The Camassa-Holm equation as a geodesic flow on the diffeomorphism group

TL;DR

This paper demonstrates that the Camassa-Holm equation with zero parameter is a geodesic flow on the diffeomorphism group, using Euler-Poincaré theory and right-trivialisation, enabling generalizations to higher dimensions.

Contribution

It rigorously applies Euler-Poincaré theory to show the CH equation as a geodesic flow on diffeomorphism groups, extending previous results to the case =0.

Findings

CH equation is a geodesic flow on diffeomorphism group for =0

Use of right-trivialisation technique to verify Euler-Poincare9 theory applicability

Facilitates generalizations of CH equation to higher-dimensional manifolds

Abstract

Misiolek has shown that the Camassa-Holm (CH) equation is a geodesic flow on the Bott-Virasoro group. In this paper it is shown that the Camassa-Holm equation for the case $\kappa =0$ is the geodesic spray of the weak Riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the $H^1$ inner product over the entire group. This paper uses the right-trivialisation technique to rigorously verify that the Euler-Poincar\'{e} theory for Lie groups can be applied to diffeomorphism groups. The observation made in this paper has led to physically meaningful generalizations of the CH-equation to higher dimensional manifolds (see Refs. \cite{HMR} and \cite{SH}).