Volume 15, Issue Supplement 2, August 2008, Pages 1 - 12
Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations
Rossen I. Ivanov
Rossen I. Ivanov
Received 21 March 2008, Available Online 1 August 2008.
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- 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.
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Cite this article
TY - JOUR AU - Rossen I. Ivanov PY - 2008 DA - 2008/08 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 - https://doi.org/10.2991/jnmp.2008.15.s2.1 ID - Ivanov2008 ER -