An integrable lattice hierarchy, associated integrable coupling, Darboux transformation and conservation laws

A new integrable lattice hierarchy is constructed from a discrete matrix spectral problem, some related properties of the new hierarchy are discussed. The Hamiltonian structures and Liouville integrability of the new hierarchy are established by using the discrete trace identity. A kind of integrable coupling for the new hierarchy is constructed through enlarging spectral problems. A Darboux transformation (DT) with two variable parameters and the infinitely many conservation laws for a typical lattice equation in the new hierarchy are constructed based on its Lax representation, the explicit solutions are obtained via the DT, the structures for those solutions are graphically investigated. All these properties might be helpful to understanding some physical phenomena.