离散可积系统

Discrete integrable system is an important part in the theory of soliton and becomes one of the main concern of the scholars of various countries in recent years. The contents of the present project will be concentrated in the following aspects. First, for some new appeared discrete integrable equations, the finite genus solutions will be calculated by the nonlinearization procedure and the theory of algebraic curve. The elementary solutions, if possible, will be studied with the help of integrable symplectic maps as well. Second, new discrete integrable systems and their solutions will be constructed based on the discretization of given continuous spectral problems. Third, we will try to extend the methods to the high dimensional integrable models.

离散可积系统是孤立子理论的主要组成部分，近年来倍受各国学者的关注。本项目的研究内容将集中在以下方面。其一，对于一些新出现的离散可积方程，运用非线性化手续和代数曲线理论计算其有限亏格解，如果可能的话，将借助可积辛映射研究其初等解。其二，基于给定的连续谱问题的离散化，构造并求解新的离散可积系统。其三，尝试将上述方法推广至高维可积模型。

近年来，离散可积系统逐渐成为活跃的分支领域。该项目主要运用Riemann面方法研究离散可积方程。受本项目资助发表论文1篇，国际会议收录1篇，完成并投稿3篇。首先，给出了离散sine-Gordon方程的初等显式解。此外，在研究离散KdV型方程时，发现不同的Liouville可积平台可通向相同的离散可积模型。最后，计算出离散pKP、离散pMKP方程的连续极限和有限亏格解，并成功求解Q1模型。为进一步研究其他情形提供了有价值的参考。