Hamiltonian systems and its integrable coupling associated with a new discrete spectral problem

Starting from a new discrete spectral problem, the corresponding positive and negative hierarchies of nonlinear lattice equations are proposed. It is shown that the both lattice soliton hierarchies possess the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, we construct two integrable coupling systems for the positive hierarchy through enlarging Lax pair method.