Integrable invariant Sobolev metrics on the Abelian extension of the diffeomorphism group of the circle and two-component generalizations of the Camassa-Holm equation

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126302In this note we classify some integrable invariant Sobolev metrics on the Abelian extension of the diffeomorphism group of the circle. We also derive a new two-component generalization of the Camassa-Holm equation. The system obtained appears to be unique bi-Hamiltonian flow on the coadiont obrit of $\Diff_+(S^1)\ltimes C^\infty(S^1)$ generalizes the Camassa-Holm flow on $\Vect(S^1)^*$.added referencesSymplectic GeometryMathematical PhysicsIntegrable invariant Sobolev metrics on the Abelian extension of the diffeomorphism group of the circle and two-component generalizations of the Camassa-Holm equationtext