Algebro-Geometric Integration of the Modified Belov—Chaltikian Lattice Hierarchy

Abstract

Using the Lenard recurrence relations and the zero-curvature equation, we derive the modified Belov—Chaltikian lattice hierarchy associated with a discrete 3×3 matrix spectral problem. Using the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a tri gonal curve K_m−2 of arithmetic genus m−2. We study the asymptotic properties of the Baker—Akhiezer function and the algebraic function carrying the data of the divisor near \(P_{\infty_{1}}\), \(P_{\infty_{2}}\), \(P_{\infty_{3}}\), and P₀ on K_m−2. Based on the theory of trigonal curves, we obtain the explicit theta-function representations of the algebraic function, the Baker—Akhiezer function, and, in particular, solutions of the entire modified Belov—Chaltikian lattice hierarchy.