Trigonal Curves And Algebro-geometric Solutions To Soliton Equations

It is an important research topic that solving partial differential equations in the meaning of theory and practical application. Especially the explicit solutions provide a powerful tool to study the variety of properties of the equations. Moreover, it is very meaningful to find the algebro-geometric solutions of the soliton equaitons. They can not only reveal the internal structure of the solutions, describe the quasi-periodic behavior of nonlinear phenomena, but also be reduced to soliton solutions, periodic solutions, elliptic function solutions and so on. In the theory of intcgrable systems, algebro-geometric method offers an approach to seek the periodic solutions and quasi-periodic solutions, which can be explicitly given byθfunctions on the Riemann surface.In this paper, we mainly study some well-known hierarchies of soliton equa-tions which associate with 3×3 matrix spectral problems by means of the algebro-geometric method, and give their algebro-geometric solutions. The hierarchies as-sociated with the 3×3 matrix spectral problems we consider here are the Hirota-Satsuma modified Boussinesq hierarchy, the Mikhailov-Shabat-Sokolov hierarchy, the Sawada-Kotcra hierarchy and the mixed Boussinesq hierarchy.First, we introduce the Lenard recursion equations, from which the hierarchy that associated with the 3×3 matrix spectral problem is constructed in view of the zero-curvature equation. Then, a trigonal curveκm-1 with degree m and arith-metic genus m-1 is brought in with the help of the characteristic polynomial of Lax matrix for the stationary equations, from which the stationary Baker-Akhiezer function and the associated meromorphic function arc given. The compactification of the curveκm-1 becomes a three-sheeted Riemann surface. With the introduction of the elliptic coordinates, the stationary equations arc decomposed into the system of Dubrovin-type ordinary differential equations. Next, we construct three kinds of Abel differentials. By studying the asymptotic properties of the three kinds of Abel differentials, Baker-Akhiezer functions and the meromorphic functions, we present the explicit Riemannθfunction representations of the stationary Baker-Akhiezer function, the meromorphic function, and in particular, that of the potentials for the entire stationary hierarchy. Finally, we extend all the analyses to the time-dependent case. The Baker-Akhiezer function, the meromorphic function and the analogs of the Dubrovin-type equations in the time-dependent case are given accordingly. The Riemannθfunction representations of the Baker-Akhiezer function, the mcromor-phic function and the algebro-gcometric solutions of the hierarchy are all extended too.