Advertisement

A Meshfree Splitting Method for Soliton Dynamics in Nonlinear Schrödinger Equations

  • Marco Caliari
  • Alexander Ostermann
  • Stefan Rainer
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 89)

Abstract

A new method for the numerical simulation of the so-called soliton dynamics arising in a nonlinear Schrödinger equation in the semi-classical regime is proposed. For the time discretization a classical fourth-order splitting method is used. For the spatial discretization, however, a meshfree method is employed in contrast to the usual choice of (pseudo) spectral methods. This approach allows one to keep the degrees of freedom almost constant as the semi-classical parameter \(\epsilon \) becomes small. This behavior is confirmed by numerical experiments.

Keywords

Meshfree discretization Splitting methods Nonlinear Schrödinger equations Soliton dynamics Semi-classical regime 

Notes

Acknowledgements

The work of Stefan Rainer was partially supported by the Tiroler Wissenschaftsfond grant UNI-0404/880.

References

  1. 1.
    A.H. Al-Mohy, N.J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33, 488–511 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    W. Bao, Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 25 1674–1697 (2004)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    W. Bao, J. Shen, A fourth-order time-splitting Laguerre–Hermite pseudospectral method for Bose–Einstein condensates. SIAM J. Sci. Comput. 26 2010–2028 (2005)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    W. Bao, D. Jaksch, P. Markowich, Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation. J. Comp. Phys. 187 318–342 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    L. Bergamaschi, M. Caliari, M. Vianello, Interpolating discrete advection-diffusion propagators at Leja sequences. J. Comput. Appl. Math. 172, 79–99 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    C. Besse, B. Bidégaray, S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40, 26–40 (2002)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    S. Blanes, P.C. Moan, Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. J. Comput. Appl. Math. 142 313–330 (2002)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    J. Bronski, R. Jerrard, Soliton dynamics in a potential. Math. Res. Lett. 7, 329–342 (2000)MathSciNetMATHGoogle Scholar
  9. 9.
    M.D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge University Press, Cambridge, 2003)MATHGoogle Scholar
  10. 10.
    M. Caliari, M. Squassina, Numerical computation of soliton dynamics for NLS equations in a driving potential. Electron. J. Differ. Equ. 89, 1–12 (2010)MathSciNetGoogle Scholar
  11. 11.
    M. Caliari, C. Neuhauser, M. Thalhammer, High-order time-splitting Hermite and Fourier spectral methods for the Gross–Pitaevskii equation. J. Comput. Phys. 228, 822–832 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    M. Caliari, A. Ostermann, S. Rainer, M. Thalhammer, A minimisation approach for computing the ground state of Gross–Pitaevskii systems. J. Comput. Phys. 228, pp. 349–360 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    M. Caliari, A. Ostermann, S. Rainer, Meshfree Exponential Integrators, to appear in SIAM J. Sci. Comput. (2011)Google Scholar
  14. 14.
    M. Caliari, A. Ostermann, S. Rainer, Meshfree integrators. Oberwolfach Rep. 8, 883–885 (2011)Google Scholar
  15. 15.
    G.E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific, Hackensack, 2007)MATHGoogle Scholar
  16. 16.
    M. Hochbruck, A. Ostermann, Exponential integrators. Acta Numer. 19, 209–286 (2010)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis (European Mathematical Society, Zürich, 2008)MATHCrossRefGoogle Scholar
  18. 18.
    R. Schaback, Creating surfaces from scattered data using radial basis functions, in Mathematical Methods for Curves and Surfaces, ed. by M. Dæhlen, T. Lyche, L.L. Schumaker (Vanderbilt University Press, Nashville, 1995), pp. 477–496Google Scholar
  19. 19.
    H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    H. Wendland, Scattered Data Approximation (Cambridge University Press, Cambridge, 2005)MATHGoogle Scholar
  21. 21.
    Z. Wu, Compactly supported positive definite radial functions. Adv. Comput. Math. 4 283–292 (1995)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marco Caliari
    • 1
  • Alexander Ostermann
    • 2
  • Stefan Rainer
    • 2
  1. 1.Dipartimento di InformaticaUniversità di VeronaVeronaItaly
  2. 2.Institut für MathematikUniversität InnsbruckInnsbruckAustria

Personalised recommendations