Uniform attractor for non-autonomous nonlinear Schrödinger equation

Previous Article
On the collapsing sandpile problem
CPAA Home
This Issue
Next Article
Global wellposedness for a transport equation with super-critial dissipation

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
Related Papers

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 & 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. 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., (1992).
[3]	J. Ball, Global attractors for damped semilinear wave equations, partial differential equations and applications,, Discrete Contin. Dyn. Syst., 10 (2004), 31. doi: doi:10.3934/dcds.2004.10.31.
[4]	T. Cazenave, "Semilinear Schrödinger Equations," vol 10,, Courant Lectures Notes in Mathematics, (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.
[6]	J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations,, Ann. Inst. Henri Poincar\'e, 5 (1988), 365.
[7]	O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, Applicable Anal., 60 (1996), 99.
[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. 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, (2008), 103.
[10]	I. Moise, R. Rosa and X. Wang, Attractors For noncompact nonautonomous systems via energy equations,, Disrete Contin. Dyn. Syst., 10 (2004), 473. 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, (2002), 885.
[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, 19 (2000), 129.
[14]	C. Sulem and P. L Sulem, "The Nonlinear Schröinger Equation. Self-focusing and Wave Collapse,", Applied Mathematical Sciences, (1999).
[15]	R. Temam, "Infinite Dimentional Dynamical Systems in Mecanics And physics,", 2nd Edition, (1997).
[16]	X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications,, Physica D, 88 (1995), 167. 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. 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., (1992).
[3]	J. Ball, Global attractors for damped semilinear wave equations, partial differential equations and applications,, Discrete Contin. Dyn. Syst., 10 (2004), 31. doi: doi:10.3934/dcds.2004.10.31.
[4]	T. Cazenave, "Semilinear Schrödinger Equations," vol 10,, Courant Lectures Notes in Mathematics, (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.
[6]	J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations,, Ann. Inst. Henri Poincar\'e, 5 (1988), 365.
[7]	O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, Applicable Anal., 60 (1996), 99.
[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. 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, (2008), 103.
[10]	I. Moise, R. Rosa and X. Wang, Attractors For noncompact nonautonomous systems via energy equations,, Disrete Contin. Dyn. Syst., 10 (2004), 473. 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, (2002), 885.
[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, 19 (2000), 129.
[14]	C. Sulem and P. L Sulem, "The Nonlinear Schröinger Equation. Self-focusing and Wave Collapse,", Applied Mathematical Sciences, (1999).
[15]	R. Temam, "Infinite Dimentional Dynamical Systems in Mecanics And physics,", 2nd Edition, (1997).
[16]	X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications,, Physica D, 88 (1995), 167. doi: doi:10.1016/0167-2789(95)00196-B.

[1]	Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2019036
[2]	Shengfan Zhou, Caidi Zhao, Yejuan Wang. Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1259-1277. doi: 10.3934/dcds.2008.21.1259
[3]	Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035
[4]	Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[5]	Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120
[6]	Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[7]	Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[8]	Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743
[9]	Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229
[10]	Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[11]	Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120
[12]	Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-21. doi: 10.3934/dcdsb.2019115
[13]	Ahmed Y. Abdallah, Rania T. Wannan. Second order non-autonomous lattice systems and their uniform attractors. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1827-1846. doi: 10.3934/cpaa.2019085
[14]	Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899
[15]	Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119
[16]	Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211
[17]	Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029
[18]	Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[19]	Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079
[20]	José A. Langa, James C. Robinson, Aníbal Rodríguez-Bernal, A. Suárez, A. Vidal-López. Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 483-497. doi: 10.3934/dcds.2007.18.483

2017 Impact Factor: 0.884

American Institute of Mathematical Sciences