Skip to main content
SpringerLink
Log in

One-stage exponential integrators for nonlinear Schrödinger equations over long times

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

Near-conservation over long times of the actions, of the energy, of the mass and of the momentum along the numerical solution of the cubic Schrödinger equation with small initial data is shown. Spectral discretization in space and one-stage exponential integrators in time are used. The proofs use modulated Fourier expansions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Bambusi, D., Grébert, B.: Birkhoff normal form for partial differential equations with tame modulus. Duke Math. J. 135(3), 507–567 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. Math. (2) 148(2), 363–439 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cano, B.: Conserved quantities of some Hamiltonian wave equations after full discretization. Numer. Math. 103(2), 197–223 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cano, B., González-Pachón, A.: Exponential time integration of solitary waves of cubic Schrödinger equations. Preprint (2011)

  5. Cano, B., González-Pachón, A.: Exponential methods for the time integration of Schrödinger equation. AIP Conf. Proc. 1281(1), 1821–1823 (2010)

    Article  Google Scholar 

  6. Castella, F., Dujardin, G.: Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation. M2AN Math. Model. Numer. Anal. 43(4), 651–676 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Celledoni, E., Cohen, D., Owren, B.: Symmetric exponential integrators with an application to the cubic Schrödinger equation. Found. Comput. Math. 8(3), 303–317 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cohen, D., Hairer, E., Lubich, C.: Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations. Numer. Math. 110(2), 113–143 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Debussche, A., Faou, E.: Modified energy for split-step methods applied to the linear Schrödinger equation. SIAM J. Numer. Anal. 47(5), 3705–3719 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dujardin, G.: Exponential Runge–Kutta methods for the Schrödinger equation. Appl. Numer. Math. 59(8), 1839–1857 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dujardin, G., Faou, E.: Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential. Numer. Math. 108(2), 223–262 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. Ann. Math. (2) 172(1), 371–435 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Faou, E., Grébert, B.: Resonances in long time integration of semi linear Hamilonian PDEs. Preprint (2009). http://www.irisa.fr/ipso/perso/faou

  14. Faou, E., Grébert, B.: Hamiltonian interpolation of splitting approximations for nonlinear PDEs. Found. Comput. Math. 11(4), 381–415 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Faou, E., Grébert, B., Paturel, E.: Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. I. Finite-dimensional discretization. Numer. Math. 114(3), 429–458 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Faou, E., Grébert, B., Paturel, E.: Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. II. Abstract splitting. Numer. Math. 114(3), 459–490 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gauckler, L.: Long-time analysis of Hamiltonian partial differential equations and their discretizations. Dissertation (doctoral thesis), Universität Tübingen (2010). http://nbn-resolving.de/urn:nbn:de:bsz:21-opus-47540

  18. Gauckler, L., Lubich, C.: Nonlinear Schrödinger equations and their spectral semi-discretizations over long times. Found. Comput. Math. 10(2), 141–169 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gauckler, L., Lubich, C.: Splitting integrators for nonlinear Schrödinger equations over long times. Found. Comput. Math. 10(3), 275–302 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. In: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin (2006)

    Google Scholar 

  21. Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19(5), 1552–1574 (1998) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lawson, J.D.: Generalized Runge–Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372–380 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  24. Minchev, B., Wright, W.M.: A review of exponential integrators for semilinear problems. Tech. rep. 2/05, Department of Mathematical Sciences, NTNU, Norway (2005). http://www.math.ntnu.no/preprint/

  25. Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol. 139. Springer, New York (1999)

    MATH  Google Scholar 

  26. Verwer, J.G., van Loon, M.: An evaluation of explicit pseudo-steady-state approximation schemes for stiff ODE systems from chemical kinetics. J. Comput. Phys. 113(2), 347–352 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We greatly appreciate the referees’ comments on an earlier version. We would like to thank Christian Lubich for interesting discussions.

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Basel, 4051, Basel, Switzerland

    David Cohen

  2. Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Ludwig Gauckler

Authors
  1. David Cohen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ludwig Gauckler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Cohen.

Additional information

Communicated by Mechthild Thalhammer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cohen, D., Gauckler, L. One-stage exponential integrators for nonlinear Schrödinger equations over long times. Bit Numer Math 52, 877–903 (2012). https://doi.org/10.1007/s10543-012-0385-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-012-0385-1

Keywords

Mathematics Subject Classification

Search

Navigation