Numerische Mathematik

, Volume 136, Issue 1, pp 315–342 | Cite as

On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit II. Analytic regularity

  • Rémi CarlesEmail author
  • Clément Gallo


We consider the time discretization based on Lie-Trotter splitting, for the nonlinear Schrödinger equation, in the semi-classical limit, with initial data under the form of WKB states. We show that both the exact and the numerical solutions keep a WKB structure, on a time interval independent of the Planck constant. We prove error estimates, which show that the quadratic observables can be computed with a time step independent of the Planck constant. The functional framework is based on time-dependent analytic spaces, in order to overcome a previously encountered loss of regularity phenomenon.


  1. 1.
    Alazard, T., Carles, R.: Supercritical geometric optics for nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 194, 315–347 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alinhac, S., Gérard, P.: Pseudo-differential operators and the Nash-Moser theorem, vol. 82 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI. Translated from the 1991 French original by Stephen S. Wilson (2007)Google Scholar
  3. 3.
    Bao, W., Jin, S., Markowich, P.A.: On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175, 487–524 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bao, W., Jin, S., Markowich, P.A.: Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes. SIAM J. Sci. Comput. 25, 27–64 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benettin, G., Giorgilli, A.: On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys. 74, 1117–1143 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Besse, C., Bidégaray, B., Descombes, S.: Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40, 26–40 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Besse, C., Carles, R., Méhats, F.: An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit. Multiscale Model. Simul. 11, 1228–1260 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carles, R.: Semi-classical analysis for nonlinear Schrödinger equations. World Scientific Publishing Co Pte. Ltd., Hackensack (2008)CrossRefzbMATHGoogle Scholar
  9. 9.
    Carles, R.: On Fourier time-splitting methods for nonlinear Schrödinger equations in the semi-classical limit. SIAM J. Numer. Anal. 51, 3232–3258 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Castella, F., Chartier, P., Méhats, F., Murua, A.: Stroboscopic averaging for the nonlinear Schrödinger equation. Found. Comput. Math. 15, 519–559 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cazenave, T., Haraux, A.: An introduction to semilinear evolution equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised by the authors (1998)Google Scholar
  12. 12.
    Chemin, J.-Y.: Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray. In: Actes des Journées Mathématiques à la Mémoire de Jean Leray, vol. 9 of Sémin. Congr., Soc. Math. pp. 99–123. France, Paris (2004)Google Scholar
  13. 13.
    Chiron, D., Rousset, F.: Geometric optics and boundary layers for nonlinear Schrödinger equations. Commun. Math. Phys. 288, 503–546 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Descombes, S., Thalhammer, M.: An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime. BIT 50, 729–749 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Descombes, S., Thalhammer, M.: The Lie-Trotter splitting for nonlinear evolutionary problems with critical parameters. A compact local error representation and application to nonlinear Schrödinger equations in the semi-classical regime. IMA J. Numer. Anal. 33, 722–745 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Faou, E.: Geometric numerical integration and Schrödinger equations. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2012)CrossRefzbMATHGoogle Scholar
  17. 17.
    Faou, E., Grébert, B.: Hamiltonian interpolation of splitting approximations for nonlinear PDEs. Found. Comput. Math. 11, 381–415 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gérard, P.: Remarques sur l’analyse semi-classique de l’équation de Schrödinger non linéaire, In: Séminaire sur les Équations aux Dérivées Partielles, 1992–1993, École; Polytech., Palaiseau, pp. Exp. No. XIII, 13 (1993).
  19. 19.
    Ginibre, J., Velo, G.: Long range scattering and modified wave operators for some Hartree type equations. III. Gevrey spaces and low dimensions. J. Differ. Equ. 175, 415–501 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Grenier, E.: Semiclassical limit of the nonlinear Schrödinger equation in small time. Proc. Am. Math. Soc. 126, 523–530 (1998)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hairer, E., Lubich, C.: Oscillations over long times in numerical Hamiltonian systems. In: Highly oscillatory problems, vol. 366 of London Math. Soc. Lecture Note Ser., pp. 1–24. Cambridge Univ. Press, Cambridge (2009)Google Scholar
  22. 22.
    Hairer, E., Lubich, C.: Modulated Fourier expansions for continuous and discrete oscillatory systems. In: Foundations of computational mathematics, Budapest, 2011, vol. 403 of London Math. Soc. Lecture Note Ser. pp. 113–128. Cambridge Univ. Press, Cambridge (2013)Google Scholar
  23. 23.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration, vol. 31 of Springer Series in Computational Mathematics, Springer, Heidelberg, 2010. Structure-preserving algorithms for ordinary differential equations. Reprint of the second (2006) edition (2010)Google Scholar
  24. 24.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations. I, vol. 8 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (1993) (Nonstiff problems) Google Scholar
  25. 25.
    Holden, H., Lubich, C., Risebro, N.H.: Operator splitting for partial differential equations with Burgers nonlinearity. Math. Comp. 82, 173–185 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lubich, C.: On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77, 2141–2153 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mouhot, C., Villani, C.: On Landau damping. Acta Math. 207, 29–201 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Thomann, L.: Instabilities for supercritical Schrödinger equations in analytic manifolds. J. Differ. Equ. 245, 249–280 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut Montpelliérain Alexander GrothendieckCNRS & Univ. MontpellierMontpellierFrance

Personalised recommendations