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Advances in Computational Mathematics

, Volume 42, Issue 2, pp 395–423 | Cite as

Multidomain spectral method for Schrödinger equations

  • Mira Birem
  • Christian KleinEmail author
Article

Abstract

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.

Keywords

Schrödinger equation Nonlinear Schrödinger equation Spectral methods Transparent boundary conditions Perfectly matched layers Rogue waves 

Mathematics Subject Classifications (2010)

65M70 35Q41 35Q55 

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References

  1. 1.
    Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Schädle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Comm. Comput. Phys. 4, 729–796 (2008)Google Scholar
  2. 2.
    Antoine, X., Bao, W., Besse, C.: Computational methods for the dynamics of the nonlinear Schrödinger/Gross- Pitaevskii equations. Comput. Phys. Commun. 184(12), 2621–2633 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Antoine, X., Besse, C.: Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation. J. Comput. Phys. 188(1), 157–175 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential. J. Comput. Phys. 228(2), 312–335 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bailung, H., Sharma, S.K., Nakamura, Y.: Observation of Peregrine solitons in a multicomponent plasma with negative ions. Phys. Rev. Lett. 255005, 107 (2011)Google Scholar
  6. 6.
    Baskakov, V.A., Popov, A.V.: Implementation of transparent boundaries for numerical solution of the Schrödinger equation. Wave Motion 14(2), 123–128 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bérenger, J.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Besse, C.: A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42(3), 934–952 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boutet de Monvel, A., Fokas, A.S., Shepelsky, D.: Analysis of the global relation for the nonlinear Schrödinger equation on the half-line. Lett. Math. Phys. 65, 199–212 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Burgnies, L., Vanbésien, O., Lippens, D.: Transient analysis of ballistic transport in stublike quantum waveguides. Appl. Phys. Lett. 71, 803–805 (1997)CrossRefGoogle Scholar
  11. 11.
    Chabchoub, A., Hoffmann, N.P., Akhmediev, N.: Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106, 204502 (2011)CrossRefGoogle Scholar
  12. 12.
    Chabchoub, A., Hoffmann, N., Onorato, M., Akhmediev, N.: Super rogue waves: observation of a higher-order breather in water waves. Phys. Rev. X 2, 011015 (2012)Google Scholar
  13. 13.
    Chabchoub, A., Hoffmann, N., Branger, H., Kharif, C., Akhmediev, N.: Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model. Phys. Fluids, 25 (2013). doi: 10.1063/1.4824706
  14. 14.
    Calini, A., Schober, C.M.: Nat. Hazards Earth Syst. Sci. 14, 1431–1440 (2014)CrossRefGoogle Scholar
  15. 15.
    Caplan, R.M., Carretero-Gonzlez, R.: A modulus-squared Dirichlet boundary condition for time-dependent complex partial differential equations and its application to the nonlinear Schrödinger equation. SIAM J. Sci. Comput. 36(1), A1–A19 (2014)CrossRefzbMATHGoogle Scholar
  16. 16.
    Claerbout, J.F.: Coarse grid calculation of waves in inhomogeneous media with application to delineation of complicated seismic structure. Geophysics 35, 407–418 (1970)CrossRefGoogle Scholar
  17. 17.
    Dubard, P., Gaillard, P., Klein, C., Matveev, V.B.: On multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation. Eur. Phys. J. Special Topics 185, 247–258 (2010)CrossRefGoogle Scholar
  18. 18.
    Duque, J.: Solving time-dependent equations of Schrödinger-type using mapped infinite elements. Int. J. Mod. Phys. C 16(2), 309–316 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Frauendiener, J.: Calculating initial data for the conformal Einstein equations by pseudo-spectral methods. J. Comput. Appl. Math. 109, 475–491 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hagstrom, T.: Radiation boundary conditions for the numerical simulation of waves. Acta Numerica 8, 47–106 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Johnson, S.G.: Notes on Perfectly Matched Layers (PMLs), http://math.mit.edu/stevenj/18.369/pml.pdf. (2010)
  22. 22.
    Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N., Dudley, J.M.: The Peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790–795 (2010)CrossRefGoogle Scholar
  23. 23.
    Klein, C.: Fourth order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equation. ETNA 29, 116–135 (2008)zbMATHGoogle Scholar
  24. 24.
    Klein, C., Roidot, K.: Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations. SIAM J. Sci. Comput. 33(6), 3333–3356 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Klein, C., Peter, R.: Numerical study of blow-up in solutions to generalized Korteweg-de Vries equations. Physica D, 52–78 (2015)Google Scholar
  26. 26.
    Ladouceur, F.: Boundaryless beam propagation. Opt. Lett. 21, 4–5 (1996)CrossRefGoogle Scholar
  27. 27.
    Lanczos, C.: Trigonometric interpolation of empirical and analytic functions. J. Math. Phys. 17, 123–199 (1938)CrossRefzbMATHGoogle Scholar
  28. 28.
    Levy, M.F.: Parabolic equation models for electromagnetic wave propagation. IEE Electromagn. Waves Series, 45 (2000)Google Scholar
  29. 29.
    McCurdy, C.W., Homer, D.A., Resigno, T.N.: Time dependent approach to collisional ionization using exterior complex scaling. Phys. Rev. A 65, 042714 (2002)CrossRefGoogle Scholar
  30. 30.
    Nissen, A., Kreiss, G.: An Optimized Perfectly Matched Layer for the Schrödinger Equation. Rapport technique, Department of Information Technology. Uppsala University (2009)Google Scholar
  31. 31.
    Grosch, C.E., Orszag, S.A.: Numerical solution of problems in unbounded regions: coordinate transforms. J. Comput. Phys. 25, 273–296 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. B 25, 16–43 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Schmidt, F., Deuflhard, P.: Discrete transparent boundary conditions for the numerical solution of Fresnel’s equation. Comput. Math. Appl 29, 53–76 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Tappert, F.D.: The parabolic approximation method. In: Keller, J.B., Papadakis, J.S. (eds.) Wave Propagation and Underwater Acoustics, Lecture Notes in Physics 70, pp. 224–287. Springer, New York (1977)CrossRefGoogle Scholar
  35. 35.
    Tsynkov, S.V.: Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math. 27(4), 465–532 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  37. 37.
  38. 38.
    Vaibhav, V.: Artificial boundary conditions for certain evolution PDEs with cubic nonlinearity for non-compactly supported initial data. J. Comput. Phys. 230(8), 3205–3229 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
  40. 40.
    Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self- modulation of waves in nonlinear media. Sov. Phys. JETP 34(1), 62–69 (1972). translated from Zh. Eksp. Teor. Fiz. 1, 118–134 (1971)MathSciNetGoogle Scholar
  41. 41.
    Zheng, C.: A perfectly matched layer approach to the nonlinear Schrödinger wave equations. J. Comput. Phys. 227, 537–556 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zheng, C.: Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations. J. Comput. Phys. 215, 552–565 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institut de Mathématiques de BourgogneUniversité de BourgogneDijon CedexFrance

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