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New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the
Darryl D Holm1, Tomasz M Tyranowski1,2
1Mathematics Department, Imperial College London, London SW7 2AZ, UK.
Abstract:
We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler-Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called 'back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with 2-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.
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