A Kind of Generalized Integrable Couplings and Their Bi-Hamiltonian Structure

We introduce a Lie algebra (g) over tilde which can be used to construct integrable couplings of some spectral problems. As two examples, the non-semisimple Lie algebra (g) over tilde is applied to enlarge the spectral problems of an extended Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem and a generalized D-Kaup-Newell (D-KN) spectral problem. It follows that we obtain two generalized integrable couplings by solving these expanded zero-curvature equations. Finally, we find that the integrable hierarchies that we obtain have bi-Hamiltonian structures of combinatorial form, thereby showing their Liouville integrability.