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

TL;DR

This paper explores the structure of Lie algebras associated with multicomponent Landau-Lifshitz systems, revealing their connection to higher genus algebraic curves and providing new insights into their integrability and classification.

Contribution

It computes the Wahlquist-Estabrook algebra for n-component Landau-Lifshitz systems, showing it is a sum of a 2D abelian algebra and an infinite-dimensional algebra linked to a higher genus curve, and presents a finite generator presentation.

Findings

The algebra is isomorphic to a direct sum involving a higher genus curve.

A finite presentation for the algebra L(n) is provided.

Constructs a family of integrable PDEs related by Miura transformations.

Abstract

The Wahlquist-Estabrook prolongation method constructs for some PDEs a Lie algebra that is responsible for Lax pairs and Backlund 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 $n\ge 3$. 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 $1+(n-3)2^{n-2}$. This curve was used by Golubchik, Sokolov, Skrypnyk, 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 Backlund 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.