与非超椭圆曲线相关的非线性演化方程拟周期解的研究

The quasi-periodic solutions of the soliton equations are a kind of very important solutions, which can not noly reveal the internal structure of the solutions, describe the quasi-periodic behavior of nonlinear phenomena, but also be reduced to soliton solutions, periodic solutions, elliptic function solutions and so on. Algebro-geometric method offers an effective approach to seek the quasi-periodic solutions. At present, there are many results about the 2×2 matrix spectral problems by using the algebro-geometric method, but, the research on the 3×3 cases is very few. In the project, we will study some well-known hierarchies of soliton equations which associate with 3×3 matrix spectral problems by means of the algebro-geometric method proposed by Gestezy and progressed by Xianguo Geng etc, and give their quasi-periodic solutions. First, the non-hyperelliptic curve is defined with the help of the characteristic polynomial of Lax matrix for the stationary equations, from which the associated Baker-Akhiezer function and meromorphic function are given. Then, we construct three kinds of Abel differentials. By using the theory of non-hyperelliptic curves and Riemann surface, we present the explicit Riemann Theta function representations of the Baker-Akhiezer 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 integrable system which associated with the three order differential operators and its application in physics.

孤子方程的拟周期解是一类非常重要的解，它不仅可以揭示解的内在结构，描述非线性现象的拟周期行为，还可以约化出孤子解、周期解、椭圆函数解等。代数几何方法提供了求拟周期解的有效途径，目前利用代数几何方法研究2×2矩阵谱问题的结果已有不少，但对3×3问题的研究还很少。本项目拟采用由Gestezy提出，耿献国等人发展的代数几何方法来求解与3×3矩阵谱问题相联系的几个著名的孤子方程族，给出它们的拟周期解。首先借助定态方程的Lax矩阵的特征多项式定义非超椭圆曲线，并引入相应的Baker-Akhiezer函数和亚纯函数；然后构造三类Abel微分，结合非超椭圆曲线和Riemann面的相关理论给出Baker-Akhiezer函数和亚纯函数，尤其是整个方程族的位势的显式Riemann Theta函数表示。本项目所研究的问题对于完善与三阶微分算子相关的可积系统的理论及其在物理等方面的应用具有重要意义。

目前，在可积系统的研究中，针对与二阶微分算子相关的孤子方程的拟周期解的研究已有不少结果，但求解与三阶微分算子相联系的孤子方程的代数几何解的研究却很少。本项目的研究目的就是借助于改进的代数几何方法，求解与三阶微分算子相联系的著名的孤子方程族，给出它们的代数几何拟周期解。受本项目资助共发表学术论文3篇，全部发表于SCI杂志上。研究成果分类如下：著名的孤子方程族的代数几何解的研究论文2篇，超可积系统的Hamilton结构和守恒率的研究论文1篇。本项目的代表性成果如下：（1）得到了著名的 three-wave resonant interaction 方程族的代数几何拟周期解，为求解代数曲线次数为3n 的方程族的代数几何解提供了很好的思路，而且为丰富相应的代数曲线理论做出了重要贡献；（2）构造了超可积耦合导数非线性 Schrodinger 方程族及其守恒率，丰富了超可积系统的家族，为进一步研究与三阶微分算子相关的超可积系统的代数几何解提供了很好的例子。