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

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.