An integrable coupling family of the Toda lattice systems, its bi-Hamiltonian structure, and a related nonisospectral integrable lattice family

Starting from a four-by-four matrix spectral problem, an integrable coupling family of the Toda lattice systems is derived. A pair of discrete Hamiltonian operators is presented, and a sequence of the corresponding Hamiltonian functionals is constructed by using the discrete variational identity. Then, the bi-Hamiltonian structure of the obtained integrable coupling family is established. Finally, a nonisospectral integrable lattice family associated with the obtained integrable coupling family is given through nonisospectral discrete zero curvature representation.