Backlund transformations for the elliptic Gaudin model and a Clebsch system

TL;DR

This paper constructs explicit, symplectic Backlund transformations for the elliptic Gaudin model, preserving integrals and mapping real solutions to real solutions, with applications to the Clebsch system.

Contribution

It introduces a two-parameter family of Backlund transformations for the elliptic Gaudin model, extending known rational and trigonometric cases and applying to the Clebsch system.

Findings

Transformations are explicit and symplectic.

They preserve integrals and map real solutions to real solutions.

Application demonstrated on the Clebsch system.

Abstract

A two-parameters family of Backlund transformations for the classical elliptic Gaudin model is constructed. The maps are explicit, symplectic, preserve the same integrals as for the continuous flows and are a time discretization of each of these flows. The transformations can map real variables into real variables, sending physical solutions of the equations of motion into physical solutions. The starting point of the analysis is the integrability structure of the model. It is shown how the analogue transformations for the rational and trigonometric Gaudin model are a limiting case of this one. An application to a particular case of the Clebsch system is given.