ELLIPTIC SOLUTIONS TO DIFFERENCE NONLINEAR EQUATIONS AND RELATED MANY-BODY PROBLEMS

We study algebro-geometric (finite-gap) and elliptic solutions of full y discretized KP or 2D Toda equations. In bilinear form they are Hirot a's difference equation for tau-functions. Starting from a given algeb raic curve, we express the tau-function and the Baker-Akhiezer functio n in terms of the Riemann theta function. We show that the elliptic so lutions, when the tau-function is an elliptic polynomial, form a subcl ass of the general algebro-geometric solutions. We construct the algeb raic curves of the elliptic solutions. The evolution of zeros of the e lliptic solutions is governed by the discrete time generalization of t he Ruijsenaars-Schneider many body system. The zeros obey equations wh ich have the form of nested Bethe-ansatz equations, known from integra ble quantum field theories. We discuss the Lax representation and the action-angle-type Variables for the many body system, We also discuss elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D To da equations and describe the loci of the zeros.