Explicit solutions for a hierarchy of differential-difference equations

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.