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

JOURNAL OF GEOMETRY AND PHYSICS

卷:68

Article

Igonin, Sergey¹  van de Leur, Johan¹  Manno, Gianni²  Trushkov, Vladimir³ 

[1] Univ Utrecht, Dept Math, NL-3508 TA Utrecht, Netherlands

[2] Univ Jena, Fak Math & Informat, Math Inst, D-07737 Jena, Germany

[3] Univ Pereslavl, Pereslavl Zalesskii 152020, Yaroslavl Regio, Russia

关键词: Wahlquist-Estabrook prolongation;    Infinite-dimensional Lie algebras;    Backlund transformations;    Multicomponent Landau-Lifshitz equations;    Algebraic curves;   

DOI  :  10.1016/j.geomphys.2013.02.002

来源: Elsevier

【 摘 要 】

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 >= 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 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. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free