N-fold Darboux transformation of the discrete PT-symmetric nonlinear Schr?dinger equation and new soliton solutions over the nonzero background

For the discrete PT-symmetric nonlinear Schrodinger (dPTNLS) equation, this paper gives a rigorous proof of the N-fold Darboux transformation (DT) and especially verifies the PT-symmetric relation between transformed potentials in the Lax pair. Meanwhile, some determinant identities are developed in completing the proof. When the tanh-function solution is directly selected as a seed for the focusing case, the onefold DT yields a three-soliton solution that exhibits the solitonic behavior with a wide range of parameter regimes. Moreover, it is shown that the solution contains three pairs of asymptotic solitons, and that each asymptotic soliton can display both the dark and antidark soliton profiles or vanish as t ->+/-infinity$t \rightarrow \pm \infty$. It indicates that the focusing dPTNLS equation admits a rich variety of soliton interactions over the nonzero background, behaving like those in the continuous counterpart.