Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-16T12:56:22.433Z Has data issue: false hasContentIssue false

Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

Published online by Cambridge University Press:  03 June 2015

Xu Qian*
Department of Mathematics and Systems Science, and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, P.R. China Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
Yaming Chen
Department of Mathematics and Systems Science, and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, P.R. China School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK
Songhe Song
Department of Mathematics and Systems Science, and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, P.R. China
Get access


In this paper, we propose a wavelet collocation splitting (WCS) method, and a Fourier pseudospectral splitting (FPSS) method as comparison, for solving one-dimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics. The two methods can preserve the intrinsic properties of original problems as much as possible. The splitting technique increases the computational efficiency. Meanwhile, the error estimation and some conservative properties are investigated. It is proved to preserve the charge conservation exactly. The global energy and momentum conservation laws can be preserved under several conditions. Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

Research Article
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1]Infeldm, E., Nonlinear Waves: From Hydrodynamics to Plasma Theory, Advances in Nonlinear Waves, Pitman, Boston, 1984.Google Scholar
[2]Nore, C., Abid, A. and Brachet, M., Small-Scale Structures in Three-dimensional Hydrodynamics and Magnetohyrodynamic Turbulence, Springer, Berlin, 1996.Google Scholar
[3]Agrawal, G.P., Nonlinear Fiber Optics, 3rd ed., Academic Press, San Diego, 2001.Google Scholar
[4]Sulem, C. and Sulem, P.L., The Nonlinear Schrödinger equation: Self-focusing and Wave Collapse, Springer, New York, 1999.Google Scholar
[5]Bao, W., Jaksch, D. and Markowich, P.A., Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys. 187 (2003), 318342.Google Scholar
[6]Stenflo, L. and Yu, M.Y., Nonlinear wave modulation in a cylindrical plasma, IEEE Trans. Plasma Sci. 25 (1997), 11551157.CrossRefGoogle Scholar
[7]Bao, W. and Shen, J., A fourth-order time-splitting Lagurre-Hermite pseudo-spectral method for Bose-Einstein condensates, SIAM. Sci. Comput. 26 (2005), 20102028.Google Scholar
[8]Lappas, D.G. and L’Huillier, A., Generation of attosecond XUV pulses in strong laser-atom interactions, Phys. Rev. A 58 (1998), 41404146.Google Scholar
[9]Zhou, X.X. and Lin, C.D., Linear-least-squares fitting method for the solution of the time-dependent Schrödinger equation: Applications to atoms in intense laser fields, Phys. Rev. A 61 (2000) 053411.CrossRefGoogle Scholar
[10]Liu, X.S., Qi, Y.Y., He, J.F. and Ding, P.Z., Recent progress in symplectic algorithms for use in quantum system, Commun. Comput. Phys. 2 (2007), 153.Google Scholar
[11]Fung, Y.C., Biodynamics: Circulation, Springer-Verlag, New York, 1981.Google Scholar
[12]Tian, B., Gao, Y. and Zhu, H., Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: Variable-coefficient bilinear form, Bäcklund transformation, brightons and symbolic computation, Phys. Lett. A 366 (2007), 223229.Google Scholar
[13]Serkin, V.N. and Hasegawa, A., Novel soliton solutions of the nonlinear Schrödinger equation model, Phys. Rev. Lett. 85 (2000), 45024505.Google Scholar
[14]Zhu, H.J., Chen, Y.M., Song, S.H. and Hu, H.Y., Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations, Appl. Numer. Math. 61 (2011), 308321.Google Scholar
[15]Delfour, M., Fortin, M. and Payre, G., Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys. 44 (1981), 277288.Google Scholar
[16]Meng, Q.J., Yin, L.P., Jin, X.Q. and Qiao, F.L., Numerical solutions of coupled nonlinear Schrödinger equations by orthogonal spline collocation method, Commun. Comput. Phys., 12 (2012), 13921416.Google Scholar
[17]Chang, Q.S., Jia, E.H. and Sun, W., Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys. 148 (1999), 397415.Google Scholar
[18]Xie, S., Li, G. and Yi, S., Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödinger equation, Comput. Methods Appl. Mech. Engrg. 198 (2009), 10521060.Google Scholar
[19]Dehghan, M. and Taleei, A., A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Comput. Phys. Commun. 181 (2010), 4351.Google Scholar
[20]Chen, J.B., Qin, M.Z. and Tang, Y.F., Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl. 43 (2002), 10951106.CrossRefGoogle Scholar
[21]Chen, J.B., A multisymplectic integrator for the periodic nonlinear Schrödinger equation, Appl. Math. Comput. 170 (2005), 13941417.Google Scholar
[22]Hong, J.L. and Kong, L.H., Novel multi-symplectic integrators for nonlinear fourth-order Schrödinger equation with trapped term, Commun. Comput. Phys. 7 (2010), 613630.Google Scholar
[23]Chen, Y.M., Zhu, H.J. and Song, S.H., Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation, Comput. Phys. Commun. 181 (2010), 12311241.Google Scholar
[24]Qian, X., Chen, Y.M., Gao, E. and Song, S.H., Multi-symplectic wavelet splitting method for the strongly coupled Schrödinger system, Chin. Phys. B 21 (2012), 120202.Google Scholar
[25]Cai, J.X., Wang, Y.S. and Liang, H., Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrödinger system, J. Comput. Phys. 239 (2013), 3050.Google Scholar
[26]Hong, J.L., Liu, Y., Munthe-Kaas, H. and Zanna, A., Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients, Appl. Numer. Math. 56 (2006), 814843.Google Scholar
[27]Gotay, M.J., A multisymplectic framework for classical field theory and the calculus of variations. I: Covariant Hamiltonian formalism, Francaviglia, M. (ed.), Mechanics, analysis and geometry: 200 years after Lagrange. Amsterdam etc.: North-Holland (1991), 203235.Google Scholar
[28]Kouranbaeva, S. and Shkoller, S., A variational approach to second-order multiisymplectic field theory, J. Geom. Phys. 35 (2000), 333366.Google Scholar
[29]Hong, J.L. and Sun, Y.J., Generating functions of multi-symplectic RK methods via DW Hamilton-Jacobi equations, Numer. Math. 110 (2008), 491519.Google Scholar
[30]Zhu, H.J., Tang, L.Y., Song, S.H., Tang, Y.F. and Wang, D.S., Symplectic wavelet collocation method for Hamiltonian wave equations, J. Comput. Phys. 229 (2010), 25502572.CrossRefGoogle Scholar
[31]Strang, G., On the construction and comparison of difference scheme, SIAM J. Numer. Anal. 5 (1968), 506517.Google Scholar
[32]Wang, H.Q., Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations, Appl. Math. Comput. 170 (2005), 1735.Google Scholar
[33]Zhu, H.J., Song, S.H. and Tang, Y.F, Multi-symplectic wavelet collocation method for the nonlinear Schrödinger equation and the Camassa-Holm equation, Comput. Phys. Commun. 182 (2011), 616627.CrossRefGoogle Scholar
[34]Bertoluzza, S. and Naldi, G., A wavelet collocation method for the numerical solution of partial differential equations, Appl. Comput. Harm. Anal. 3 (1996), 19.Google Scholar
[35]Zhou, Y.L., Application of Discrete Functional Analysis to the Finite Difference Method, International Academic Press, Hong Kong, 1990.Google Scholar
[36]Qian, X., Song, S.H., Gao, E. and Li, W.B., Explicit multi-symplectic method for the Zakharov-Kuznetsov equation, Chin. Phys. B 21 (2012), 070206.Google Scholar