# 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), 45-48. 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., vol 25, North Holland, Amsterdam 1992.  Google Scholar [3] J. Ball, Global attractors for damped semilinear wave equations, partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: doi:10.3934/dcds.2004.10.31.  Google Scholar [4] T. Cazenave, "Semilinear Schrödinger Equations," vol 10, Courant Lectures Notes in Mathematics, NYU Courant Institute of Mathematical Sciences, New York, 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), 279-333.  Google Scholar [6] J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations, Ann. Inst. Henri Poincaré, 5 (1988), 365-405.  Google Scholar [7] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119.  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), 357-369. 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, Vol. 4 (2008) 103-200.  Google Scholar [10] I. Moise, R. Rosa and X. Wang, Attractors For noncompact nonautonomous systems via energy equations, Disrete Contin. Dyn. Syst., 10 (2004), 473-496. 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, 885-982, North-Holland, Amsterdam, 2002.  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, 1999), Mat. Contemp., 19 (2000), 129-152.  Google Scholar [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.  Google Scholar [15] R. Temam, "Infinite Dimentional Dynamical Systems in Mecanics And physics," 2nd Edition, Springer-Verlag, 1997.  Google Scholar [16] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications, Physica D, 88 (1995), 167-175. 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), 45-48. 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., vol 25, North Holland, Amsterdam 1992.  Google Scholar [3] J. Ball, Global attractors for damped semilinear wave equations, partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: doi:10.3934/dcds.2004.10.31.  Google Scholar [4] T. Cazenave, "Semilinear Schrödinger Equations," vol 10, Courant Lectures Notes in Mathematics, NYU Courant Institute of Mathematical Sciences, New York, 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), 279-333.  Google Scholar [6] J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations, Ann. Inst. Henri Poincaré, 5 (1988), 365-405.  Google Scholar [7] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119.  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), 357-369. 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, Vol. 4 (2008) 103-200.  Google Scholar [10] I. Moise, R. Rosa and X. Wang, Attractors For noncompact nonautonomous systems via energy equations, Disrete Contin. Dyn. Syst., 10 (2004), 473-496. 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, 885-982, North-Holland, Amsterdam, 2002.  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, 1999), Mat. Contemp., 19 (2000), 129-152.  Google Scholar [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.  Google Scholar [15] R. Temam, "Infinite Dimentional Dynamical Systems in Mecanics And physics," 2nd Edition, Springer-Verlag, 1997.  Google Scholar [16] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications, Physica D, 88 (1995), 167-175. doi: doi:10.1016/0167-2789(95)00196-B.  Google Scholar
 [1] Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. 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, 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] 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 [5] 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 [6] 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 [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, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229 [10] Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088 [11] 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 [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, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115 [13] 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, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120 [14] Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231 [15] 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 [16] 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 [17] Xueli Song, Jianhua Wu. Non-autonomous 2D Newton-Boussinesq equation with oscillating external forces and its uniform attractor. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020102 [18] 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 [19] Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211 [20] Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029

2019 Impact Factor: 1.105

## Metrics

• PDF downloads (42)
• HTML views (0)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]