On convergence and stability of a numerical scheme of coupled nonlinear Schrödinger equations

Abstract

We consider numerical solution of Coupled Nonlinear Schrödinger Equation. We prove stability and convergence in the $L_2$ space for an explicit scheme which estimations is used for implicit scheme and compare both method. As a test we compare numerical solutions of Manakov system with known analytical solitonic solutions and as example of general system - evolution of two impulses with different group velocity (model of pulses interaction in optic fibers). Last example, a rectangular pulse evolution, shows asymptotic behavior typical for Nonlinear Schrödinger Equation asymptotics with the same initial condition.