American Institute of Mathematical Sciences

Advanced
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 - 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]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[2]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[3]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[4]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247
[5]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017
[6]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020
[7]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[8]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[9]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[10]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[11]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[12]

Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
[13]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[14]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[15]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[16]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454
[17]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[18]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021
[19]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[20]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences