Communications on Pure and Applied Analysis (CPAA)

Discrete Schrödinger equations and dissipative dynamical systems

Pages: 211 - 227, Volume 7, Issue 2, March 2008      doi:10.3934/cpaa.2008.7.211

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Mostafa Abounouh - Universite Cadi Ayyad, Faculte des Sciences et Techniques, Avenue Abdelkrim Khattabi, BP 549, Marrakech, Morocco (email)
H. Al Moatassime - Université Cadi Ayyad, Faculté des sciences et techniques, Gueliz, BP 549 Marrakech, Morocco (email)
J. P. Chehab - Laboratoire de Mathématiques Paul Painlevé, CNRS, UMR 8524, Bât. M2, Université de Lille 1, 59655 Villeneuve d'Ascq cedex, France (email)
S. Dumont - LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex, France (email)
Olivier Goubet - Universite de Picardie Jules Verne, LAMFA UMR 7352, 33 rue Saint-Leu, 80039 Amiens cedex, France (email)

Abstract: We introduce a Crank-Nicolson scheme to study numerically the long-time behavior of solutions to a one dimensional damped forced nonlinear Schrödinger equation. We prove the existence of a smooth global attractor for these discretized equations. We also provide some numerical evidences of this asymptotical smoothing effect.

Keywords:  Finite differences, stability, multilevel decomposition.
Mathematics Subject Classification:  35B41, 35Q55, 65M06.

Received: February 2007;      Revised: July 2007;      Available Online: December 2007.