Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations

TL;DR

This paper introduces an algebraic discretization of the Camassa-Holm and Hunter-Saxton equations, revealing their structure as integrable tops with infinite momentum components and analyzing their phase space and momentum map.

Contribution

It provides a novel algebraic discretization method for CH and HS equations, connecting them to integrable tops and exploring their phase space structure.

Findings

Discretization leads to integrable top models with infinite momentum components.

The phase space and momentum map structures are characterized.

The approach links CH and HS equations to well-known integrable systems.

Abstract

The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the $H^1$ and $\dot{H}^1$ right-invariant metrics correspondingly. There is an analogy 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 CH and HS equations are integrable bi-hamiltonian equations and one of their Hamiltonian structures is associated to the Virasoro algebra. The parallel with the integrable SO(3) top is made explicit by a discretization of both equation based on Fourier modes expansion. The obtained equations represent integrable tops with infinitely many momentum components. An emphasis is given on the structure of the phase space of these equations, the momentum map and the space of canonical variables.