Discrete-time Calogero-Moser Model and Lattice KP Equations

TL;DR

This paper develops a discrete-time integrable Calogero-Moser model derived from lattice KP equations, providing new solutions, invariants, and a connection to the continuum model.

Contribution

It introduces a novel integrable discrete Calogero-Moser model from lattice KP equations, including Lax pairs, invariants, and exact solutions.

Findings

Discrete Calogero-Moser model reduces to the classical model in continuum limit.

Constructed Lax pair and invariants for the discrete model.

Provided exact solutions for the rational discrete-time case.

Abstract

We introduce an integrable time-discretized version of the classical Calogero-Moser model, which goes to the original model in a continuum limit. This discrete model is obtained from pole solutions of a semi-discretized version of the Kadomtsev-Petviashvili equation, leading to a finite-dimensional symplectic mapping. Lax pair, symplectic structure and sufficient set of invariants of the discrete Calogero-Moser model are constructed for both the rational and elliptic cases. The classical $r$-matrix is the same as for the continuum model. An exact solution of the initial value problem is given for the rational discrete-time Calogero-Moser model. The pole-expansion and elliptic solutions of the fully discretized Kadomtsev-Petviashvili equation are also discussed.