Uniform attractor for  non-autonomous nonlinear Schrödinger equation

Olivier Goubet; Wided Kechiche

doi:10.3934/cpaa.2011.10.639

Article Contents

2011, Volume 10, Issue 2: 639-651. Doi: 10.3934/cpaa.2011.10.639

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Uniform attractor for non-autonomous nonlinear Schrödinger equation

Olivier Goubet^1, and
Wided Kechiche^2,

1.
Universite de Picardie Jules Verne, LAMFA UMR 7352, 33 rue Saint-Leu, 80039 Amiens cedex
2.
Département de Mathématiques, Faculté des Sciences de Monastir, Av. de l'environement, 5000 Monastir

Received: February 28, 2010

Revised: August 31, 2010

Published: February 2011

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Abstract

We consider a weakly coupled system of nonlinear Schrödinger equations which models a Bose Einstein condensate with an impurity. The first equation is dissipative, while the second one is conservative. We consider this dynamical system into the framework of non-autonomous dynamical systems, the solution to the conservative equation being the symbol of the semi-process. We prove that the first equation possesses a uniform attractor, which attracts the solutions for the weak topology of the underlying energy space. We then study the limit of this attractor when the coupling parameter converges towards $0$.

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Mathematics Subject Classification: 35Q55, 35B41, 37L30.

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