearly dependent nonlocal conservation laws. Likewise, α degenerates (is gauge equivalent to the one that satisfies a_{12}b_{21} = a_{21}b_{12}) whenever at least one of the off-diagonal coefficients a_{12}, a_{21}, b_{12}, b_{21} is zero (see [17]).

Let C = χ_{1} or any other χ_{i} that is nonzero (if all are zero, then α is trivial). Assuming α non-degenerate, by the main result of [17] we have exactly two possible normal forms for the matrices C, A:

– Nilpotent case

(16)  C = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}), A = (\begin{matrix}0 & a_{12} \\ a_{21} & 0 \end{matrix}).

– Diagonal case

(17)  C = (\begin{matrix}r & 0 \\ 0 & −r \end{matrix}), A = (\begin{matrix}a_{11} & a_{12} \\ 1 & −a_{11} \end{matrix}).

In general, no further reduction (of B or any other χ_{i}) is possible.

With Proposition 7 in mind, let us consider 𝔰𝔩_{2}^{Δ}-matrices of the block form

C̃ = (\begin{matrix}C & 0 \\ C′ & C \end{matrix}),  Ã = (\begin{matrix}A & 0 \\ A′ & A \end{matrix}),

where C, A were introduced above and C′, A′ are arbitrary 𝔰𝔩_{2}-matrices.

Proposition 8. Assuming α nontrivial and non-degenerate, possible normal forms for the matrices C̃, Ã are

– Nilpotent case

(18)  C̃ = (\begin{matrix}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & s & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}) , Ã = (\begin{matrix}0 & a_{12} & 0 & 0 \\ a_{21} & 0 & 0 & 0 \\ 0 & a_{32} & 0 & a_{12} \\ a_{41} & 0 & a_{21} & 0 \end{matrix}) .

– Diagonal case

(19) C̃ = (\begin{matrix}r & 0 & 0 & 0 \\ 0 & −r & 0 & 0 \\ s & 0 & r & 0 \\ 0 & −s & 0 & −r \end{matrix}) , Ã = (\begin{matrix}a_{11} & a_{12} & 0 & 0 \\ 1 & −a_{11} & 0 & 0 \\ a_{31} & a_{32} & a_{11} & a_{12} \\ 0 & −a_{31} & 1 & −a_{11} \end{matrix}) .

Proof. Provided r ≠ 0 and a_{12} ≠ 0, both covered by our assumptions, it is easy to find a matrix S̃ = (E, S′) ∈ SL_{2}^{Δ} that transforms C̃ to its normal form by conjugation and then a matrix T̃ = (E, T′) ∈ SL_{2}^{Δ} that transforms Ã to its normal form by gauge equivalence while preserving the normal form of C̃. Details are omitted. □

To compute α̃, one substitutes the normal forms of Proposition 8 in the determining system (14)eq, (15).

Remark 9. The method using the system (14), (15) does not guarantee non-triviality of results. For instance, in the nilpotent case one always obtains a trivial solution

(\begin{matrix}0 & a_{12} \\ −a_{21} & 0 \end{matrix}) dx + (\begin{matrix}0 & b_{12} \\ −b_{21} & 0 \end{matrix}) dy

which coincides with

d̄Q − [α, Q] with Q = (\begin{matrix}\frac{1}{2} & 0 \\ 0 & −\frac{1}{2} \end{matrix}),

and should be always excluded from the solution set.

5. Example

Example 10. For the mKdV equation u_{t} +u_{xxx} −6u^{2}u_{x} = 0, the ZCR α = A(λ) dx + B(λ) dt was found in [2]; we write it here in the normal form

A(λ) = (\begin{matrix}u & λ \\ 1 & −u \end{matrix}),
B(λ) = (\begin{matrix}−u_{xx} + 2u^{3} − 4λu & 2λu_{x} + 2λu^{2} − 4λ^{2} \\ −2u_{x} + 2u^{2} − 4λ & u_{xx} − 2u^{3} + 4λu \end{matrix}).

Then A_{t} − B_{x} + [A, B] = (u_{t} + u_{xxx} − 6u^{2}u_{x}) · C with C = ({1 \atop 0} {0 \atop −1}) and hence C is the characteristic matrix for this ZCR by formula (5). Let us compute H̄_{α}^{1} for λ = 1, i.e., with

A = A(1) = (\begin{matrix}u & 1 \\ 1 & −u \end{matrix}),
B = B(1) = (\begin{matrix}−u_{xx} + 2u^{3} − 4u & 2u_{x} + 2u^{2} − 4 \\ −2u_{x} + 2u^{2} − 4 & u_{xx} − 2u^{3} + 4u \end{matrix}).

As C, A are already in normal form (17), the next step is to write conditions (2) and (4) for Ã, C̃ in normal form (19). We obtain the following system of six linear equations in total derivatives for the six unknowns a_{31}, a_{32}, b_{31}, b_{32}, b_{41}, s:

0 = −D_{t}a_{31} + 2(u_{x} − u^{2} + 2)a_{32} + D_{x}b_{31} + b_{32} − b_{41},
0 = −4(u_{x} + u^{2} − 2)a_{31} − (D_{t} + 2u_{xx} − 4u^{3} + 8u)a_{32} + 2b_{31}
+ (D_{x} − 2u)b_{32},
0 = 4(u^{2} − u_{x} − 2)a_{31} − 2b_{31} + (D_{x} + 2u)b_{41},
− [D_{t} + D_{xxx} + 6(u^{2} − 2)D_{x}]s − 6D_{x}a_{32},
0 = −6(D_{xx} − 2uD_{x})s + 4(D_{x} − 4u)a_{31}
− 2(D_{xx} − 4uD_{x} − 2u_{x} − 2u^{2} + 8)a_{32} − 2b_{32},
0 = 6(D_{xx} + 2uD_{x})s + 4(D_{x} + 4u)a_{31} + 8a_{32} + 2b_{41}.

Solution vectors depending on second-order jet coordinates at most are multiples of (a_{31}, a_{32}, b_{31}, b_{32}, b_{41}) = (0, \frac{1}{2}, −2u, u_{x} + u^{2} − 4, −2), while s = const. This solution obviously satisfies A′ = ∂A/∂λ|_{λ=1}, B′ = ∂B/∂λ|_{λ=1}, hence corresponds to the nonremovable parameter λ in α.

Appendix

Let us demonstrate that the series of nonlocal conservation laws of the Burgers equation as generated in [21, Ch. I, Example 7.10a] is actually finite. First we recall briefly the procedure [21, Ch. I, Prop. 7.5].

For any ZCR

(\begin{matrix}\frac{1}{2}a_{2} & \frac{1}{2}a_{1} − \frac{1}{2}a_{3} \\ \frac{1}{2}a_{1} + \frac{1}{2}a_{3} & −\frac{1}{2}a_{2} \end{matrix}) dx + (\begin{matrix}\frac{1}{2}b_{2} & \frac{1}{2}b_{1} − \frac{1}{2}b_{3} \\ \frac{1}{2}b_{1} + \frac{1}{2}b_{3} & −\frac{1}{2}b_{2} \end{matrix}) dt,

the equations

(20)

ϕ_{x} = a_{1} sin ϕ + a_{2} cos ϕ + a_{3},
ϕ_{t} = b_{1} sin ϕ + b_{2} cos ϕ + b_{3}

are compatible and the 1-form

ω = (a_{1} cos ϕ − a_{2} sin ϕ) dx + (b_{1} cos ϕ − b_{2} sin ϕ) dt

is d̄-closed and hence represents a nonlocal conservation law. Considering a_{i}, b_{i} to be analytic functions of the parameter η, one may expand ϕ into a power series ϕ = ∑ ϕ_{i}η^{i}; and likewise ω = ∑ ω_{i}η^{i}. Coefficients ω_{i} then form an infinite series of conservation laws, although trivial if the parameter η is removable.

For the Burgers equation we have a_{1} = \frac{1}{2}u, a_{2} = η, a_{3} = −η, b_{1} = \frac{1}{2}u_{x} + \frac{1}{4}u^{2}, b_{2} = \frac{1}{2}ηu, b_{3} = −\frac{1}{2}ηu. Upon introducing two nonlocal variables v = ∫ u dx, z = ∫ e^{v/2} dx equations (20) on ϕ are easily solved as

ϕ = 2 arctan \frac{e^{v/2}}{ηz + 1}.

The variables v, z are potentials of two nontrivial and independent conservation laws

γ_{1} = u dx + (u_{x} + \frac{1}{2}u^{2}) dt, γ_{2} = e^{v/2}(dx + \frac{1}{2} dt).

The Taylor expansion of ϕ around η = 0 is ϕ = ∑_{i} ϕ_{i}η^{i} = 2 arctan e^{v/2}− 2e^{v/2}(e^{v} +1)^{−1}zη+4e^{v/2}(e^{v} +1)^{−2}z^{2}η^{2}+4e^{v/2}(e^{v} −3)(e^{v} +1)^{−3}z^{3}η^{3}+· · ·. Coefficients ϕ_{i} coincide with those found in [21] modulo integration constants. But one readily observes that ω is a coboundary, namely ω = d̄μ for

μ = \frac{1}{2}v − ln(η^{2}z^{2} + 2ηz + 1 + e^{v}).

Since ω is exact, so is each of its coefficients: ω_{i} = d̄μ_{i}, where μ = ∑_{i} μ_{i}η^{i}. Thus, the only nontrivial conservation laws of the Burgers equation that are generated by the procedure are the two γ’s above. In particular, the potential h introduced in [21] is related to v as h = \frac{1}{2}v − ln(e^{v} + 1).

Acknowledgements

Above all, I would like to thank I.S. Krasil’shchik for his influence on the development on these ideas. Thanks are also due to Artur Sergyeyev for suggestions on improving readability of the manuscript.

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E-mail address: Michal.Marvan@math.slu.cz

Mathematical Institute, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic