Integrable couplings of two expanded non-isospectral soliton hierarchies and their bi-Hamiltonian structures

Abstract

Starting from two matrix spectral problems associated with the real special orthogonal Lie algebra [Formula: see text], two non-isospectral hierarchies of Ablowitz–Kaup–Newell–Segur (AKNS) type and Kaup–Newell (KN) type are constructed. In addition, we enlarge the two spectral problems and then obtain non-isospectral [Formula: see text] integrable couplings of AKNS type and KN type by solving the expanded non-isospectral zero curvature equation. We find that the obtained four hierarchies have the bi-Hamiltonian structure of the combined form. It follows that all equations in the resulting soliton hierarchy are integrable in the sense of Liouville.