Holm DD, Tyranowski TM. New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map.
Proc Math Phys Eng Sci 2018;
474:20180052. [PMID:
29887752 DOI:
10.1098/rspa.2018.0052]
[Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2018] [Accepted: 04/09/2018] [Indexed: 11/12/2022] Open
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.
Collapse