The Camassa-Holm equation as a geodesic flow for the $H^1$ right-invariant metric

TL;DR

This paper explores the Camassa-Holm equation as a geodesic flow on the diffeomorphism group with an $H^1$ metric, highlighting its geometric structure and connections to Lie group theory.

Contribution

It provides a geometric interpretation of the Camassa-Holm equation as a geodesic flow, including explicit parametrization of the Virasoro group and analysis of the momentum map.

Findings

CH is a geodesic flow on the diffeomorphism group with $H^1$ metric

Explicit parametrization of the Virasoro group related to CH solutions

Connection between CH and Lie group geometric structures

Abstract

The fundamental role played by the Lie groups in mechanics, and especially by the dual space of the Lie algebra of the group and the coadjoint action are illustrated through the Camassa-Holm equation (CH). In 1996 Misio{\l}ek observed that CH is a geodesic flow equation on the group of diffeomorphisms, preserving the $H^1$ metric. This example is analogous to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinite-dimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a left-invariant metric on SO(3). The momentum map and an explicit parametrization of the Virasoro group, related to recently obtained solutions for the CH equation are presented.