# American Institute of Mathematical Sciences

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

## Uniform attractor for non-autonomous nonlinear Schrödinger equation

 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

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 $0$.
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.  Google Scholar [2] A. Babin and M. Vishik, "Attractors of Evolution Equations," Nauka, Moscow 1989; English transl., Stud. Math. Appl., (1992).   Google Scholar [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.  Google Scholar [4] T. Cazenave, "Semilinear Schrödinger Equations," vol 10,, Courant Lectures Notes in Mathematics, (2003).   Google Scholar [5] V. V. Chepyzhov and M. I. Vishik, Attractor of non-autonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279.   Google Scholar [6] J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations,, Ann. Inst. Henri Poincar\'e, 5 (1988), 365.   Google Scholar [7] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, Applicable Anal., 60 (1996), 99.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [11] W. Kechiche, Ph D thesis,, in preparation., ().   Google Scholar [12] G. Raugel, "Global Attractor in Partial Differential Equations. Handbook of Dynamical Systems,", Vol. 2, (2002), 885.   Google Scholar [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.   Google Scholar [14] C. Sulem and P. L Sulem, "The Nonlinear Schröinger Equation. Self-focusing and Wave Collapse,", Applied Mathematical Sciences, (1999).   Google Scholar [15] R. Temam, "Infinite Dimentional Dynamical Systems in Mecanics And physics,", 2nd Edition, (1997).   Google Scholar [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.  Google Scholar

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.  Google Scholar [2] A. Babin and M. Vishik, "Attractors of Evolution Equations," Nauka, Moscow 1989; English transl., Stud. Math. Appl., (1992).   Google Scholar [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.  Google Scholar [4] T. Cazenave, "Semilinear Schrödinger Equations," vol 10,, Courant Lectures Notes in Mathematics, (2003).   Google Scholar [5] V. V. Chepyzhov and M. I. Vishik, Attractor of non-autonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279.   Google Scholar [6] J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations,, Ann. Inst. Henri Poincar\'e, 5 (1988), 365.   Google Scholar [7] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, Applicable Anal., 60 (1996), 99.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [11] W. Kechiche, Ph D thesis,, in preparation., ().   Google Scholar [12] G. Raugel, "Global Attractor in Partial Differential Equations. Handbook of Dynamical Systems,", Vol. 2, (2002), 885.   Google Scholar [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.   Google Scholar [14] C. Sulem and P. L Sulem, "The Nonlinear Schröinger Equation. Self-focusing and Wave Collapse,", Applied Mathematical Sciences, (1999).   Google Scholar [15] R. Temam, "Infinite Dimentional Dynamical Systems in Mecanics And physics,", 2nd Edition, (1997).   Google Scholar [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.  Google Scholar
 [1] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383 [2] Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080 [3] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [4] Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $p$-Laplacian equations on $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265 [5] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [6] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436 [7] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456 [8] Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260 [9] Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461 [10] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 [11] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [12] Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121 [13] Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297 [14] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [15] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [16] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376 [17] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 [18] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [19] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [20] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

2019 Impact Factor: 1.105