本项目基于三叶和四叶Riemann面理论，利用Baker-Akhiezer函数来研究孤子方程的有限亏格解，内容主要包括：(一)深入研究三叶Riemann面的几何性质，借助Baker-Akhiezer函数的theta函数表示构造出与3×3矩阵谱问题相联系的孤子方程的有限亏格解；(二)通过对3×3矩阵谱问题的分析建立Baker-Akhiezer函数的Riemann-Hilbert(RH)问题，通过对RH问题的变形和求解，借助于RH问题解的渐近展式得到孤子方程的有限亏格解表示；(三)研究四叶Riemann面的基本性质，无穷远点的分支情况、三类Abel微分的结构形式、亏格的计算等，借助亚纯函数考察Baker-Akhiezer函数的因子和渐近展式，研究并发展出一套有效的方法来构造与4×4矩阵谱问题相联系的孤子方程的有限亏格解。上述研究具有重要的理论意义和应用价值。

Based on the theory of three and four sheeted Riemann surfaces, this project uses the Baker-Akhiezer function to study finite genus solutions to soliton equations. The project includes three parts. Part one aims to make an intensive study on the geometric property of three sheeted Riemann surface and construct finite genus solutions to soliton equations associated with 3×3 matrix spectral problems by virtue of the theta representation of Baker-Akhiezer function. Part two concentrates to establish the Riemann-Hilbert (RH) problem of Baker-Akhiezer function based on analysis of the matrix spectral problem. By deforming and solving the RH problem, finite genus solution to soliton equation can be obtained from asymptotic expansion of the solution for RH problem. Part three devotes to do research on the basic property of four sheeted Riemann surface including ramification conditions of infinite points, the structural forms of three kinds of Abel differentials, calculation of genus and so on. We investigate the divisor and asymptotic expansions of Baker-Akhiezer function by means of meromorphic functions and develop an effective way to construct finite genus solutions to soliton equations associated with 4×4 matrix spectral problems. The study above is of considerable theoretical significance and practical value.