Finite genus solutions of the generalized Merola–Ragnisco–Tu lattice hierarchy

Abstract

Resorting to the zero-curvature equation and the Lenard recursion equations, the generalized Merola–Ragnisco–Tu lattice hierarchy associated with a 3 × 3 discrete matrix spectral problem is derived. With the aid of the characteristic polynomial of the Lax matrix for the generalized Merola–Ragnisco–Tu lattice hierarchy, a trigonal curve is defined, on which we construct the Baker–Akhiezer function, two meromorphic functions, three kinds of Abelian differentials, and Riemann theta function. By analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions, especially their asymptotic expansions near three infinite points and three zero points, we obtain their essential singularities and divisors. Finally, we obtain the finite genus solutions of the generalized Merola–Ragnisco–Tu lattice hierarchy in terms of the Riemann theta function.