A hierarchy of nonlinear differential-difference equations and a new Bargmann type integrable system

A hierarchy of nonlinear differential-difference equations is derived from a new discrete spectral problem. The Hamiltonian structure of the resulting hierarchy is constructed by means of a trace identity formula. Moreover, a new Bargmann type integrable system associated with the hierarchy is presented by applying the binary nonlinearization approach of Lax pairs. Based on the symmetry constraints and the generating function of integrals of motion, the resulting system is further proved to be completely integrable Hamiltonian system in the Liouville sense.