On shifted reverse space nonlocal semi-discrete Nonlinear Schrödinger equation

In this article, we explore the application of the Ablowitz-Ladik scheme, which is a renowned method for solving nonlinear differential-difference equations, to derive both shifted and unshifted versions of the semi-discrete nonlinear Schrödinger equation in a nonlocal context. This is accomplished by utilizing various symmetry reductions, which allow us to simplify and study different configurations of the equation. Using the Darboux transformation, multi-soliton solutions of space shifted nonlocal semi-discrete NLS equation are obtained. Specifically, one- and two soliton solutions are constructed and demonstrated graphically, providing insights into their behavior in both shifted and unshifted cases. It is noted that the one-soliton solution in both shifted and unshifted cases behaves identically, representing a simple translation of the soliton. However, the second-order soliton solutions undergo independent nonlocal shifts, and a clear distinction between shifted and unshifted solutions is explained. Furthermore, the interactions among singular and a second-order bound states are also investigated.