混合Boussinesq 方程的有限亏格解

借助于Lenard 递推方程和定态零曲率方程, 我们给出与3 × 3 矩阵谱问题相联系的一族混合Boussinesq 方程. 利用Lax 矩阵的特征多项式, 引入一条三角曲线K_m-1, 由此构造出相应的Baker-Akhiezer 函数、亚纯函数和Dubrovin- 型方程. 混合Boussinesq 流在Abel 映射下被拉直. 基于三角曲线和三类Abel 微分的理论, 我们得到了Baker-Akhiezer 函数、亚纯函数的Riemann θ 函数表示, 特别地, 给出了混合Boussinesq 方程的有限亏格解.

Finite genus solutions to the mixed Boussinesq equation

The mixed Boussinesq hierarchy associated with a 3 × 3 matrix spectral problem is proposed in view of Lenard recursion equations and the stationary zero-curvature equation. A trigonal curve Km-1 is introduced with the help of the characteristic polynomial of the Lax matrix, from which we construct closely related Baker- Akhiezer function, the meromorphic function and Dubrovin-type equations. Moreover, the flows are straighten out under the Abel map. Based on the basic knowledge of the trigonal curves and three kinds of Abelian differential forms, the Riemann 0 function representations of the Baker-Akhiezer function, the meromorphic function, and in particular, that of finite genus solutions for the mixed Boussinesq equation are obtained.