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

We consider the numerical solution of Coupled Nonlinear Schrodinger Equations. We prove the stability and convergence in the L"2 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 Schrodinger Equation asymptotics with the same initial conditions.

[1]  M. Ablowitz,et al.  Discrete vector spatial solitons in a nonlinear waveguide array. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  APPROXIMATE SOLUTION FOR EULER EQUATIONS OF STRATIFIED WATER VIA NUMERICAL SOLUTION OF COUPLED KdV SYSTEM , 2003 .

[3]  M. Radziunas,et al.  On Convergence and Stability of the Explicit Difference Method for Solution of Nonlinear Schrödinger Equations , 1999 .

[4]  Feliksas Ivanauskas,et al.  Finite Difference Solution Methods for a System of the Nonlinear Schrödinger Equations , 2004 .

[5]  Allan W. Snyder,et al.  Modes of optical waveguides , 1978 .

[6]  Alessandro Verri,et al.  Absolute depth estimate in stereopsis , 1986 .

[7]  E. M. Dianov,et al.  Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines , 1991 .

[8]  M. Ablowitz,et al.  Nonlinear differential–difference equations and Fourier analysis , 1976 .

[9]  M. S. Ismail,et al.  Highly accurate finite difference method for coupled nonlinear Schrödinger equation , 2004, Int. J. Comput. Math..

[10]  Weizhong Dai,et al.  An unconditionally stable three-level explicit difference scheme for the Schro¨dinger equation with a variable coefficient , 1992 .

[11]  Lixin Wu,et al.  DuFort--Frankel-Type Methods for Linear and Nonlinear Schrödinger Equations , 1996 .

[12]  C. Menyuk.  Nonlinear pulse propagation in birefringent optical fibers , 1987 .

[13]  D. Faddeev,et al.  Computational methods of linear algebra , 1959 .

[14]  Numerical integration of a coupled Korteweg-de Vries system , 2002, math/0211453.

[15]  J. Gordon,et al.  Solitons in Optical Fibers: Fundamentals and Applications , 2006 .

[16]  Qianshun Chang,et al.  Difference Schemes for Solving the Generalized Nonlinear Schrödinger Equation , 1999 .

[17]  Mark J. Ablowitz,et al.  A Nonlinear Difference Scheme and Inverse Scattering , 1976 .

[18]  C. Pask,et al.  Nonlinear effects in elliptical-core few-mode optical fibers , 1992 .

[19]  Govind P. Agrawal,et al.  Nonlinear Fiber Optics , 1989 .

[20]  A. C. Bajpai,et al.  Numerical Methods for Engineers and Scientists. , 1978 .

[21]  K. Porsezian.  Soliton models in resonant and nonresonant optical fibers , 2001 .

[22]  Claude Brezinski,et al.  Numerical Methods for Engineers and Scientists , 1992 .

[23]  B. Reichel,et al.  Mode interaction in few-mode optical fibres with Kerr effect , 2008 .

[24]  J. M Varah,et al.  Computational methods in linear algebra , 1984 .

[25]  Jian-Qiang Sun,et al.  Numerical study of the soliton waves of the coupled nonlinear Schrödinger system , 2004 .