Explicit solutions for a hierarchy of differential–difference equations

Abstract

A new discrete isospectral problem and the corresponding hierarchy of nonlinear differential–difference equations are proposed. On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the hierarchy of differential–difference equations are straightened out using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the corresponding hierarchy of differential–difference equations are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function, the Baker–Akhiezer function and the hyperelliptic curve.