A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

This article explores adapted mathematical methods to solve the coupled nonlinear Schrodinger (C-NLS) equation through analytical and numerical methods. To obtain exact solutions for the (C-NLS) equation, we utilize the improved modified, extended tanh-function method. By separating the Schrodinger equation into real and imaginary parts, we can obtain four coupled equations, which we then analyze using the generalized tanh method to extract exact solutions. This system of equations is essential for understanding the behavior of quantum systems and has various applications in quantum mechanics. We obtain an analytical solution and demonstrate numerical solutions using implicit finite difference. Studies have shown that this scheme is second-order in space and time, and the von Neumann stability analysis confirms its unconditional stability. We introduce the comparison between numerical and exact solutions.