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

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

2006-10-26

arxiv.org/abs/math/0610786v2

Math. Notes 81, 149--152 (2007)

Mathematics

Symplectic Geometry

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Scientific paper

In 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 $\Vect(S^1)^*$.

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