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.

19 pages

References in corpus (4)

• Solution properties of a 3D stochastic Euler fluid equation
• Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics
• Multisymplectic formulation of fluid dynamics using the inverse map
• Continuous and discrete Clebsch variational principles

Cited by in corpus (3)

• Berry Phases in the Reconstructed KdV Equation
• Stochastic variational principles for the collisional Vlasov-Maxwell and Vlasov-Poisson equations
• Stochastic multisymplectic PDEs and their structure-preserving numerical methods