A new 3-component Degasperis-Procesi hierarchy

TL;DR

This paper explores the bi-Hamiltonian structures of a 3-component generalization of the Degasperis-Procesi equation, revealing local Hamiltonian functionals, constructing Liouville transformations, and relating the hierarchy to other integrable systems.

Contribution

It introduces new bi-Hamiltonian structures for the 3-DP hierarchy and constructs two Liouville transformations linking it to modified Yajima-Oikawa and mKdV hierarchies.

Findings

All Hamiltonian functionals are homogeneous.

Hamiltonian functionals in the negative hierarchy are local.

Liouville transformations relate the hierarchy to known integrable systems.

Abstract

We study the bi-Hamiltonian structures for the hierarchy of a 3-component generalization of the Degasperis-Procesi (3-DP) equation. We show that all Hamiltonian functionals in the hierarchy are homogenous, and Hamiltonian functionals of the hierarchy in the negative direction are local. We construct two different Liouville transformations by construct a reciprocal transformation. The associated system for the first one is a reduction of a negative flow in a modified Yajima-Oikawa hierarchy. The associated system for the second one and the associated system for another 3-component CH type system under a equivalent transformation are shown to be different reductions of a negative generalized mKdV equation, besides the Hamiltonian structures of the 3-DP equation under this Liouville transformation are considered. In addition, we consider a limit for the 3-DP equation.