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

Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement

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.
Citation: Florian Méhats, Christof Sparber. Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5097-5118. doi: 10.3934/dcds.2016021
References:
[1]

P. Antonelli, R. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement,, Commun. Math. Phys., 334 (2015), 367.  doi: 10.1007/s00220-014-2166-y.  Google Scholar

[2]

P. Antonelli, D. Marahrens and C. Sparber, On the Cauchy problem for nonlinear Schrödinger equations with rotation,, Discrete Contin. Dyn. Syst., 32 (2012), 703.  doi: 10.3934/dcds.2012.32.703.  Google Scholar

[3]

W. Bao, N. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement,, SIAM J. Math. Anal., 44 (2012), 1713.  doi: 10.1137/110850451.  Google Scholar

[4]

W. Bao, P. A. Markowich, C. Schmeiser and R. M. Weishäupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments,, Math. Models Meth. Appl. Sci., 15 (2005), 767.  doi: 10.1142/S0218202505000534.  Google Scholar

[5]

N. Ben Abdallah, Y. Cai, F. Castella and F. Méhats, Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential,, Kinet. Relat. Models, 4 (2011), 831.  doi: 10.3934/krm.2011.4.831.  Google Scholar

[6]

N. Ben Abdallah, F. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity,, J. Differential Equ., 245 (2008), 154.  doi: 10.1016/j.jde.2008.02.002.  Google Scholar

[7]

N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential,, SIAM J. Math. Anal., 37 (2005), 189.  doi: 10.1137/040614554.  Google Scholar

[8]

R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials., Discrete Contin. Dyn. Syst., 13 (2005), 385.  doi: 10.3934/dcds.2005.13.385.  Google Scholar

[9]

R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications,, SIAM J. Math. Anal., 35 (2003), 823.  doi: 10.1137/S0036141002416936.  Google Scholar

[10]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[11]

F. Delebecque-Fendt and F. Méhats, An effective mass theorem for the bidimensional electron gas in a strong magnetic field,, Comm. Math. Phys., 292 (2009), 829.  doi: 10.1007/s00220-009-0868-3.  Google Scholar

[12]

S. Flügge, Practical Quantum Mechanics,, Classics in Mathematics, (1999).   Google Scholar

[13]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Cambridge University Press, (1988).   Google Scholar

[14]

B. Helffer, Théorie Spectrale Pour Des Opérateurs Globalement Elliptiques,, Société mathématique de France, (1984).   Google Scholar

[15]

H. Kitada, On a construction of the fundamental solution for Schrödinger equations,, J. Fac. Sci. Univ. Tokyo Sec. IA, 27 (1980), 193.   Google Scholar

[16]

E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and its Condensation,, Oberwolfach Seminars, (2005).   Google Scholar

[17]

E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases,, Comm. Math. Phys., 264 (2006), 505.  doi: 10.1007/s00220-006-1524-9.  Google Scholar

[18]

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases,, Cambridge University Press, (2002).   Google Scholar

[19]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation,, Clarendon Press, (2003).   Google Scholar

[20]

N. Tzvetkov and N. Visciglia, Small data scattering for the nonlinear Schrödinger equation on product spaces,, Commun. Partial Differ. Equ., 37 (2012), 125.  doi: 10.1080/03605302.2011.574306.  Google Scholar

show all references

References:
[1]

P. Antonelli, R. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement,, Commun. Math. Phys., 334 (2015), 367.  doi: 10.1007/s00220-014-2166-y.  Google Scholar

[2]

P. Antonelli, D. Marahrens and C. Sparber, On the Cauchy problem for nonlinear Schrödinger equations with rotation,, Discrete Contin. Dyn. Syst., 32 (2012), 703.  doi: 10.3934/dcds.2012.32.703.  Google Scholar

[3]

W. Bao, N. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement,, SIAM J. Math. Anal., 44 (2012), 1713.  doi: 10.1137/110850451.  Google Scholar

[4]

W. Bao, P. A. Markowich, C. Schmeiser and R. M. Weishäupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments,, Math. Models Meth. Appl. Sci., 15 (2005), 767.  doi: 10.1142/S0218202505000534.  Google Scholar

[5]

N. Ben Abdallah, Y. Cai, F. Castella and F. Méhats, Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential,, Kinet. Relat. Models, 4 (2011), 831.  doi: 10.3934/krm.2011.4.831.  Google Scholar

[6]

N. Ben Abdallah, F. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity,, J. Differential Equ., 245 (2008), 154.  doi: 10.1016/j.jde.2008.02.002.  Google Scholar

[7]

N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential,, SIAM J. Math. Anal., 37 (2005), 189.  doi: 10.1137/040614554.  Google Scholar

[8]

R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials., Discrete Contin. Dyn. Syst., 13 (2005), 385.  doi: 10.3934/dcds.2005.13.385.  Google Scholar

[9]

R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications,, SIAM J. Math. Anal., 35 (2003), 823.  doi: 10.1137/S0036141002416936.  Google Scholar

[10]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[11]

F. Delebecque-Fendt and F. Méhats, An effective mass theorem for the bidimensional electron gas in a strong magnetic field,, Comm. Math. Phys., 292 (2009), 829.  doi: 10.1007/s00220-009-0868-3.  Google Scholar

[12]

S. Flügge, Practical Quantum Mechanics,, Classics in Mathematics, (1999).   Google Scholar

[13]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Cambridge University Press, (1988).   Google Scholar

[14]

B. Helffer, Théorie Spectrale Pour Des Opérateurs Globalement Elliptiques,, Société mathématique de France, (1984).   Google Scholar

[15]

H. Kitada, On a construction of the fundamental solution for Schrödinger equations,, J. Fac. Sci. Univ. Tokyo Sec. IA, 27 (1980), 193.   Google Scholar

[16]

E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and its Condensation,, Oberwolfach Seminars, (2005).   Google Scholar

[17]

E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases,, Comm. Math. Phys., 264 (2006), 505.  doi: 10.1007/s00220-006-1524-9.  Google Scholar

[18]

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases,, Cambridge University Press, (2002).   Google Scholar

[19]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation,, Clarendon Press, (2003).   Google Scholar

[20]

N. Tzvetkov and N. Visciglia, Small data scattering for the nonlinear Schrödinger equation on product spaces,, Commun. Partial Differ. Equ., 37 (2012), 125.  doi: 10.1080/03605302.2011.574306.  Google Scholar

[1]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[2]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[3]

Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017

[4]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[5]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[6]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[7]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[8]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[9]

Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043

[10]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

[11]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303

[12]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[13]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[14]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[15]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265

[16]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[17]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[18]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[19]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[20]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]