与gl(3)相关的Lax矩阵产生的Lie-Poisson Hamilton系统的分离变量

Along with the deepening and development of soliton theory, a lot of integrable systems have been found. The finite dimensional integrable systems that generated from 3×3 Lax matrices that related to gl(3) are getting more and more attention. The project is mainly to investigate separation of variables for Lie-Poisson Hamiltonian systems generated from Lax matrices associated with gl(3). We would mainly be working on the following 3 topics: 1) With the help of Poisson structure and co-adjoint representation theory, the relationship between Lie-Poisson Hamiltonian systems and canonical Hamiltonian systems are studied. 2) in the framework of Lie-Poisson structure, separation of variables on the common level set of Casimir functions are constructed. 3) Based on the Hamilton-Jacobi theory, action-angle variables are introduced to establish the Jacobi inversion problems for the Lie-Poisson Hamiltonian systems and soliton equations. The accomplishment of this project is of great significance for richening and deepening integrable system theory. A new approach will be given for further studying integrable systems.

随着孤子理论的深入发展，发现了许多可积系统，与gl(3)相关的Lax矩阵产生的可积系统受到越来越多的关注。本项目主要探讨在Lie-Poisson结构下与 Lie 代数 gl(3) 相关的有限维可积系统的分离变量。具体将关注以下3个方面1) 研究 Lie-Poisson 结构下有限维可积系统与标准辛结构下有限维可积系统的关系。2) 在与孤子方程相关的Lie-Poisson Hamilton系统的Casimir函数的公共水平集上，构造可分离变量。3) 利用 Hamilton-Jacobi 理论给出作用-角变量并得到孤子方程的 Jacobi 反演问题。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，将为进一步研究可积系统提供新的途径。

在可积系统的研究中，针对与超椭圆曲线相关的可积系统的研究已有不少，但与非超椭圆曲线相关的可积系统研究却很少。本项目的研究目的就是在Lie-Poisson结构下，构造与非超椭圆曲线相关的可积系统的可分离变量及作用角变量。受本项目的资助完成并投稿3篇论文。主要研究成果如下：1)构造了与三波方程相关的Lie-Poisson Hamilton系统。2）在Casimir函数的公共水平集上，构造与三波方程相关的Lie-Poisson Hamilton 系统的分离变量。3）进一步利用守恒积分母函数的Hamilton–Jacobi 理论，得到Lie-Poisson Hamilton 系统的作用-角变量，从而得到三波方程的Jacobi反演问题。