Numerical analysis of a multi-symplectic scheme for a strongly coupled Schrödinger system

Abstract In this paper, we analyze a multi-symplectic scheme for a strongly coupled nonlinear Schrodinger equations. We derive a general box scheme which is equivalent to the multi-symplectic scheme by reduction method. Based on the general box scheme, we prove that the multi-symplectic scheme preserves not only the multi-symplectic structure of the equation but also conservation law of mass. In general, the multi-symplectic schemes are not conservative to energy in the nonlinear case, so it is difficult to obtain the estimates of numerical solutions in ‖ · ‖ ∞ norm. Hence proofs of convergence and stability are difficult for multi-symplectic schemes of nonlinear equations. A deduction argument and the energy analysis method are used to prove that the numerical solution is stable for initial values, and second order convergent to the exact solutions in ‖ · ‖ 2 norm. A fixed point theorem is introduced and used to prove the unique solvability of the numerical solutions.

[1]  Thiab R. Taha,et al.  Numerical simulation of coupled nonlinear Schrödinger equation , 2001 .

[2]  Christo I. Christov,et al.  Strong coupling of Schrödinger equations: Conservative scheme approach , 2005, Math. Comput. Simul..

[3]  Brian E. Moore,et al.  Backward error analysis for multi-symplectic integration methods , 2003, Numerische Mathematik.

[4]  Ying Liu,et al.  Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients , 2006 .

[5]  Jerrold E. Marsden,et al.  Variational Methods, Multisymplectic Geometry and Continuum Mechanics , 2001 .

[6]  S. Reich.  Multi-Symplectic Runge—Kutta Collocation Methods for Hamiltonian Wave Equations , 2000 .

[7]  Robert A. Minlos,et al.  The one-particle energy spectrum of weakly coupled quantum rotators , 2000 .

[8]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[9]  Ying Liu,et al.  Multisymplecticity of the centred box scheme for a class of hamiltonian PDEs and an application to quasi-periodically solitary waves , 2004 .

[10]  Mengzhao Qin,et al.  Construction of symplectic schemes for wave equations via hyperbolic functions sinh(x), cosh(x) and tanh(x) , 1993 .

[11]  M. Qin,et al.  Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation , 2000 .

[12]  E. Gross,et al.  Hydrodynamics of a Superfluid Condensate , 1963 .

[13]  Zhi-Zhong Sun,et al.  A second order accurate difference scheme for the heat equation with concentrated capacity , 2004, Numerische Mathematik.

[14]  Tingting Liu,et al.  Multisymplectic geometry and multisymplectic Preissman scheme for the KP equation , 2002 .

[15]  Ben-yu Guo,et al.  The convergence of numerical method for nonlinear Schro¨dinger equation , 1986 .

[16]  Jian-Qiang Sun,et al.  Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system , 2003 .

[17]  Ya Juan Sun,et al.  Construction of multisymplectic schemes of any finite order for modified wave equations , 2000 .

[18]  Tang,et al.  SYMPLECTIC COMPUTATION OF HAMILTONIAN SYSTEMS (I) , 2002 .

[19]  T. Bridges.  Multi-symplectic structures and wave propagation , 1997, Mathematical Proceedings of the Cambridge Philosophical Society.

[20]  Zhi-zhong Sun,et al.  A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation , 1995 .

[21]  E. Gross.  Structure of a quantized vortex in boson systems , 1961 .

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

[23]  Thiab R. Taha,et al.  A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation , 2007, Math. Comput. Simul..

[24]  Chun Li,et al.  Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations , 2006 .

[25]  Ruxun Liu,et al.  Numerical simulation of interaction between Schrödinger field and Klein-Gordon field by multisymplectic method , 2006, Appl. Math. Comput..

[26]  Masato Hisakado,et al.  A Coupled Nonlinear Schrodinger Equation and Optical Solitons , 1992 .

[27]  S. Reich,et al.  Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity , 2001 .

[28]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

[29]  Ernst Hairer,et al.  Invariant tori of dissipatively perturbed Hamiltonian systems under symplectic discretization , 1999 .

[30]  Bin Wang,et al.  Numerical implementation of the multisymplectic Preissman scheme and its equivalent schemes , 2004, Appl. Math. Comput..

[31]  Ying Liu,et al.  A novel numerical approach to simulating nonlinear Schro"dinger equations with varying coefficients , 2003, Appl. Math. Lett..