Numerical solution of nonlinear Schrodinger equation using Crank-Nicolson method

Abstract

The nonlinear Schrödinger equations (NLS) are used in modeling several physical phenomena such as Bose-Einstein condensation, laser beam transmissions, deep water turbulence, and solitary wave propagation in optical fibers. Solving NLS equation resulting in a wave function that makes it easier to examine the behavior and performance of physical systems or chemical reactions. There are several methods that can be used to solve the nonlinear Schrödinger equation. In this study, one dimensional nonlinear Schrödinger equation was solved by Crank-Nicolson method with Dirichlet boundary condition and symmetric cyclic tridiagonal matrix using MATLAB. The Crank-Nicolson scheme is used as it is one of the adaptable, fast, and robust techniques for integrating the time-dependent Schrödinger equation. In addition, the operational framework for finite difference scheme and stability analysis for this technique are presented. Lastly, the performance of the Crank-Nicolson scheme is analyzed by computing the error between the estimated and the exact solution. It is shown that both results from numerical scheme and exact solution have good agreement.