Lie群和Lie代数方法在可积系统中的应用

项目名称： Lie群和Lie代数方法在可积系统中的应用

项目编号： No.11271337

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 杜殿楼

作者单位： 郑州大学

项目金额： 60万元

中文摘要： 随着孤立子理论的深入和发展，发现了为数众多的可积系统。如何用一个统一的框架来处理可积系统已成为一个重要的课题。本项目将以Lie群、Lie代数为工具来探讨这一问题。在二阶谱问题对应可积系统Lie-Poissn结构的基础上，本项目拟从以下几个方面开展工作：建立高阶谱问题对应可积系统的Lie-Poisson结构和Poisson几何理论；用Lie群和Lie代数表示理论来阐明对应于同一无限维可积系统的有限维约化可积系统之间的关系，并给出分类；构造保Poisson结构的典则变换寻找可分离的典则坐标，利用Hamilton-Jacobi理论给出作用-角变量；借助代数几何知识，通过Jacobi反演用Riemann-Theta函数表示原系统的解。本项目的实现对丰富和深化可积系统的理论具有重要意义，其结果将有助于可积系统理论的进一步研究和应用。

中文关键词： Poisson结构；可积系统；代数几何解；Lie群；Lie 代数

英文摘要： Along with the deepening and development of soliton theory, a lot of integrable systems have been found. How to use a unified framework to deal with integrable systems has become an important topic. The project is to apply Lie groups and Lie algebras to investigate this subject. Based on the Lie-Poisson structures of the integrable systems associated with second order spectral problems, in this project, we plan to carry out researches in the following aspects: To construct the Lie-Poisson structures and Poisson geometry theory for the integrable systems generated by higher order spectral problems. To discuss the relationship among the finite dimensional integrable systems reduced from one infinite dimensional integrable system by using the representation theory of Lie groups and Lie algebras, and to classify these finite dimensional integrable systems. To introduce a series of canonical transformations to get separable canonical coordinates, and to derive the action-angle variables by the Hamilton-Jacobi theory. To use the Jacobi inversions to obtain the Riemann-Theta function solutions of the original systems through the theory of algebraic g

英文关键词： Poisson structure；integrable system；algebro-geometric solution；Lie group；Lie algebra