Lie群和Lie代数方法在可积系统中的应用

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 geometry. .The accomplishment of this project is of great significance for richening and deepening integrable system theory, and is to help us to further study integrable systems.

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。

可积系统的结构和可积系统之间的关系一直是孤立子理论的重要研究课题。本项目就此问题展开了研究，主要得到以下结果：. (1)解决了二阶谱问题对应有限维可积系统之间的关系问题;. (2)建立了高阶谱问题对应可积系统的Lie-Poisson结构和Poisson几何理论;. (3)构造出保Poisson结构的典则变换找到可分离的典则坐标，利用守恒积分母函数的Hamilton-Jacobi理论给出作用-角变量；. (4)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；. (5)利用Lie代数扩展理论引入相应Poisson结构，并由此构造新的有限维可积系统，得到了扩展的Gaudin模型、Garnier模型及Neumann模型，为获得新的有限维可积系统提供了新途径。. 这些结果对丰富和深化可积系统的理论具有重要意义，其方法将有助于可积系统理论的进一步研究和应用。