混合Boussinesq方程的有限亏格解

Finite genus solutions to the mixed Boussinesq equation

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Km-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Riemannθ函数表示,特别地,给出了混合Boussinesq方程的有限亏格解.

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.