Lie algebras responsible for zero-curvature representations of scalar evolution equations

Abstract

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs.For any (1+1)-dimensional scalar evolution equation E, we define a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations.As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a Bäcklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for Bäcklund transformations are presented in other publications as well.In our approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation E. The algebras F(E) generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.In this paper we describe general properties of F(E) and present generators and relations for these algebras. In other publications we study the structure of F(E) for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt, Sawada-Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.2010 Mathematics Subject Classification. 37K30, 37K35.

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

“…Zero-curvature representations belong to the main tools in the theory of integrable PDEs (see, e.g., [31,3]). This preprint is a sequel of [7] and is part of a research program on investigating the structure of zero-curvature representations (ZCRs) for partial differential equations (PDEs) of various types.…”

“…The ZCR (7) is said to be gauge equivalent to the ZCR (3), (4). For a given equation (1), formulas (7) determine an action of the group of gauge transformations on the set of ZCRs of this equation.…”

“…Recall that the infinite prolongation E of equation (1) is an infinite-dimensional manifold with the coordinates x, t, u k for k ∈ Z ≥0 . The precise definition of the manifold E is given in [7].…”

“…For each p ∈ Z ≥0 and each a ∈ E, the paper [7] defines a Lie algebra F p (E, a) so that the following property holds. For every finite-dimensional Lie algebra g, on a neighborhood of a ∈ E, any g-valued ZCR (3), (4) of order ≤ p is locally gauge equivalent to the ZCR arising from a homomorphism F p (E, a) → g. (We suppose that the g-valued functions (3) are defined on a neighborhood of a ∈ E.)…”

“…The algebra F p (E, a) is defined in [7] in terms of generators and relations, using a normal form for ZCRs with respect to the action of the group of local gauge transformations. The definition of F p (E, a) from [7] is recalled in Section 2 of the present preprint. (To clarify the main idea, in Example 3 below we consider the case p = 1.…”