Manuscript submitted to Website: http: AIMsciences org AIMS' Journals Volume Number Xxxx XXXX pp

mijec - Paul Painleve

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18 pages

English

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Niveau: Supérieur, Doctorat, Bac+8
Manuscript submitted to Website: AIMS' Journals Volume 00, Number 0, Xxxx XXXX pp. 000–000 DISCRETE SCHRODINGER EQUATIONS AND DISSIPATIVE DYNAMICAL SYSTEMS M. Abounouh, H. Al Moatassime Universite Cadi Ayyad, Faculte des sciences et techniques Gueliz, BP 549 Marrakech, Maroc J.-P. Chehab Laboratoire de Mathematiques Paul Painleve CNRS, UMR 8524, Bat. M2 Universite de Lille 1 59655 Villeneuve d'Ascq cedex, France and SIMPAF project, INRIA futurs S. Dumont, O. Goubet LAMFA CNRS UMR 6140 Universite de Picardie Jules Verne 33 rue Saint-Leu 80039 Amiens cedex, France (Communicated by ) Abstract. We introduce a Crank-Nicolson scheme to study numerically the long- time behavior of solutions to a one dimensional damped forced nonlinear Schrodinger 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. 1. Introduction. Weakly damped nonlinear Schrodinger equations provide examples of infinite-dimensional dynamical systems, in the framework described in [18], [10], [17]. For these infinite-dimensional dynamical systems the major issues are: does it exist a global attractor for the dissipative dynamical system under consideration ? does this global attractor has finite Haussdorf and fractal dimension ? is this global attractor regular ? Let us give an overview of the previous results for weakly damped nonlinear Schrodinger equations, that are equations that read ut + ?u + iuxx + i|u|2u = f.

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global attractor

∆t ?

attractor has finite

scheme

crank-nicolson scheme

dynamical system

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Painlevé

Scheme

Crank?Nicolson method

Dynamical system

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Publié par	mijec
Nombre de lectures	23
Langue	English

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ManuscriptsubmittdeoteWsbti:eth:/tpIM/Aiescesncgro.SMIAuoJ’lanrume0sVolmber0,NuxxXX,0xX0.00XXpp00–0

M. Abounouh, H. Al MoatassimeUniversit´eCadiAyyad,Faculte´dessciencesettechniquesGueliz, BP 549 Marrakech, MarocJ.-P. ChehabLaboratoiredeMath´ematiquesPaulPainleve´CNRS,UMR8524,Bˆat.M2Universit´edeLille159655 Villeneuve d’Ascq cedex, FranceandSIMPAFproject,INRIA futursS. Dumont, O. GoubetLAMFA CNRS UMR 6140Universite´dePicardieJulesVerne33 rue Saint-Leu 80039 Amiens cedex, France(Communicated by )

¨DISCRETE SCHRODINGER EQUATIONS AND DISSIPATIVEDYNAMICAL SYSTEMS

Abstract.We introduce a Crank-Nicolson scheme to study numerically the long-timebehaviorofsolutionstoaonedimensionaldampedforcednonlinearSchro¨dingerequation. We prove the existence of asmoothglobal attractor for these discretizedequations. We also provide some numerical evidences of this asymptotical smoothingeﬀect.1.Introduction.WeaklydampednonlinearSchro¨dingerequationsprovideexamplesof inﬁnite-dimensional dynamical systems, in the framework described in [18], [10], [17].For these inﬁnite-dimensional dynamical systems the major issues are: does it exist aglobal attractorfor the dissipative dynamical system under consideration ? does thisglobal attractor has ﬁnite Haussdorf and fractal dimension ? is this global attractorregular?Let us give an overview of the previous results for weakly damped nonlinearSchro¨dingerequations,thatareequationsthatreadut+αu+iuxx+i|u|2u=f.(1)Here the unknownu(t, x) mapsRt×TxintoC. We mean thatuis a periodic functionwith respect tox. Actuallyα >0 is the damping parameter and the external forcef,that does not depend tot, belongs toL2(T). The pioneering work [6] proved the existenceof a ﬁnite dimensionalweakglobal attractorAfor dissipative NLS. By weak attractorwe mean that the attractor attracts the trajectories for the weak topology in the Hilbert2000Mathematics Subject Classiﬁcation.35B41, 35Q55, 65M06.