Uniform attractor for non-autonomous nonlinear Schrödinger equation

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March 2011, 10(2): 639-651. doi: 10.3934/cpaa.2011.10.639

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, Tunisia

Received March 2010 Revised September 2010 Published December 2010

Abstract
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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 $0$.

Keywords: upper semicontinuity, non-autonomous dynamical systems., Uniform attractor, nonlinear Schrodinger equations.

Mathematics Subject Classification: 35Q55, 35B41, 37L3.

Citation: Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639

References:

[1]	N. Akroune, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on $\mathbb R$, Appl. Math. Lett, 12 (1999), 45-48. doi: doi:10.1016/S0893-9659(98)00170-0.
[2]	A. Babin and M. Vishik, "Attractors of Evolution Equations," Nauka, Moscow 1989; English transl. Stud. Math. Appl., vol 25, North Holland, Amsterdam 1992.
[3]	J. Ball, Global attractors for damped semilinear wave equations, partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: doi:10.3934/dcds.2004.10.31.
[4]	T. Cazenave, "Semilinear Schrödinger Equations," vol 10, Courant Lectures Notes in Mathematics, NYU Courant Institute of Mathematical Sciences, New York, 2003.
[5]	V. V. Chepyzhov and M. I. Vishik, Attractor of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
[6]	J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations, Ann. Inst. Henri Poincaré, 5 (1988), 365-405.
[7]	O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119.
[8]	P. Lauren\c cot, Long-time behavior for weakly damped driven nonlinear Schrödinger equation in $\mathbb R^N, N\leq 3$, No DEA, 2 (1995), 357-369. doi: doi:10.1007/BF01261181.
[9]	A. Miranville and S. Zelik, "Attractors For Dissipative Partial Differential Equations In Bounded And Unbounded Domains," Handbook of Differential Equations: Evolutionary Equations, Vol. 4 (2008) 103-200.
[10]	I. Moise, R. Rosa and X. Wang, Attractors For noncompact nonautonomous systems via energy equations, Disrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: doi:10.3934/dcds.2004.10.473.
[11]	W. Kechiche, Ph D thesis,, in preparation., ().
[12]	G. Raugel, "Global Attractor in Partial Differential Equations. Handbook of Dynamical Systems," Vol. 2, 885-982, North-Holland, Amsterdam, 2002.
[13]	R. Rosa, The global attractor of a weakly damped, forced Korteweg-De Vries equation in $H^1(R)$, VI workshop on partial differential equations, Part II (Rio de Janeiro, 1999), Mat. Contemp., 19 (2000), 129-152.
[14]	C. Sulem and P. L Sulem, "The Nonlinear Schröinger Equation. Self-focusing and Wave Collapse," Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.
[15]	R. Temam, "Infinite Dimentional Dynamical Systems in Mecanics And physics," 2nd Edition, Springer-Verlag, 1997.
[16]	X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications, Physica D, 88 (1995), 167-175. doi: doi:10.1016/0167-2789(95)00196-B.

show all references

References:

[1]	N. Akroune, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on $\mathbb R$, Appl. Math. Lett, 12 (1999), 45-48. doi: doi:10.1016/S0893-9659(98)00170-0.
[2]	A. Babin and M. Vishik, "Attractors of Evolution Equations," Nauka, Moscow 1989; English transl. Stud. Math. Appl., vol 25, North Holland, Amsterdam 1992.
[3]	J. Ball, Global attractors for damped semilinear wave equations, partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: doi:10.3934/dcds.2004.10.31.
[4]	T. Cazenave, "Semilinear Schrödinger Equations," vol 10, Courant Lectures Notes in Mathematics, NYU Courant Institute of Mathematical Sciences, New York, 2003.
[5]	V. V. Chepyzhov and M. I. Vishik, Attractor of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
[6]	J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations, Ann. Inst. Henri Poincaré, 5 (1988), 365-405.
[7]	O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119.
[8]	P. Lauren\c cot, Long-time behavior for weakly damped driven nonlinear Schrödinger equation in $\mathbb R^N, N\leq 3$, No DEA, 2 (1995), 357-369. doi: doi:10.1007/BF01261181.
[9]	A. Miranville and S. Zelik, "Attractors For Dissipative Partial Differential Equations In Bounded And Unbounded Domains," Handbook of Differential Equations: Evolutionary Equations, Vol. 4 (2008) 103-200.
[10]	I. Moise, R. Rosa and X. Wang, Attractors For noncompact nonautonomous systems via energy equations, Disrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: doi:10.3934/dcds.2004.10.473.
[11]	W. Kechiche, Ph D thesis,, in preparation., ().
[12]	G. Raugel, "Global Attractor in Partial Differential Equations. Handbook of Dynamical Systems," Vol. 2, 885-982, North-Holland, Amsterdam, 2002.
[13]	R. Rosa, The global attractor of a weakly damped, forced Korteweg-De Vries equation in $H^1(R)$, VI workshop on partial differential equations, Part II (Rio de Janeiro, 1999), Mat. Contemp., 19 (2000), 129-152.
[14]	C. Sulem and P. L Sulem, "The Nonlinear Schröinger Equation. Self-focusing and Wave Collapse," Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.
[15]	R. Temam, "Infinite Dimentional Dynamical Systems in Mecanics And physics," 2nd Edition, Springer-Verlag, 1997.
[16]	X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications, Physica D, 88 (1995), 167-175. doi: doi:10.1016/0167-2789(95)00196-B.

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