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

TL;DR

This paper investigates the multi-Hamiltonian structures of the $r$-KdV-CH hierarchy, establishing their properties, transformations under reciprocal maps, and connections to Frobenius manifolds, advancing the understanding of integrable systems.

Contribution

It clarifies properties of the hierarchy's bihamiltonian structures, proves their semisimplicity, and derives transformation formulas under reciprocal transformations.

Findings

Proved the semisimplicity of bihamiltonian structures.

Derived transformation formulas for Hamiltonian structures.

Explored relations to Frobenius manifolds.

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.