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

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.