代数几何方法在离散可积系统中的应用

The theory of integrable systems is one of the important subjects in the research of mathematics and physics. In recent years, the discrete integrable systems attracted more and more interests. the construction of an discrete integrable system and their solutions have many importance in the integrable systems. We are mainly concerned with the method of algebraic-geometric in studying the quasi-periodic solutions for discrete soliton equations. With the help of Baker-Akhiezer functio and meromorphhic function, we discuss the algebro-geometric constructions of the soliton equations associated with 3rd order or higher order matrix problems, from which we develop an efficient way to construct the quasi-periodic solutions of the soliton equations associated with 3rd order or higher order discrete matrix problems.

可积系统理论是数学和物理中的重要研究领域。近年来，离散可积系统的研究得到越来越多的重视。离散可积方程的构造及求解在可积系统理论中有十分重要的意义。本项目拟应用代数几何方法研究离散孤子方程的拟周期解。考虑三阶或高阶离散矩阵谱问题，构造与其相联系的离散孤子方程，基于三角曲线理论，利用Baker-Akhiezer 函数和亚纯函数，讨论离散孤子方程的代数几何构造，由此发展一条有效的途径构造与三阶或高阶离散矩阵谱问题相联系的孤子方程的拟周期解。

代数几何方法在可积系统尤其是在离散可积系统中的应用和研究是一项难度大很有意义的研究。具有二阶矩阵谱问题的可积系统的代数几何解构造在过去的半个世纪里已获得成功，但三阶及以上矩阵谱问题的可积系统的代数几何构造进展甚慢，一直是学术界的关注热点。本项目研究了几个三阶矩阵谱问题的离散孤立子方程族的代数几何解的构造，研究涉及到三次曲线理论和高阶的Baker-Ａkhierzer函数构造，具有相当的难度和创新。