New variational and multisymplectic formulations of the Euler-Poincar\'e equation on the Virasoro-Bott group using the inverse map

TL;DR

This paper introduces new variational and multisymplectic formulations for Euler-Poincaré equations on the Virasoro-Bott group, unifying several soliton equations and opening avenues for future research on nonlinearity, dispersion, and noise.

Contribution

It presents a novel variational principle, momentum map, and multisymplectic formulation for a family of equations on the Virasoro-Bott group using the inverse map, unifying known soliton equations.

Findings

Derived a new variational principle for Euler-Poincaré equations.

Established a new momentum map and multisymplectic formulation.

Unified several soliton equations within this framework.

Abstract

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincar\'e 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.