Darboux Transformation and Exact Solutionsfor a Semi-discrete Coupled Local Nonlinear Schrdinger Equation

In recent years, significant progress has been made in the study of nonlinear Schrdinger equations within the fields of mathematical physics and nonlinear wave dynamics. However, investigations on the semi-discrete form of coupled local nonlinear Schrdinger equations remain relatively limited. In this work, within the framework of semi-discrete systems, we impose local reduction conditions on the coupled nonlinear Schrdinger equation and propose a new integrable semi-discrete two-component coupled local nonlinear Schrdinger equation. Starting from a semi-discrete44matrix Lax pair, and with the aid of a gauge transformation, explicit formulas for the Darboux transformation of the system are derived. By selecting appropriate seed solutions, exact solutions of the system on the zero background are obtained through the constructed Darboux transformation. Furthermore, the dynamical properties of these solutions are visualized using mathematical software. The results reveal that the system admits typical breather-type soliton solutions, which exhibit periodic oscillations in the temporal direction and localization in the spatial direction. This study not only deepens the understanding of soliton dynamics in semi-discrete local nonlinear systems but also provides an effective approach and theoretical reference for analyzing similar nonlinear wave equations.