本项目研究四方曲线的理论及其在可积系统中应用，探索求解与4×4矩阵谱问题相联系的孤子方程族拟周期解的理论框架。研究与4×4矩阵谱问题相联系的孤子方程族的Lax矩阵导出的四方曲线及其紧化产生的四叶Riemann面。讨论四叶Riemann面上的Baker–Akhiezer函数和亚纯函数的性质。引入广义椭圆变量并讨论它们满足的方程。构造相应四叶Riemann面上一组全纯微分基和Abel影射。研究三类Abel微分在无穷远点和零点的渐近式。基于亚纯函数的因子及其代数几何结构，构造亚纯函数和Baker–Akhiezer函数的Riemannθ函数表示。计算Baker-Akhiezer函数与亚纯函数及其Riemannθ函数表示在无穷远点的展式，进而得到与4×4矩阵谱问题相联系的孤子方程族解的Riemannθ函数表示。作为上述方法的应用将导出一批与4×4矩阵谱问题相联系孤子方程族的拟周期解。

The principal aim of this project is to study the theory of tetragonal curves and the application to integrable systems. We explore a theoretical framework to construct quasi-periodic solutions of soliton equations associated with 4×4 matrix spectral problems. It is studied that the Lax matrices associated with 4×4 matrix spectral problems lead to tetragonal curves and four-sheeted Riemann surfaces by their compactification. We discuss the properties of the Baker–Akhiezer functions and meromorphic functions on the four-sheeted Riemann surfaces. Then we introduce generalized elliptic variables and give that they satisfy the equations. A basis for holomorphic differentials and Abel maps are constructed on the four-sheeted Riemann surfaces. The asymptotic expansions for three kinds of Abel differentials are investigated near infinite points and zero ones. On basis of the divisors of meromorphic functions and their algebraic and geometric structures, we derive the Riemann theta function representation for meromorphic functions and Baker–Akhiezer functions. The Riemann theta function representations of solutions for soliton hierarchies associated with 4×4 matrix spectral problems are obtained by using the expansions of the Riemann theta function representations for the Baker–Akhiezer functions and meromorphic functions near infinite points. As applications, quasi-periodic solutions of some soliton hierarchies associated with 4×4 matrix spectral problems are derived.