A family of integrable differential–difference equations, its bi-Hamiltonian structure and binary nonlinearization of the Lax pairs and adjoint Lax pairs

A family of integrable differential–difference equations is derived by the method of Lax pairs. A discrete Hamiltonian operator involving two arbitrary real parameters is introduced. When the parameters are suitably selected, a pair of discrete Hamiltonian operators is presented. Bi-Hamiltonian structure of obtained family is established by discrete trace identity. Then, Liouville integrability for the obtained family is proved. Ultimately, through the binary nonlinearization of the Lax pairs and adjoint Lax pairs, every differential–difference equation in obtained family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.

► We deduce a family of integrable differential–difference equations.
► We present a discrete Hamiltonian operator involving two arbitrary real parameters.
► We establish the bi-Hamiltonian structure for obtained integrable family.
► Liouvolle integrability of the obtained family is demonstrated.
► Every equation in obtained family is factored through the binary nonlinearization.