Riemann surface and Riemann theta function solutions of the discrete integrable hierarchy

Abstract

A hierarchy of discrete nonlinear evolution equations associated with a discrete 3 × 3 matrix spectral problem with two potentials is proposed by means of the Lenard recursion equations and zero-curvature equation. Based on the characteristic polynomial of Lax matrix for the hierarchy, we introduce a trigonal curve and study the properties of the corresponding three-sheeted Riemann surface, especially including arithmetic genus, holomorphic differentials. Base on the essential properties of the meromorphic functions ϕ2, ϕ3 and the Baker–Akhiezer function ψ1, and their asymptotic behavior, we obtain Riemann theta function solutions of the entire discrete integrable hierarchy.