Stability analysis of the implicit finite difference schemes for nonlinear Schrödinger equation

This paper analyzes the stability of numerical solutions for a nonlinear Schrödinger equation that is widely used in several applications in quantum physics, optical business, etc. One of the most popular approaches to solving nonlinear problems is the application of a linearization scheme. In this paper, two linearization schemes—Newton and Picard methods were utilized to construct systems of linear equations and finite difference methods. Crank-Nicolson and backward Euler methods were used to establish numerical solutions to the corresponding linearized problems. We investigated the stability of each system when a finite difference discretization is applied, and the convergence of the suggested approximation was evaluated to verify theoretical analysis.