A Lagrangian view on complete integrability of the two-component Camassa-Holm system

TL;DR

This paper demonstrates that the two-component Camassa-Holm system remains completely integrable when reformulated in Lagrangian coordinates, providing a new perspective on its mathematical structure and integrability.

Contribution

It introduces a Lagrangian framework for the two-component Camassa-Holm system, showing its complete integrability through reparametrizations of the isospectral problem.

Findings

Lagrangian coordinates correspond to different normalizations of the isospectral problem

The two-component Camassa-Holm system is completely integrable in Lagrangian form

Reparametrizations reveal the underlying integrable structure

Abstract

We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa-Holm system can be understood in terms of certain reparametrizations of the underlying isospectral problem. The respective coordinates correspond to different normalizations of an associated first order system. In particular, we will see that the two-component Camassa-Holm system in Lagrangian variables is completely integrable as well.