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

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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

Abstract
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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.

Keywords: fractal dimension., global attractor, Bose-Einstein equation.

Mathematics Subject Classification: Primary: 35L05, 35Q55; Secondary: 76B0.

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.
[2]	N. Akroune, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on $\R$,, Applied Mathematics Lettres, 12 (1999), 45. doi: 10.1016/S0893-9659(98)00170-0.
[3]	J. Ball, Global attractors for damped semilinear wave equations,, Partial differential equations and applications, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31.
[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. doi: 10.1103/PhysRevLett.78.985.
[5]	R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential,, Annales Henri Poincare, 3 (2002), 757. doi: 10.1007/s00023-002-8635-4.
[6]	T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations,", Textos de M\'etodos Matem\`aticos \textbf{26}, 26 (1989).
[7]	T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, (2003), 0.
[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. doi: 10.1016/j.jmaa.2005.07.008.
[9]	G. B. Folland, "Fourier Analysis And Its Applications,", The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, (1992), 0.
[10]	J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations,, Ann. Inst. H. Poincar?Anal. Non Lin閍ire, 5 (1988), 365.
[11]	J. M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schrödinger equations,, Attractors, 23 (1989), 433.
[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. doi: 10.1016/j.na.2010.02.013.
[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. doi: 10.1063/1.1654836.
[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. doi: 10.1007/BF01261181.
[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. doi: 10.1016/j.amc.2005.11.024.
[16]	K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation,, Physica D: Nonlinear Phenomena, 21 (1986), 381. doi: 10.1016/0167-2789(86)90012-6.
[17]	K. Promislow and N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator,, Nonlinearity, 13 (2000), 675. doi: 10.1088/0951-7715/13/3/310.
[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, (2001), 0.
[19]	B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$,, J. Funct. Anal., 219 (2005), 340. doi: 10.1016/j.jfa.2004.06.013.
[20]	R. Temam, "Infinite-Dimensional Dynamical Systems In Mechanics And Physics,", Springer applied mathmatical sciences, 68 (1997), 0.
[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. doi: 10.1016/0167-2789(95)00196-B.

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.
[2]	N. Akroune, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on $\R$,, Applied Mathematics Lettres, 12 (1999), 45. doi: 10.1016/S0893-9659(98)00170-0.
[3]	J. Ball, Global attractors for damped semilinear wave equations,, Partial differential equations and applications, 10 (2004), 31. doi: 10.3934/dcds.2004.10.31.
[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. doi: 10.1103/PhysRevLett.78.985.
[5]	R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential,, Annales Henri Poincare, 3 (2002), 757. doi: 10.1007/s00023-002-8635-4.
[6]	T. Cazenave, "An Introduction to Nonlinear Schrödinger Equations,", Textos de M\'etodos Matem\`aticos \textbf{26}, 26 (1989).
[7]	T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, (2003), 0.
[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. doi: 10.1016/j.jmaa.2005.07.008.
[9]	G. B. Folland, "Fourier Analysis And Its Applications,", The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, (1992), 0.
[10]	J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations,, Ann. Inst. H. Poincar?Anal. Non Lin閍ire, 5 (1988), 365.
[11]	J. M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schrödinger equations,, Attractors, 23 (1989), 433.
[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. doi: 10.1016/j.na.2010.02.013.
[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. doi: 10.1063/1.1654836.
[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. doi: 10.1007/BF01261181.
[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. doi: 10.1016/j.amc.2005.11.024.
[16]	K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation,, Physica D: Nonlinear Phenomena, 21 (1986), 381. doi: 10.1016/0167-2789(86)90012-6.
[17]	K. Promislow and N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator,, Nonlinearity, 13 (2000), 675. doi: 10.1088/0951-7715/13/3/310.
[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, (2001), 0.
[19]	B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$,, J. Funct. Anal., 219 (2005), 340. doi: 10.1016/j.jfa.2004.06.013.
[20]	R. Temam, "Infinite-Dimensional Dynamical Systems In Mechanics And Physics,", Springer applied mathmatical sciences, 68 (1997), 0.
[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. doi: 10.1016/0167-2789(95)00196-B.

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