Finite Genus Solutions to a Hierarchy of Integrable Semi-Discrete Equations

Abstract

Resorting to the discrete zero-curvature equation and the Lenard recursion equations, a hierarchy of integrable semi-discrete nonlinear evolution equations is derived from a $3 \times 3$ matrix spectral problem with three potentials. Based on the characteristic polynomial of the Lax matrix for the hierarchy, a trigonal curve is introduced, and the properties of the corresponding three-sheeted Riemann surface are studied, including the genus, three kinds of Abelian differentials, Riemann theta functions. The asymptotic properties of the Baker-Akhiezer function and fundamental meromorphic functions defined on the trigonal curve are analyzed with the established theory of trigonal curves. As a result, finite genus solutions of the whole integrable semi-discrete nonlinear evolution hierarchy are obtained.