American Institute of Mathematical Sciences

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March  2008, 7(2): 211-227. doi: 10.3934/cpaa.2008.7.211

Discrete Schrödinger equations and dissipative dynamical systems

Mostafa Abounouh 1, , H. Al Moatassime 2, , J. P. Chehab 3, , S. Dumont 4, and  Olivier Goubet 5,

1. 

Universite Cadi Ayyad, Faculte des Sciences et Techniques, Avenue Abdelkrim Khattabi, BP 549, Marrakech
2. 

Université Cadi Ayyad, Faculté des sciences et techniques, Gueliz, BP 549 Marrakech, Morocco
3. 

Laboratoire de Mathématiques Paul Painlevé, CNRS, UMR 8524, Bât. M2, Université de Lille 1, 59655 Villeneuve d'Ascq cedex, France
4. 

LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex, France
5. 

Universite de Picardie Jules Verne, LAMFA UMR 7352, 33 rue Saint-Leu, 80039 Amiens cedex

Received  February 2007 Revised  July 2007 Published  December 2007

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, multilevel decomposition., stability.
Mathematics Subject Classification: 35B41, 35Q55, 65M0.
Citation: Mostafa Abounouh, H. Al Moatassime, J. P. Chehab, S. Dumont, Olivier Goubet. Discrete Schrödinger equations and dissipative dynamical systems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 211-227. doi: 10.3934/cpaa.2008.7.211
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