Lie algebras and Hamiltonian structures of multi-component Ablowitz–Kaup–Newell–Segur hierarchy

Abstract

Isospectral and non-isospectral hierarchies of multi-component Ablowitz–Kaup–Newell–Segur (AKNS) are obtained from a matrix spectral problem, then by means of the zero curvature representations of the isospectral flows {Km} and non-isospectral flows {σn}, we construct the symmetries and their algebraic structures for isospectral multi-component AKNS hierarchies, demonstrate the recursive operator L is a strong and hereditary symmetry for the isospectral hierarchy. We also derive that there are implectic operator θ and symplectic operator J such that L = θJ, and discuss the multi-Hamiltonian structures and the Liouville integrability of the isospectral hierarchies.