A generalized Ablowitz–Ladik hierarchy, multi-Hamiltonian structure and Darboux transformation

Starting from a discrete spectral problem, we derive a hierarchy of nonlinear discrete equations which includes the Ablowitz–Ladik hierarchy and a new hierarchy as special cases. Especially, we investigate in detail the integrability and resolvability of the new hierarchy. It is shown that the new hierarchy is integrable in Liouville’s sense and possesses multi-Hamiltonian structure. A Darboux transformation is established for a typical discrete system in the new hierarchy with the help of the gauge transformation of its Lax pair. As applications of the Darboux transformation, new exact solutions of the discrete system are explicitly given.