与高亏格曲线相关的无限维延拓李代数和多分量Landau-Lifshitz系统
Infinite-dimensional prolongation Lie algebras and multicomponent Landau-Lifshitz systems associated with higher genus curves
摘要：
Wahlquist-Estabrook延拓方法为某些偏微分方程构造了一个李代数，它负责某些类型的拉克斯对和Backlund变换。给出了（1+1）维发展偏微分方程的Wahlquist-Estabrook代数的一些一般性质，并计算了Golubchik和Sokolov的n-分量Landau-Lifshitz系统对任意$n\ge 3$的Wahlquist-Estabrook代数.我们证明了所得到的代数同构于亏格为$1+(n-3)2^{n-2}$的代数曲线上的2维交换李代数与某些矩阵值函数的无限维李代数L（n）的直和.这条曲线被Golubchik，Sokolov，Skrypnyk，Holod在拉克斯对的建设。此外，我们找到了代数L（n）在有限数量的生成元和关系方面的一个表示。这些结果有助于获得一个部分答案的问题的分类多组分Landau-Lifshitz系统的Backlund变换。进一步，我们构造了一族可积发展偏微分方程，它们通过上述曲线参数化的Miura型变换与n分量Landau-Lifshitz系统相联系.这些偏微分方程的一些解决方案的描述。
英文摘要：
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 $n\ge 3$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
