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PLANE WAVE STABILITY OF THE SPLIT-STEP FOURIER METHOD FOR THE NONLINEAR SCHRÖDINGER EQUATION

Part of: Numerical problems in dynamical systems

Published online by Cambridge University Press:  10 April 2014

ERWAN FAOU ,
LUDWIG GAUCKLER  and
CHRISTIAN LUBICH
ERWAN FAOU
Affiliation:
INRIA and ENS Cachan Bretagne, Avenue Robert Schumann, F-35170 Bruz, [email protected] Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, F-75230 Paris Cedex 05, France
LUDWIG GAUCKLER
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, [email protected]
CHRISTIAN LUBICH
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, [email protected]

Abstract

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Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a nonresonance condition. They can both be verified in the case of a spatially constant plane wave if the time step-size is restricted by a Courant–Friedrichs–Lewy condition (CFL condition). The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.

MSC classification

Secondary: 65P10: Hamiltonian systems including symplectic integrators 65P40: Nonlinear stabilities
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e5
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

References

Agrawal, G. P., Nonlinear Fiber Optics, 5th ed. (Academic Press, 2013).Google Scholar
Bambusi, D., ‘On long time stability in Hamiltonian perturbations of nonresonant linear PDEs’, Nonlinearity 12 (1999), 823–850.Google Scholar
Bambusi, D. and Grébert, B., ‘Birkhoff normal form for partial differential equations with tame modulus’, Duke Math. J. 135 (2006), 507–567.Google Scholar
Cano, B. and González-Pachón, A., Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation, Preprint, 2013. http://hermite.mac.cie.uva.es/bego/cgp3.pdf.Google Scholar
Cohen, D., Hairer, E. and Lubich, Ch., ‘Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions’, Arch. Ration. Mech. Anal. 187 (2008), 341–368.Google Scholar
Dahlby, M. and Owren, B., ‘Plane wave stability of some conservative schemes for the cubic Schrödinger equation’, M2AN Math. Model. Numer. Anal. 43 (2009), 677–687.Google Scholar
Faou, E., Geometric Numerical Integration and Schrödinger Equations, Zurich Lectures in Advanced Mathematics (European Mathematical Society (EMS), Zürich, 2012).Google Scholar
Faou, E., Gauckler, L. and Lubich, Ch., ‘Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equations on a torus’, Comm. Partial Differential Equations 38 (2013), 1123–1140.Google Scholar
Faou, E., Grébert, B. and Paturel, E., ‘Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. I. Finite-dimensional discretization’, Numer. Math. 114 (2010), 429–458.Google Scholar
Faou, E., Grébert, B. and Paturel, E., ‘Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. II. Abstract splitting’, Numer. Math. 114 (2010), 459–490.Google Scholar
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.Google Scholar
Gauckler, L., Hairer, E. and Lubich, Ch., ‘Energy separation in oscillatory Hamiltonian systems without any non-resonance condition’, Comm. Math. Phys. 321 (2013), 803–815.Google Scholar
Gauckler, L. and Lubich, Ch., ‘Splitting integrators for nonlinear Schrödinger equations over long times’, Found. Comput. Math. 10 (2010), 275–302.Google Scholar
Gauckler, L., Hairer, E., Lubich, Ch. and Weiss, D., ‘Metastable energy strata in weakly nonlinear wave equations’, Comm. Partial Differential Equations 37 (2012), 1391–1413.Google Scholar
Hairer, E. and Lubich, Ch., ‘Long-time energy conservation of numerical methods for oscillatory differential equations’, SIAM J. Numer. Anal. 38 (2000), 414–441. (electronic).Google Scholar
Hairer, E. and Lubich, Ch., ‘On the energy distribution in Fermi–Pasta–Ulam lattices’, Arch. Ration. Mech. Anal. 205 (2012), 993–1029.Google Scholar
Hairer, E., Lubich, Ch. and Wanner, G., Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed, Springer Series in Computational Mathematics vol. 31 (Springer-Verlag, Berlin, 2006).Google Scholar
Hani, Z., ‘Long-time instability and unbounded Sobolev orbits for some periodic nonlinear Schrödinger equations’, Arch. Ration. Mech. Anal. 211 (2014), 929–964.Google Scholar
Hardin, R. H. and Tappert, F. D., ‘Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations’, SIAM Rev. 15 (1973), 423.Google Scholar
Jin, S., Markowich, P. and Sparber, Ch., ‘Mathematical and computational methods for semiclassical Schrödinger equations’, Acta Numer. 20 (2011), 121–209.Google Scholar
Khanamiryan, M., Nevanlinna, O. and Vesanen, T., Long-term behavior of the numerical solution of the cubic nonlinear Schrödinger equation using Strang splitting method, Preprint, 2012. http://www.damtp.cam.ac.uk/user/na/people/Marianna/papers/NLS.pdf.Google Scholar
Lakoba, T. I., ‘Instability of the split-step method for a signal with nonzero central frequency’, J. Opt. Soc. Am. B 30 (2013), 3260–3271.Google Scholar
Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical Recipes. The Art of Scientific Computing, 3rd ed. (Cambridge University Press, Cambridge, 2007).Google Scholar
Taha, T. R. and Ablowitz, M. J., ‘Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation’, J. Comput. Phys. 55 (1984), 203–230.Google Scholar
Weideman, J. A. C. and Herbst, B. M., ‘Split-step methods for the solution of the nonlinear Schrödinger equation’, SIAM J. Numer. Anal. 23 (1986), 485–507.CrossRefGoogle Scholar