孤子方程中的代数曲线方法

We are mainly concerned with the method of algebraic curve in studying the quasi-periodic solutons for soliton equations. On one hand, with the help of Baker-Akhiezer function，meromorphic function and the theory of algebraic curve, we discuss the algebro-geometric constructions of the soliton equations associated with different kinds of 3rd order or higher order matrix problems, from which we develop an efficient way to construct the quasi-periodic solutons of the soliton equations associated with 3rd order or higher order matrix problems. On the other hand, based on the method of algebraic curve in the classical integrable system, we study the quasi-periodic solutons of the super soliton equations combined with the knowledge of algebraic curve in super integrable system.

本项目拟应用代数曲线方法研究孤子方程的拟周期解。一方面，考虑不同类型的三阶或高阶矩阵谱问题，利用Baker-Akhiezer函数、亚纯函数及代数曲线理论，讨论与其相联系的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矩阵谱问题相联系的孤子方程的拟周期。另一方面，参考经典可积系统中的代数曲线方法，结合超可积系统的代数曲线理论，讨论超孤子方程的拟周期解。

本项目利用三角曲线理论及Baker-Akhiezer函数、亚纯函数的渐近性质，深入细致地研究了几个与三阶矩阵谱问题相联系的孤子方程族的代数几何结构，并给出其Riemann-theta函数表示的拟周期解。最终，我们形成了一套系统有效的研究与三阶矩阵谱问题相联系的孤子方程族拟周期解的方法。在此基础上，探讨了超孤子方程族的代数曲线特征，为进一步讨论其拟周期解做了一些必要的准备。