New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map

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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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Holm, D. D., & Tyranowski, T. M. (2018). New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 474(2213). https://doi.org/10.1098/rspa.2018.0052

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