On exact solution of a classical 3D integrable model

Abstract

We investigate some classical evolution model in the discrete 2+1 space-time. A map, giving an one-step time evolution, may be derived as the compatibility condition for some systems of linear equations for a set of auxiliary linear variables. Dynamical variables for the evolution model are the coefficients of these systems of linear equations. Determinant of any system of linear equations is a polynomial of two numerical quasimomenta of the auxiliary linear variables. For one, this determinant is the generating functions of all integrals of motion for the evolution, and on the other hand it defines a high genus algebraic curve. The dependence of the dynamical variables on the space-time point (exact solution) may be expressed in terms of theta functions on the jacobian of this curve. This is the main result of our paper.