2012/09/30 by Sergey Igonin, S. Igonin, Johan van de Leur +5
Mathematics · Physics and Astronomy · #Abelian group #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Genus #Integrable system #Lie algebra #Lie conformal algebra #Mathematics #Nonlinear Waves and Solitons #Pure mathematics #Type (biology) #math-ph #math.MP #math.RA #msc:37K30 #msc:37K35 #nlin.SI
paper · pdf · doi:10.1016/j.geomphys.2013.02.002
published as J. Geom. Phys. 68 (2013), 1--26 · 30 pages; v2: minor changes
arxiv created 2012/10/11 · openalex publication_date 2013/02/13 · arxiv updated 2015/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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, 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.