Elliptic solutions to difference non-linear equations and related many-body problems

TL;DR

This paper explores elliptic and algebro-geometric solutions of discretized integrable equations like KP and Toda, linking their zeros' dynamics to many-body systems and Bethe-Ansatz equations.

Contribution

It introduces a new class of elliptic solutions for discrete integrable equations and connects their zero dynamics to generalized many-body systems.

Findings

Elliptic solutions form a subclass of algebro-geometric solutions.

Zeros of solutions follow equations akin to nested Bethe-Ansatz.

Constructed algebraic curves for elliptic solutions.

Abstract

We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for $\tau$-functions. Starting from a given algebraic curve, we express the $\tau$-function and the Baker-Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the $\tau$-function is an elliptic polynomial, form a subclass of the general algebro-geometric solutions. We construct the algebraic curves of the elliptic solutions. The evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the Ruijsenaars-Schneider many body system. The zeros obey equations which have the form of nested Bethe-Ansatz equations, known from integrable 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 Toda equations and describe the loci of the zeros.