Hamiltonian Structures and Reciprocal Transformations for the $r$-KdV-CH Hierarchy

Details Hamiltonian Structures and Reciprocal Transformations for the $r$-KdV-CH Hierarchy Hamiltonian Structures and Reciprocal Transformations for the $r$-KdV-CH Hierarchy

2008-09-02

arxiv.org/abs/0809.0397v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

37 pages

Scientific paper

The $r$-KdV-CH hierarchy is a generalization of the Korteweg-de Vries and Camassa-Holm hierarchies parametrized by $r+1$ constants. In this paper we clarify some properties of its multi-Hamiltonian structures, prove the semisimplicity of the associated bihamiltonian structures and the formula for their central invariants. By introducing a class of generalized Hamiltonian structures, we give in a natural way the transformation formulae of the Hamiltonian structures of the hierarchy under certain reciprocal transformation, and prove the formulae at the level of its dispersionless limit. We also consider relations of the associated bihamiltonian structures to Frobenius manifolds.

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