New hierarchies of integrable lattice equations and associated properties: Darboux transformation, conservation laws and integrable coupling

Starting from a discrete spectral problem, new hierarchies of integrable lattice equations are presented. Some associated properties are discussed. By applying the discrete trace identity, the Hamiltonian structures for a new hierarchy are derived, it is shown that the resulting hierarchy is integrable in the Liouville sense. Moreover, a Darboux transformation with four variable functions for a typical equation coming from the new hierarchy is constructed based on its Lax pairs, the explicit solutions are obtained with the Darboux transformation, the structures for those obtained solutions are graphically investigated. Further, the infinitely many conservation laws for that typical equation are given. Finally, an integrable coupling system of the resulting hierarchy is constructed through enlarging spectral problems. All these properties may be helpful to explam some physical phenomena.