Infinite-dimensional prolongation Lie algebras and multicomponent Landau–Lifshitz systems associated with higher genus curves

Abstract

The Wahlquist–Estabrook prolongation method constructs for some PDEs a Lie algebra that is responsible for Lax pairs and Bäcklund transformations of certain type. We present some general properties of Wahlquist–Estabrook algebras for (1+1)-dimensional evolution PDEs and compute this algebra for the n-component Landau–Lifshitz system of Golubchik and Sokolov for any n≥3. We prove that the resulting algebra is isomorphic to the direct sum of a 2-dimensional abelian Lie algebra and an infinite-dimensional Lie algebra L(n) of certain matrix-valued functions on an algebraic curve of genus 1+(n−3)2n−2. This curve was used by Golubchik, Sokolov, Skrypnyk, and Holod in constructions of Lax pairs. Also, we find a presentation for the algebra L(n) in terms of a finite number of generators and relations. These results help to obtain a partial answer to the problem of classification of multicomponent Landau–Lifshitz systems with respect to Bäcklund transformations.Furthermore, we construct a family of integrable evolution PDEs that are connected with the n-component Landau–Lifshitz system by Miura type transformations parametrized by the above-mentioned curve. Some solutions of these PDEs are described.