Trigonal Curve And Quasi-periodic Solutions Of The Discrete Soliton Hierarchies

The theory of integrable systems is one of the important subjects in the research of mathematics and physics. In recent years, the discrete integrable systems have attracted more and more interests. The integrable discrete equations(including semi-discrete and full-discrete) have important applications in many fields such as nonlinear lattice dynamics, ladder type electric circuit and Volterra system, now an important task for the study of the discrete integrable systems is to find and develop some new tools. The construction of an discrete integrable system and their solutions have many importance in the integrable systems.In this paper, by means of the algebro-geometric method, with the help of the theory of trigonal curve, we mainly study quasi-periodic solutions to four discrete(semi-discrete in this paper) soliton hierarchies in terms of Riemann theta function representation. These hierarchies here are Blaszak-Marciniak lattice hierarchy,the hierarchy of coupled Toda lattice, Belov-Chaltikian lattice hierarchy, discrete coupled nonlinear Schr(?)dinger hierarchy.With the aid of the zero-curvature equation, the discrete soliton hierarchy associated with the third-order discrete spectral problems are proposed. A three sheeted compact Riemann surface Kgof arithmetric genus g is introduced by using the characteristic polynomial of Lax matrix for the stationary soliton equations, from which the Baker-Akhiezer function and the associated meromorphic functions are given. We introduce elliptic variables, then the soliton equations are decomposed into the system of Dubrovin-type ordinary differential equations. Furthermore, in accordance with the properties of the zeros and poles of the meromorphic function and Baker-Akhiezer function, we get their Riemann theta function representations by means of three kinds of Abel differentials and Riemann-Roch theorem. Combining the Riemann theta function representations of the meromorphic function and the Baker-Akhiezer function with their asymptotic properties, we finally obtain the quasi-periodic solutions of discrete soliton equations.