On the high frequency limit of the LLL equation
Authors:
Pierre Germain and Laurent Thomann
Journal:
Quart. Appl. Math. 74 (2016), 633-641
MSC (2010):
Primary 37Kxx
DOI:
https://doi.org/10.1090/qam/1435
Published electronically:
June 17, 2016
MathSciNet review:
3539025
Full-text PDF Free Access
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References |
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Additional Information
Abstract: We derive heuristically an integro-differential equation, as well as a shell model, governing the dynamics of the Lowest Landau Level equation in a high frequency regime.
References
- A. Aftalion, X. Blanc, and F. Nier, Lowest Landau level functional and Bargmann spaces for Bose-Einstein condensates, J. Funct. Anal. 241 (2006), no. 2, 661–702. MR 2271933, DOI 10.1016/j.jfa.2006.04.027
- Amandine Aftalion and Sylvia Serfaty, Lowest Landau level approach in superconductivity for the Abrikosov lattice close to $H_{c_2}$, Selecta Math. (N.S.) 13 (2007), no. 2, 183–202. MR 2361092, DOI 10.1007/s00029-007-0043-7
- Luca Biferale, Shell models of energy cascade in turbulence, Annual review of fluid mechanics, Vol. 35, Annu. Rev. Fluid Mech., vol. 35, Annual Reviews, Palo Alto, CA, 2003, pp. 441–468. MR 1967019, DOI 10.1146/annurev.fluid.35.101101.161122
- L. O. Baksmaty, S. J. Woo, S. Choi, and N. P. Bigelow, Tkachenko waves in rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett. 92 (2004), 160405.
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 (2010), no. 1, 39–113. MR 2651381, DOI 10.1007/s00222-010-0242-2
- James E. Colliander, Jeremy L. Marzuola, Tadahiro Oh, and Gideon Simpson, Behavior of a model dynamical system with applications to weak turbulence, Exp. Math. 22 (2013), no. 3, 250–264. MR 3171091, DOI 10.1080/10586458.2013.793110
- M. Escobedo and J. Velazquez, On the theory of weak turbulence for the nonlinear Schrödinger equation. Preprint, arXiv:1305.5746.
- E. Faou, P. Germain, and Z. Hani, The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation, J. Amer. Math. Soc., published electronically on October 20, 2015.
- P. Germain, P. Gérard, and L. Thomann, On the Lowest Landau Level equation. In preparation.
- P. Germain, Z. Hani and L. Thomann, On the continuous resonant equation for NLS. I. Deterministic analysis. J. Math. Pures Appl. 9 105 (2016), no. 1, 131-163.
- Pierre Germain, Zaher Hani, and Laurent Thomann, On the continuous resonant equation for NLS, II: Statistical study, Anal. PDE 8 (2015), no. 7, 1733–1756. MR 3399137, DOI 10.2140/apde.2015.8.1733
- E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion. Sov. Phys. Dokl. 18, 216–217.
- Z. Hani and L. Thomann, Asymptotic behavior of the nonlinear Schrödinger with harmonic trapping, Comm. Pure Appl. Math. To appear. Preprint, arXiv:1408.6213.
- S. Herr and J. Marzuola, On discrete rarefaction waves in a nonlinear Schrödinger equation toy model for weak turbulence. Preprint, arXiv:1307.1873 (2013).
- Sergey Nazarenko, Wave turbulence, Lecture Notes in Physics, vol. 825, Springer, Heidelberg, 2011. MR 3014432, DOI 10.1007/978-3-642-15942-8
- F. Nier, Bose-Einstein condensates in the lowest Landau level: Hamiltonian dynamics, Rev. Math. Phys. 19 (2007), no. 1, 101–130. MR 2293086, DOI 10.1142/S0129055X07002900
- Koji Ohkitani and Michio Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully-developed model turbulence, Progr. Theoret. Phys. 81 (1989), no. 2, 329–341. MR 997440, DOI 10.1143/PTP.81.329
- V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff, and E. A. Cornell, Rapidly rotating Bose-Einstein condensates in and near the Lowest Landau Level, Phys. Rev. Lett. 92 (2004), 040404.
- E. B. Sonin, Ground state and Tkachenko modes of a rapidly rotating Bose Einstein condensate in the lowest Landau level state, Phys. Rev. A 72 (2005), 021606(R).
- Kehe Zhu, Analysis on Fock spaces, Graduate Texts in Mathematics, vol. 263, Springer, New York, 2012. MR 2934601, DOI 10.1007/978-1-4419-8801-0
References
- A. Aftalion, X. Blanc, and F. Nier, Lowest Landau level functional and Bargmann spaces for Bose-Einstein condensates, J. Funct. Anal. 241 (2006), no. 2, 661–702. MR 2271933 (2008c:82052), DOI 10.1016/j.jfa.2006.04.027
- Amandine Aftalion and Sylvia Serfaty, Lowest Landau level approach in superconductivity for the Abrikosov lattice close to $H_{c_2}$, Selecta Math. (N.S.) 13 (2007), no. 2, 183–202. MR 2361092 (2008m:82093), DOI 10.1007/s00029-007-0043-7
- Luca Biferale, Shell models of energy cascade in turbulence, Annual review of fluid mechanics, Vol. 35, Annu. Rev. Fluid Mech., vol. 35, Annual Reviews, Palo Alto, CA, 2003, pp. 441–468. MR 1967019 (2004b:76074), DOI 10.1146/annurev.fluid.35.101101.161122
- L. O. Baksmaty, S. J. Woo, S. Choi, and N. P. Bigelow, Tkachenko waves in rapidly rotating Bose-Einstein condensates, Phys. Rev. Lett. 92 (2004), 160405.
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 (2010), no. 1, 39–113. MR 2651381 (2011f:35320), DOI 10.1007/s00222-010-0242-2
- James E. Colliander, Jeremy L. Marzuola, Tadahiro Oh, and Gideon Simpson, Behavior of a model dynamical system with applications to weak turbulence, Exp. Math. 22 (2013), no. 3, 250–264. MR 3171091, DOI 10.1080/10586458.2013.793110
- M. Escobedo and J. Velazquez, On the theory of weak turbulence for the nonlinear Schrödinger equation. Preprint, arXiv:1305.5746.
- E. Faou, P. Germain, and Z. Hani, The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation, J. Amer. Math. Soc., published electronically on October 20, 2015.
- P. Germain, P. Gérard, and L. Thomann, On the Lowest Landau Level equation. In preparation.
- P. Germain, Z. Hani and L. Thomann, On the continuous resonant equation for NLS. I. Deterministic analysis. J. Math. Pures Appl. 9 105 (2016), no. 1, 131-163.
- Pierre Germain, Zaher Hani, and Laurent Thomann, On the continuous resonant equation for NLS, II: Statistical study, Anal. PDE 8 (2015), no. 7, 1733–1756. MR 3399137, DOI 10.2140/apde.2015.8.1733
- E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion. Sov. Phys. Dokl. 18, 216–217.
- Z. Hani and L. Thomann, Asymptotic behavior of the nonlinear Schrödinger with harmonic trapping, Comm. Pure Appl. Math. To appear. Preprint, arXiv:1408.6213.
- S. Herr and J. Marzuola, On discrete rarefaction waves in a nonlinear Schrödinger equation toy model for weak turbulence. Preprint, arXiv:1307.1873 (2013).
- Sergey Nazarenko, Wave turbulence, Lecture Notes in Physics, vol. 825, Springer, Heidelberg, 2011. MR 3014432, DOI 10.1007/978-3-642-15942-8
- F. Nier, Bose-Einstein condensates in the lowest Landau level: Hamiltonian dynamics, Rev. Math. Phys. 19 (2007), no. 1, 101–130. MR 2293086 (2008h:82056), DOI 10.1142/S0129055X07002900
- Koji Ohkitani and Michio Yamada, Temporal intermittency in the energy cascade process and local Lyapunov analysis in fully-developed model turbulence, Progr. Theoret. Phys. 81 (1989), no. 2, 329–341. MR 997440 (90j:76065), DOI 10.1143/PTP.81.329
- V. Schweikhard, I. Coddington, P. Engels, V. P. Mogendorff, and E. A. Cornell, Rapidly rotating Bose-Einstein condensates in and near the Lowest Landau Level, Phys. Rev. Lett. 92 (2004), 040404.
- E. B. Sonin, Ground state and Tkachenko modes of a rapidly rotating Bose Einstein condensate in the lowest Landau level state, Phys. Rev. A 72 (2005), 021606(R).
- Kehe Zhu, Analysis on Fock spaces, Graduate Texts in Mathematics, vol. 263, Springer, New York, 2012. MR 2934601, DOI 10.1007/978-1-4419-8801-0
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Additional Information
Pierre Germain
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012-1185
MR Author ID:
758713
Email:
pgermain@cims.nyu.edu
Laurent Thomann
Affiliation:
Institut Élie Cartan, Université de Lorraine, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy Cedex, France
MR Author ID:
794415
Email:
laurent.thomann@univ-lorraine.fr
Keywords:
Lowest Landau Level,
shell model
Received by editor(s):
September 29, 2015
Published electronically:
June 17, 2016
Additional Notes:
The first author was partially supported by NSF grant DMS-1101269, a start-up grant from the Courant Institute, and a Sloan fellowship.
The second author was partially supported by the grant “ANAÉ” ANR-13-BS01-0010-03.
Article copyright:
© Copyright 2016
Brown University
