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Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain
1. | Unité de recherche: Ondelettes et Fractals, Faculté des Sciences de Monastir, Av. de l'environnement, 5000 Monastir |
References:
[1] |
B. Alouini, Long-time behavior of a Bose-Einstein equation in a two dimensional thin domain,, \emph{Communications in Pure and Applied Analysis}, 10 (2011), 1629.
doi: 10.3934/cpaa.2011.10.1629. |
[2] |
B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain,, \emph{Discrete Continuous Dynam. Systems - B}, 19 (2014), 651.
doi: 10.3934/dcdsb.2014.19.651. |
[3] |
J. M. Ball, Global attractors for damped semilinear wave equations,, \emph{Discrete Continuous Dynam. Systems - A}, 10 (2004), 31.
doi: 10.3934/dcds.2004.10.31. |
[4] |
B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator,, \emph{Proc. Indian. Acad. Sci. (Math. Sci.)}, 116 (2006), 337.
doi: 10.1007/BF02829750. |
[5] |
C. C. Bradlay, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: observation of limited condensate number,, \emph{Phys. Rev. Lett.}, 78 (1997), 985.
doi: 10.1103/PhysRevLett.78.985. |
[6] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011).
|
[7] |
R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential,, \emph{Annales Henri Poincare}, 3 (2002), 757.
doi: 10.1007/s00023-002-8635-4. |
[8] |
T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, 10 (2003).
|
[9] |
I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems,, University Lectures in Contemporary Mathematics, (2002). Google Scholar |
[10] |
I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping,, Memoirs of the American Mathematical Society, 195 (2008).
|
[11] |
G. B. Folland, Fourier Analysis and Its Applications,, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, (1992).
|
[12] |
O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$,, \emph{Advances in Differential Equations}, 3 (1998), 337.
|
[13] |
O. Goubet and L. Legry, Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential,, \emph{Nonlinear Analysis}, 72 (2010), 4397.
doi: 10.1016/j.na.2010.02.013. |
[14] |
A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anormalous dispersion,, \emph{Applied Physics Lettres}, 23 (1973), 14.
doi: 10.1063/1.1654836. |
[15] |
P. Lauren\ccot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $R^N,N\leq 3$,, \emph{NoDEA}, 2 (1995), 357.
doi: 10.1007/BF01261181. |
[16] |
Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential,, \emph{Appl. Math. Comput.}, 177 (2006), 482.
|
[17] |
K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation,, \emph{Physica D: Nonlinear Phenomena}, 21 (1986), 381.
doi: 10.1016/0167-2789(86)90012-6. |
[18] |
K. Promislow and J. N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator,, \emph{Nonlinearity}, 13 (2000), 675.
doi: 10.1088/0951-7715/13/3/310. |
[19] |
J.C. Robinson, Infinite-Dimensionel Dynamical Systems, An Introduction To Dissipative Parabolic PDEs And The Theorie Of Global Attractors,, Cambridge Texts in Applied Mathematics. Cambridge University Press, (2001).
|
[20] |
B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$,, \emph{J. Funct. Anal.}, 219 (2005), 340.
doi: 10.1016/j.jfa.2004.06.013. |
[21] |
R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics,, Springer applied mathmatical sciences, 68 (1997).
|
[22] |
X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors,, \emph{Physica D: Nonlinear Phenomena}, 88 (1995), 167.
doi: 10.1016/0167-2789(95)00196-B. |
show all references
References:
[1] |
B. Alouini, Long-time behavior of a Bose-Einstein equation in a two dimensional thin domain,, \emph{Communications in Pure and Applied Analysis}, 10 (2011), 1629.
doi: 10.3934/cpaa.2011.10.1629. |
[2] |
B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain,, \emph{Discrete Continuous Dynam. Systems - B}, 19 (2014), 651.
doi: 10.3934/dcdsb.2014.19.651. |
[3] |
J. M. Ball, Global attractors for damped semilinear wave equations,, \emph{Discrete Continuous Dynam. Systems - A}, 10 (2004), 31.
doi: 10.3934/dcds.2004.10.31. |
[4] |
B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator,, \emph{Proc. Indian. Acad. Sci. (Math. Sci.)}, 116 (2006), 337.
doi: 10.1007/BF02829750. |
[5] |
C. C. Bradlay, C. A. Sackett and R. G. Hulet, Bose-Einstein condensation of lithium: observation of limited condensate number,, \emph{Phys. Rev. Lett.}, 78 (1997), 985.
doi: 10.1103/PhysRevLett.78.985. |
[6] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011).
|
[7] |
R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential,, \emph{Annales Henri Poincare}, 3 (2002), 757.
doi: 10.1007/s00023-002-8635-4. |
[8] |
T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, 10 (2003).
|
[9] |
I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems,, University Lectures in Contemporary Mathematics, (2002). Google Scholar |
[10] |
I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping,, Memoirs of the American Mathematical Society, 195 (2008).
|
[11] |
G. B. Folland, Fourier Analysis and Its Applications,, The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, (1992).
|
[12] |
O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $R^2$,, \emph{Advances in Differential Equations}, 3 (1998), 337.
|
[13] |
O. Goubet and L. Legry, Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential,, \emph{Nonlinear Analysis}, 72 (2010), 4397.
doi: 10.1016/j.na.2010.02.013. |
[14] |
A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anormalous dispersion,, \emph{Applied Physics Lettres}, 23 (1973), 14.
doi: 10.1063/1.1654836. |
[15] |
P. Lauren\ccot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $R^N,N\leq 3$,, \emph{NoDEA}, 2 (1995), 357.
doi: 10.1007/BF01261181. |
[16] |
Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential,, \emph{Appl. Math. Comput.}, 177 (2006), 482.
|
[17] |
K. Nosaki and N. Bekki, Low-Dimentional chaos in a driven damped nonlinear Schrödinger equation,, \emph{Physica D: Nonlinear Phenomena}, 21 (1986), 381.
doi: 10.1016/0167-2789(86)90012-6. |
[18] |
K. Promislow and J. N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of optical parametric oscillator,, \emph{Nonlinearity}, 13 (2000), 675.
doi: 10.1088/0951-7715/13/3/310. |
[19] |
J.C. Robinson, Infinite-Dimensionel Dynamical Systems, An Introduction To Dissipative Parabolic PDEs And The Theorie Of Global Attractors,, Cambridge Texts in Applied Mathematics. Cambridge University Press, (2001).
|
[20] |
B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $R^2$,, \emph{J. Funct. Anal.}, 219 (2005), 340.
doi: 10.1016/j.jfa.2004.06.013. |
[21] |
R. Temam, Infinite-Dimensional Dynamical Systems In Mechanics and Physics,, Springer applied mathmatical sciences, 68 (1997).
|
[22] |
X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors,, \emph{Physica D: Nonlinear Phenomena}, 88 (1995), 167.
doi: 10.1016/0167-2789(95)00196-B. |
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