Poisson structures for difference equations

TL;DR

This paper investigates the existence and construction of log-canonical Poisson structures preserved by certain difference equations, with applications to integrable systems like KP maps.

Contribution

It introduces methods to identify and construct Poisson structures for difference equations, including inverse problem solutions and examples related to integrable systems.

Findings

Log-canonical Poisson structures can be preserved by specific difference equations.

Quadratic Poisson structures are identified for KP-type maps.

The inverse problem of finding difference equations from Poisson structures is addressed.

Abstract

We study the existence of log-canonical Poisson structures that are preserved by difference equations of special form. We also study the inverse problem, given a log-canonical Poisson structure to find a difference equation preserving this structure. We give examples of quadratic Poisson structures that arise for the Kadomtsev-Petviashvili (KP) type maps which follow from a travelling-wave reduction of the corresponding integrable partial difference equation.