Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

In this paper, we develop symplectic and multi-symplectic wavelet collocation methods to solve the two-dimensional nonlinear Schrodinger equation in wave propagation problems and the two-dimensional time-dependent linear Schrodinger equation in quantum physics. The Hamiltonian and the multi-symplectic formulations of each equation are considered. For both formulations, wavelet collocation method based on the autocorrelation function of Daubechies scaling functions is applied for spatial discretization and symplectic method is used for time integration. The conservation of energy and total norm is investigated. Combined with splitting scheme, splitting symplectic and multi-symplectic wavelet collocation methods are also constructed. Numerical experiments show the effectiveness of the proposed methods.

[1]  Jason Frank,et al.  Linear PDEs and Numerical Methods That Preserve a Multisymplectic Conservation Law , 2006, SIAM J. Sci. Comput..

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

[3]  Songhe Song,et al.  Multi-symplectic wavelet collocation method for the nonlinear Schrödinger equation and the Camassa-Holm equation , 2011, Comput. Phys. Commun..

[4]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[5]  G. Beylkin.  On the representation of operators in bases of compactly supported wavelets , 1992 .

[6]  M. Qin,et al.  MULTI-SYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHR ¨ ODINGER EQUATION , 2001 .

[7]  Songhe Song,et al.  Symplectic wavelet collocation method for Hamiltonian wave equations , 2010, J. Comput. Phys..

[8]  Víctor M. Pérez-García,et al.  Symplectic methods for the nonlinear Schrödinger equation , 1996 .

[9]  Alvaro L. Islas,et al.  Multi-symplectic methods for generalized Schrödinger equations , 2003, Future Gener. Comput. Syst..

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

[11]  Eberly,et al.  One-dimensional model of a negative ion and its interaction with laser fields. , 1993, Physical review. A, Atomic, molecular, and optical physics.

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

[13]  Catherine Sulem,et al.  Numerical simulation of singular solutions to the two‐dimensional cubic schrödinger equation , 1984 .

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

[15]  Numerical studies of the dynamics of multiphoton processes with arbitrary field polarization: Methodological considerations , 1997 .

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

[17]  Jingjing Zhang,et al.  Splitting multisymplectic integrators for Maxwell's equations , 2010, J. Comput. Phys..

[18]  Songhe Song,et al.  Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation , 2010, Comput. Phys. Commun..

[19]  Anne L'Huillier,et al.  Generation of attosecond xuv pulses in strong laser-atom interactions , 1998 .

[20]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[21]  U. Ascher,et al.  Multisymplectic box schemes and the Korteweg{de Vries equation , 2004 .

[22]  J. Frank.  Conservation of wave action under multisymplectic discretizations , 2006 .

[23]  Ignacio Martín Llorente,et al.  A numerical scheme for the simulation of blow-up in the nonlinear Schrödinger equation , 2003, Appl. Math. Comput..

[24]  S. Bertoluzza,et al.  A Wavelet Collocation Method for the Numerical Solution of Partial Differential Equations , 1996 .

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

[26]  Víctor M. Pérez-García,et al.  Numerical methods for the simulation of trapped nonlinear Schrödinger systems , 2003, Appl. Math. Comput..

[27]  D. B. Duncan,et al.  Sympletic Finite Difference Approximations of the Nonlinear Klein--Gordon Equation , 1997 .

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

[29]  Tao Ma,et al.  The multisymplectic numerical method for Gross-Pitaevskii equation , 2008, Comput. Phys. Commun..

[30]  O. Vasilyev,et al.  A Fast Adaptive Wavelet Collocation Algorithm for Multidimensional PDEs , 1997 .

[31]  Chi-Wang Shu,et al.  Local discontinuous Galerkin methods for nonlinear Schrödinger equations , 2005 .

[32]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[33]  C. D. Lin,et al.  Linear-least-squares fitting method for the solution of the time-dependent Schrodinger equation: Applications to atoms in intense laser fields , 2000 .

[34]  Y. Qi,et al.  Recent Progress in Symplectic Algorithms for Use in Quantum Systems , 2007 .