Lobatto IIIA-IIIB discretization of the strongly coupled nonlinear Schrödinger equation

In this paper, we construct a second order semi-explicit multi-symplectic integrator for the strongly coupled nonlinear Schrodinger equation based on the two-stage Lobatto IIIA-IIIB partitioned Runge-Kutta method. Numerical results for different solitary wave solutions including elastic and inelastic collisions, fusion of two solitons and with periodic solutions confirm the excellent long time behavior of the multi-symplectic integrator by preserving global energy, momentum and mass.

[1]  Ayhan Aydin,et al.  Multisymplectic integration of N-coupled nonlinear Schrödinger equation with destabilized periodic wave solutions , 2009 .

[2]  William Kahan,et al.  Unconventional Schemes for a Class of Ordinary Differential Equations-With Applications to the Korteweg-de Vries Equation , 1997 .

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

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

[5]  Robert I. McLachlan,et al.  On Multisymplecticity of Partitioned Runge-Kutta Methods , 2008, SIAM J. Sci. Comput..

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

[7]  Bülent Karasözen,et al.  Symplectic and multisymplectic Lobatto methods for the “good” Boussinesq equation , 2008 .

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

[9]  Luming Zhang,et al.  Numerical simulation of a nonlinearly coupled Schrödinger system: A linearly uncoupled finite difference scheme , 2008, Math. Comput. Simul..

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

[11]  Bülent Karasözen,et al.  Multi-symplectic integration of coupled non-linear Schrödinger system with soliton solutions , 2009, Int. J. Comput. Math..

[12]  Jason Frank,et al.  On the multisymplecticity of partitioned Runge–Kutta and splitting methods , 2007, Int. J. Comput. Math..

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

[14]  Luming Zhang,et al.  Numerical analysis of a multi-symplectic scheme for a strongly coupled Schrödinger system , 2008, Appl. Math. Comput..

[15]  Bülent Karasözen,et al.  Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions , 2007, Comput. Phys. Commun..

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

[17]  Michail D. Todorov,et al.  Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations , 2007 .