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September  2016, 36(9): 5097-5118. doi: 10.3934/dcds.2016021

Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement

Florian Méhats 1, and  Christof Sparber 2,

1. 

IRMAR, Université de Rennes 1 and IPSO, INRIA Rennes, 35042 Rennes Cedex
2. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, 851 South Morgan Street, Chicago, Illinois 60607

Received  July 2015 Revised  February 2016 Published  May 2016

We consider the three-dimensional time-dependent Gross-Pitaevskii equation arising in the description of rotating Bose-Einstein condensates and study the corresponding scaling limit of strongly anisotropic confinement potentials. The resulting effective equations in one or two spatial dimensions, respectively, are rigorously obtained as special cases of an averaged three dimensional limit model. In the particular case where the rotation axis is not parallel to the strongly confining direction the resulting limiting model(s) include a negative, and thus, purely repulsive quadratic potential, which is not present in the original equation and which can be seen as an effective centrifugal force counteracting the confinement.
Keywords: Gross-Pitaevskii equation, angular momentum operator., averaging, dimension reduction, Bose-Einstein condensation.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B2.
Citation: 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
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show all references

References:
[1]

Commun. Math. Phys., 334 (2015), 367-396. doi: 10.1007/s00220-014-2166-y.  Google Scholar

[2]

Discrete Contin. Dyn. Syst., 32 (2012), 703-715. doi: 10.3934/dcds.2012.32.703.  Google Scholar

[3]

SIAM J. Math. Anal., 44 (2012), 1713-1741. doi: 10.1137/110850451.  Google Scholar

[4]

Math. Models Meth. Appl. Sci., 15 (2005), 767-782. doi: 10.1142/S0218202505000534.  Google Scholar

[5]

Kinet. Relat. Models, 4 (2011), 831-856. doi: 10.3934/krm.2011.4.831.  Google Scholar

[6]

J. Differential Equ., 245 (2008), 154-200. doi: 10.1016/j.jde.2008.02.002.  Google Scholar

[7]

SIAM J. Math. Anal., 37 (2005), 189-199. doi: 10.1137/040614554.  Google Scholar

[8]

Discrete Contin. Dyn. Syst., 13 (2005), 385-398. doi: 10.3934/dcds.2005.13.385.  Google Scholar

[9]

SIAM J. Math. Anal., 35 (2003), 823-843. doi: 10.1137/S0036141002416936.  Google Scholar

[10]

Courant Lecture Notes in Mathematics, Vol. 10, American Mathematical Society, Providence, RI, 2003.  Google Scholar

[11]

Comm. Math. Phys., 292 (2009), 829-870. doi: 10.1007/s00220-009-0868-3.  Google Scholar

[12]

Classics in Mathematics, Springer-Verlag, Berlin, 1999.  Google Scholar

[13]

Cambridge University Press, Cambridge, 1988.  Google Scholar

[14]

Société mathématique de France, 1984.  Google Scholar

[15]

J. Fac. Sci. Univ. Tokyo Sec. IA, 27 (1980), 193-226.  Google Scholar

[16]

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[17]

Comm. Math. Phys., 264 (2006), 505-537. doi: 10.1007/s00220-006-1524-9.  Google Scholar

[18]

Cambridge University Press, 2002. Google Scholar

[19]

Clarendon Press, Oxford, 2003.  Google Scholar

[20]

Commun. Partial Differ. Equ., 37 (2012), 125-135. doi: 10.1080/03605302.2011.574306.  Google Scholar

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