两种三角曲线在可积系统中的应用

In the study of the integrable system, the funding of the solutions to the soliton equations is an important research area, especially to find the algebro geometric solutions of them. This kind of solutions can reveal the internal structure of the solutions and describe the quasi-periodic behavior of nonlinear phenomena. Algebro geometric method offers an effective approach to seek the algebro geometric solutions. At present, there are many results on the research of the matrix spectral problems associated with the hyperelliptic curves, and a few on the soliton equations related to the cyclic trigonal curves, but, the research on the other cases is very few. In the project, we will study some hierarchies of soliton equations which associated with the trigonal curves that have two infinite points or have the complete form by means of the algebro geometric method, proposed by Gesztesy and progressed by professor Xianguo Geng etc.,give their algebro geometric solutions, and discover the properties of the solutions. Firt, the trigonal curve is defined with the help of the characteristic polynomial for the stationary equations, from which the associated Baker function and meromorphic function are given. Then, we construct Abel differentials. By using the theory of trigonal curves and Riemann surface, we present the explicit Theta function representations of the Baker function, the meromorphic function, and in particular, that of the potentials for the entire hierarchy. Researches in the project are important for completing the theory of soliton equations associated with three order differential operators in integrable system and the trigonal curve theory in algebra geometry.

在可积系统的研究中，寻找孤子方程的解是一个重要的研究领域，尤其是寻找其代数几何解，这类解可以揭示解的内在结构，描述非线性现象的拟周期行为。代数几何方法提供了有效的研究途径，目前该方法在研究与超椭圆曲线相联系的矩阵谱问题时结果很多，也有少量与循环三角曲线相关的孤子方程解的研究，但对其他类型的三角曲线的研究结果还很少。本项目拟采用由Gesztesy提出、耿献国教授等人改进的代数几何方法来求解与具有两个无穷远点和包含完整曲线形式的三角曲线相联系的孤子方程族，给出它们的代数几何解并研究其性质。首先借助定态方程的特征多项式定义三角曲线，引入Baker函数和亚纯函数；然后构造Abel微分，结合三角曲线和Riemann面的相关理论给出Baker函数和亚纯函数，尤其是整个方程族的解的显式Theta函数表示。本项目的研究有助于完善可积系统中与三阶微分算子相关的孤子方程理论和代数几何中的三角曲线理论。

在可积系统的研究中，寻找孤子方程的解是一个重要的研究领域，尤其是寻找其代数几何解。本项目采用改进的代数几何方法，利用三角曲线理论及 Baker-Akhiezer 函数和亚纯函数的渐近性质，深入研究了与具有两个（或一个）无穷远点的特殊三角曲线和包含完整曲线形式的三角曲线相联系的孤子方程族，主要包括 MNW 方程族，MSS 方程族，HS 方程族，约束 KP 方程族，给出了它们的代数几何解或可积性质。本项目的研究完善了可积系统中与三阶微分算子相关的孤子方程理论和代数几何中的三角曲线理论。