代数曲线与孤子方程的有限亏格解

孤子方程的求解及解的研究是可积系统领域研究的重要组成部分，本项目基于代数曲线理论来研究孤子方程的有限亏格解，内容主要包括：（一）构造与二阶微分算子相联系的孤子方程的有限亏格解，并研究解的约化及应用；（二）研究与三阶微分算子相联系的孤子方程，利用驻定零曲率方程定义三叶紧致非奇异Riemann面，研究该Riemann面在一些有限点以及无穷远点处的性质，根据这些性质来选取局部坐标，在局部坐标下考察亚纯函数以及Baker-Akhiezer函数的因子和渐近展式，结合其theta函数表示构造出孤子方程的有限亏格解。上述研究在实际应用中具有重要的价值，同时在理论研究上也是对代数几何方法的进一步完善。

It is an important part to do research on solving soliton equations and the solutions in the area of integrable systems. Based on the theory of algebraic curve, the project studies finite genus solutions to soliton equations. The project includes two parts. Part one aims to construct finite genus solutions to soliton equations associated with the second order differential operators and study the reduction and application of the solutions. Part two devotes to do research on soliton equations associated with the third order differential operators. A three sheeted compact nonsingular Riemann surface can be defined by using the stationary zero-curvature equation. Investigating properties near some finite points and infinite points, the local coordinates can be decided accordingly. Under these local coordinates, we discuss the divisors and asymptotic expansions of meromorphic functions and Baker-Akhiezer function. Together with the theta functional representation of these functions, the finite genus solutions to soliton equations can be derived. The project not only possesses an important value in practical application, but also makes the algebro-geometric method more completed in theoretical study.