Dynamical R-Matrices for Integrable Maps

TL;DR

This paper demonstrates the integrability of certain discrete symplectic maps using a dynamical $r$-matrix approach, revealing their Poisson structures and separation variables, thus extending integrability concepts to discrete systems.

Contribution

It introduces a dynamical $r$-matrix framework for integrable maps, linking Lax representations to Poisson structures and separation variables in discrete systems.

Findings

Established integrability of maps via $r$-matrix approach

Identified dynamical $r$-matrix as key to discrete integrability

Connected Lax matrices to separation variables in discrete dynamics

Abstract

The integrability of two symplectic maps, that can be considered as discrete-time analogs of the Garnier and Neumann systems is established in the framework of the $r$-matrix approach, starting from their Lax representation. In contrast with the continuous case, the $r$-matrix for such discrete systems turns out to be of dynamical type; remarkably, the induced Poisson structure appears as a linear combination of compatible ``more elementary" Poisson structures. It is also shown that the Lax matrix naturally leads to define separation variables, whose discrete and continuous dynamics is investigated.