A new 3-component Degasperis–Procesi hierarchy

Abstract

We propose a new 3-component Degasperis–Procesi (3-DP) system. By investigating the bi-Hamiltonian structure of the corresponding hierarchy, we find that the Hamiltonian functionals of the 3-DP hierarchy are local in the negative direction and homogeneous in both directions. We show that the 3-DP system is related to the first negative flow in a modified Yajima–Oikawa hierarchy via a reciprocal transformation. With the help of this reciprocal transformation and a Darboux transformation of the associated 3-DP system, we construct a Bäcklund transformation for the 3-DP system. Furthermore, we present a short-wave model of the 3-DP system.