Conformal structure-preserving method for damped nonlinear Schrödinger equation*

In this paper, we propose a conformal momentum-preserving method to solve a damped nonlinear Schrodinger (DNLS) equation. Based on its damped multi-symplectic formulation, the DNLS system can be split into a Hamiltonian part and a dissipative part. For the Hamiltonian part, the average vector field (AVF) method and implicit midpoint method are employed in spatial and temporal discretizations, respectively. For the dissipative part, we can solve it exactly. The proposed method conserves the conformal momentum conservation law in any local time–space region. With periodic boundary conditions, this method also preserves the total conformal momentum and the dissipation rate of momentum exactly. Numerical experiments are presented to demonstrate the conservative properties of the proposed method.

[1]  Chun Li,et al.  Multi-symplectic Runge-Kutta-Nyström methods for nonlinear Schrödinger equations with variable coefficients , 2007, J. Comput. Phys..

[2]  Songhe Song,et al.  The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs , 2011, J. Comput. Appl. Math..

[3]  Yifa Tang,et al.  Symplectic and multi-symplectic methods for the nonlinear Schrodinger equation , 2002 .

[4]  Brian E. Moore,et al.  Conformal conservation laws and geometric integration for damped Hamiltonian PDEs , 2013, J. Comput. Phys..

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

[6]  Jiaxiang Cai.  Multisymplectic schemes for strongly coupled schrödinger system , 2010, Appl. Math. Comput..

[7]  Ivan H. Deutsch,et al.  Reduction of quantum noise in soliton propagation by phase-sensitive amplification , 1994 .

[8]  Linghua Kong,et al.  Symplectic Euler Method for Nonlinear High Order Schr ¨ odinger Equation with a Trapped Term , 2009 .

[9]  Elphick,et al.  Localized structures in surface waves. , 1989, Physical review. A, General physics.

[10]  S. Reich,et al.  Multi-symplectic spectral discretizations for the Zakharov–Kuznetsov and shallow water equations , 2001 .

[11]  R. McLachlan,et al.  Conformal Hamiltonian systems , 2001 .

[12]  Wang Yu-shun,et al.  Numerical Simulation of Rogue Waves by the Local Discontinuous Galerkin Method , 2014 .

[13]  Brian E. Moore.  Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations , 2009, Math. Comput. Simul..

[14]  Brian E. Moore,et al.  Multi-symplectic integration methods for Hamiltonian PDEs , 2003, Future Gener. Comput. Syst..

[15]  Yushun Wang,et al.  Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs , 2014, J. Comput. Phys..

[16]  C. Schober,et al.  Geometric integrators for the nonlinear Schrödinger equation , 2001 .

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