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

We consider the numerical solution of Coupled Nonlinear Schrödinger Equations. We prove the stability and convergence in the L2L2 space for an explicit scheme the estimations of which are used for the implicit scheme and compare both methods. As a test we compare the numerical solutions of the Manakov system with known analytical solitonic solutions and as an example of the general system — evolution of two impulses with different group velocity (model of interaction of pulses in optic fibers). As a last example, a rectangular pulse evolution, shows asymptotic behavior typical for Nonlinear Schrödinger Equation asymptotics with the same initial conditions.