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

Ivanov, Rossen I LU (2008) In Journal of Nonlinear Mathematical Physics 15. p.1-12
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... (More)
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. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Nonlinear Mathematical Physics
volume
15
pages
1 - 12
publisher
Bokförlaget Atlantis
external identifiers
  • wos:000263517200002
  • scopus:51149105279
ISSN
1402-9251
language
English
LU publication?
yes
id
b62c130f-16fb-40c1-8e70-f6d03cf97161 (old id 1375226)
date added to LUP
2009-05-07 13:09:35
date last changed
2017-10-22 04:15:18
@article{b62c130f-16fb-40c1-8e70-f6d03cf97161,
  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.},
  author       = {Ivanov, Rossen I},
  issn         = {1402-9251},
  language     = {eng},
  pages        = {1--12},
  publisher    = {Bokförlaget Atlantis},
  series       = {Journal of Nonlinear Mathematical Physics},
  title        = {Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations},
  volume       = {15},
  year         = {2008},
}