On convergence and stability of a numerical scheme of Coupled Nonlinear Schrödinger Equations

We consider numerical solution of Coupled Nonlinear Schrödinger Equation. We prove stability and convergence in the L2 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.