Dual Hierarchies of a Multi-Component Camassa–Holm System**Supported by the National Natural Science Foundation of China under Grant Nos. 11275072 and 11375090, Research Fund for the Doctoral Program of Higher Education of China under No. 20120076110024, the Innovative Research Team Program of the National Natural Science Foundation of China under Grant No. 61321064, Shanghai Knowledge Service Platform for Trustworthy Internet of

Abstract

In this paper, we derive the bi-Hamiltonian structure of a multi-component Camassa–Holm system, which associates with the multi-component AKNS hierarchy and multi-component KN hierarchy via the tri-Hamiltonian duality method. Furthermore, the spectral problems of the dual hierarchies may be obtained.