Algebraic Discretization of the CamassaHolm and HunterSaxton Equations
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Abstract
The CamassaHolm (CH) and HunterSaxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H1 and. H1 rightinvariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a rightinvariant metric on the infinitedimensional 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 bihamiltonian 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.
Details
Authors  

Organisations  
Research areas and keywords  Subject classification (UKÄ) – MANDATORY

Original language  English 

Pages (fromto)  112 
Journal  Journal of Nonlinear Mathematical Physics 
Volume  15 
Publication status  Published  2008 
Publication category  Research 
Peerreviewed  Yes 