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

DOI

10.2991/jnmp.2008.15.s2.1How to use a DOI?

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 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.

Copyright

© 2008, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Rossen I. Ivanov
PY  - 2008
DA  - 2008/08/01
TI  - Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 1
EP  - 12
VL  - 15
IS  - Supplement 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2008.15.s2.1
DO  - 10.2991/jnmp.2008.15.s2.1
ID  - Ivanov2008
ER  -