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

Abstract

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.

37 pages

References in corpus (9)

• An integrable shallow water equation with peaked solitons
• A 2-Component Generalization of the Camassa-Holm Equation and Its Solutions
• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• Deformations of Semisimple Bihamiltonian Structures of Hydrodynamic Type
• Compatible Poisson brackets of hydrodynamic type
• Hamiltonian operators and l*-coverings
• Flat pencils of metrics and Frobenius manifolds
• Reciprocal transformations of Hamiltonian operators of hydrodynamic type: nonlocal Hamiltonian formalism for linearly degenerate systems
• Integrable peakon equations with cubic nonlinearity

Cited by in corpus (1)

• Multi-component generalizations of the {CH} equation: Geometrical Aspects, Peakons and Numerical Examples