可积差分方程的有限亏格解

Finite genus solutions of integrable difference equations is an important part of the theory of discrete integrable systems. The current project will concentrate on the following aspects. First, new discrete spectral problems are constructed, by which integrable symplectic maps are got through a nonlinearization procedure. Second, resorting to these maps and the commutativity of the discrete phase flows sharing the same Liouville platform, some integrable difference equations are derived. Third, finite genus solutions are obtained with the help of the theory of algebraic curve and Baker functions.

可积差分方程的有限亏格解是离散可积系统理论的重要组成部分，本项目将对其展开研究。首先，构造新的离散谱问题，经非线性化得到可积辛映射。其次，运用这些映射和同一个Liouville可积平台上之离散相流的可换性，推导出可积差分方程。最后，借助代数曲线理论和Baker函数给出方程的有限亏格解。

可积差分方程可描述统计物理、等离子体物理、非线性光学等物理学领域中大量的离散现象，近年来倍受各国学者的关注。本项目的研究目的是构造新的离散谱问题，运用可积辛映射和Riemann面方法，推导并求解有意义的可积差分方程。受本项目资助完成并投稿3篇，其中一篇给出离散sine-Gordon方程的初等解，其余两篇分别计算出离散Schwarzian KdV方程（Q0模型）和离散pMKdV方程（特殊H3模型）的有限亏格解，为进一步研究其他情形提供了有价值的参考。