# American Institute of Mathematical Sciences

• Previous Article
Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains
• CPAA Home
• This Issue
• Next Article
A generalization of $H$-measures and application on purely fractional scalar conservation laws
November  2011, 10(6): 1629-1643. doi: 10.3934/cpaa.2011.10.1629

## Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain

 1 Unité de recherche: Ondelettes et Fractals, Faculté des Sciences de Monastir, Av. de l'environnement, 5000 Monastir, Tunisia

Received  March 2010 Revised  November 2010 Published  May 2011

We study the long-time behavior of the solutions to a nonlinear Schrödinger equation with a zero order dissipation and a quadratic potential when they are driven by an external force on a thin canal. We show that this behavior is described by a regular attractor which captures all the trajectories and have a finite Fractal dimension.
Citation: Brahim Alouini. Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1629-1643. doi: 10.3934/cpaa.2011.10.1629
##### References:
 [1] M. Abounouh and O. Goubet, Attractor for a damped cubic Schrödinger equation on a two-dimensional thin domain, Differential Integral Equations, 13 (2000), 311-340.  Google Scholar [2] N. Akroune, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on $\R$, Applied Mathematics Lettres, 12 (1999), 45-48. doi: 10.1016/S0893-9659(98)00170-0.  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: 10.3934/dcds.2004.10.31.  Google Scholar [4] C. C. Bradlay, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: observation of limited condensate number, Phys. Rev. Lett., 78 (1997), 985-989. doi: 10.1103/PhysRevLett.78.985.  Google Scholar [5] R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Annales Henri Poincare, 3 (2002), 757-772. doi: 10.1007/s00023-002-8635-4.  Google Scholar [6] T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations," Textos de Métodos Matemàticos 26, Rio de Janeiro, 1989. Google Scholar [7] T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp. ISBN: 0-8218-3399-5.  Google Scholar [8] G. Chen and J. Zhang, Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential, J. Math. Anal. Appl., 320 (2006), 591-598. doi: 10.1016/j.jmaa.2005.07.008.  Google Scholar [9] G. B. Folland, "Fourier Analysis And Its Applications," The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. x+433 pp. ISBN: 0-534-17094-3.  Google Scholar [10] J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. H. Poincar?Anal. Non Linaire, 5 (1988), 365-405.  Google Scholar [11] J. M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schrödinger equations, Attractors, inertial manifolds and their approximation (Marseille-Luminy, 1987). RAIRO Modl. Math. Anal. Numr., 23 (1989), 433-443.  Google Scholar [12] O. Goubet and L. Legry, Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential, Nonlinear Analysis, 72 (2010), 4397-4406. doi: 10.1016/j.na.2010.02.013.  Google Scholar [13] A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers.I.Anormalous dispersion, Applied Physics Lettres, 23 (1973), 14-24. doi: 10.1063/1.1654836.  Google Scholar [14] P. Laurencot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $R^N, N\leq 3$, NoDEA: Nonlinear Differential Equations and Applications, 2 (1995), 357-369. doi: 10.1007/BF01261181.  Google Scholar [15] Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput., 177 (2006), 482-487. doi: 10.1016/j.amc.2005.11.024.  Google Scholar [16] K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation, Physica D: Nonlinear Phenomena, 21 (1986), 381-393. doi: 10.1016/0167-2789(86)90012-6.  Google Scholar [17] K. Promislow and N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator, Nonlinearity, 13 (2000), 675-698. doi: 10.1088/0951-7715/13/3/310.  Google Scholar [18] J. C. Robinson, "Infinite-Dimensional Dynamical Systems, An Introduction To Dissipative Parabolic PDEs And The Theorie Of Global Attractors," Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001. xviii+461 pp. ISBN: 0-521-63204-8.  Google Scholar [19] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.  Google Scholar [20] R. Temam, "Infinite-Dimensional Dynamical Systems In Mechanics And Physics," Springer applied mathmatical sciences, volume 68, Springer-Verlag, New York, 1997. xxii+648 pp. ISBN: 0-387-94866-X .  Google Scholar [21] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D, 88 (1995), 167-175. doi: 10.1016/0167-2789(95)00196-B.  Google Scholar

show all references

##### References:
 [1] M. Abounouh and O. Goubet, Attractor for a damped cubic Schrödinger equation on a two-dimensional thin domain, Differential Integral Equations, 13 (2000), 311-340.  Google Scholar [2] N. Akroune, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on $\R$, Applied Mathematics Lettres, 12 (1999), 45-48. doi: 10.1016/S0893-9659(98)00170-0.  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: 10.3934/dcds.2004.10.31.  Google Scholar [4] C. C. Bradlay, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: observation of limited condensate number, Phys. Rev. Lett., 78 (1997), 985-989. doi: 10.1103/PhysRevLett.78.985.  Google Scholar [5] R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Annales Henri Poincare, 3 (2002), 757-772. doi: 10.1007/s00023-002-8635-4.  Google Scholar [6] T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations," Textos de Métodos Matemàticos 26, Rio de Janeiro, 1989. Google Scholar [7] T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp. ISBN: 0-8218-3399-5.  Google Scholar [8] G. Chen and J. Zhang, Remarks on global existence for the supercritical nonlinear Schrödinger equation with a harmonic potential, J. Math. Anal. Appl., 320 (2006), 591-598. doi: 10.1016/j.jmaa.2005.07.008.  Google Scholar [9] G. B. Folland, "Fourier Analysis And Its Applications," The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992. x+433 pp. ISBN: 0-534-17094-3.  Google Scholar [10] J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Ann. Inst. H. Poincar?Anal. Non Linaire, 5 (1988), 365-405.  Google Scholar [11] J. M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schrödinger equations, Attractors, inertial manifolds and their approximation (Marseille-Luminy, 1987). RAIRO Modl. Math. Anal. Numr., 23 (1989), 433-443.  Google Scholar [12] O. Goubet and L. Legry, Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential, Nonlinear Analysis, 72 (2010), 4397-4406. doi: 10.1016/j.na.2010.02.013.  Google Scholar [13] A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers.I.Anormalous dispersion, Applied Physics Lettres, 23 (1973), 14-24. doi: 10.1063/1.1654836.  Google Scholar [14] P. Laurencot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $R^N, N\leq 3$, NoDEA: Nonlinear Differential Equations and Applications, 2 (1995), 357-369. doi: 10.1007/BF01261181.  Google Scholar [15] Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput., 177 (2006), 482-487. doi: 10.1016/j.amc.2005.11.024.  Google Scholar [16] K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation, Physica D: Nonlinear Phenomena, 21 (1986), 381-393. doi: 10.1016/0167-2789(86)90012-6.  Google Scholar [17] K. Promislow and N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator, Nonlinearity, 13 (2000), 675-698. doi: 10.1088/0951-7715/13/3/310.  Google Scholar [18] J. C. Robinson, "Infinite-Dimensional Dynamical Systems, An Introduction To Dissipative Parabolic PDEs And The Theorie Of Global Attractors," Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001. xviii+461 pp. ISBN: 0-521-63204-8.  Google Scholar [19] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013.  Google Scholar [20] R. Temam, "Infinite-Dimensional Dynamical Systems In Mechanics And Physics," Springer applied mathmatical sciences, volume 68, Springer-Verlag, New York, 1997. xxii+648 pp. ISBN: 0-387-94866-X .  Google Scholar [21] X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D, 88 (1995), 167-175. doi: 10.1016/0167-2789(95)00196-B.  Google Scholar
 [1] Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781 [2] Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651 [3] Florian Méhats, Christof Sparber. Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5097-5118. doi: 10.3934/dcds.2016021 [4] Weizhu Bao, Loïc Le Treust, Florian Méhats. Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime. Kinetic & Related Models, 2017, 10 (3) : 553-571. doi: 10.3934/krm.2017022 [5] Xuguang Lu. Long time strong convergence to Bose-Einstein distribution for low temperature. Kinetic & Related Models, 2018, 11 (4) : 715-734. doi: 10.3934/krm.2018029 [6] P.G. Kevrekidis, Dimitri J. Frantzeskakis. Multiple dark solitons in Bose-Einstein condensates at finite temperatures. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1199-1212. doi: 10.3934/dcdss.2011.4.1199 [7] Weizhu Bao, Yongyong Cai. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic & Related Models, 2013, 6 (1) : 1-135. doi: 10.3934/krm.2013.6.1 [8] Pedro J. Torres, R. Carretero-González, S. Middelkamp, P. Schmelcher, Dimitri J. Frantzeskakis, P.G. Kevrekidis. Vortex interaction dynamics in trapped Bose-Einstein condensates. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1589-1615. doi: 10.3934/cpaa.2011.10.1589 [9] Vadym Vekslerchik, Víctor M. Pérez-García. Exact solution of the two-mode model of multicomponent Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 179-192. doi: 10.3934/dcdsb.2003.3.179 [10] Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306 [11] Vladimir S. Gerdjikov. Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1181-1197. doi: 10.3934/dcdss.2011.4.1181 [12] Yongming Luo, Athanasios Stylianou. On 3d dipolar Bose-Einstein condensates involving quantum fluctuations and three-body interactions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3455-3477. doi: 10.3934/dcdsb.2020239 [13] Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119 [14] Kui Li, Zhitao Zhang. A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 851-860. doi: 10.3934/dcds.2016.36.851 [15] Anne de Bouard, Reika Fukuizumi, Romain Poncet. Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2793-2817. doi: 10.3934/dcdsb.2015.20.2793 [16] Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887 [17] Uta Renata Freiberg. Einstein relation on fractal objects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 509-525. doi: 10.3934/dcdsb.2012.17.509 [18] Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939 [19] Delin Wu and Chengkui Zhong. Estimates on the dimension of an attractor for a nonclassical hyperbolic equation. Electronic Research Announcements, 2006, 12: 63-70. [20] Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165

2019 Impact Factor: 1.105