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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

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