Abstract
We study the cubic wave equation in AdS d+1 (and a closely related cubic wave equation on S 3) in a weakly nonlinear regime. Via time-averaging, these systems are accurately described by simplified infinite-dimensional quartic Hamiltonian systems, whose structure is mandated by the fully resonant spectrum of linearized perturbations. The maximally rotating sector, comprising only the modes of maximal angular momentum at each frequency level, consistently decouples in the weakly nonlinear regime. The Hamiltonian systems obtained by this decoupling display remarkable periodic return behaviors closely analogous to what has been demonstrated in recent literature for a few other related equations (the cubic Szegő equation, the conformal flow, the LLL equation). This suggests a powerful underlying analytic structure, such as integrability. We comment on the connection of our considerations to the Gross-Pitaevskii equation for harmonically trapped Bose-Einstein condensates.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Bizon and A. Rostworowski, On weakly turbulent instability of anti-de Sitter space, Phys. Rev. Lett. 107 (2011) 031102 [arXiv:1104.3702] [INSPIRE].
B. Craps and O. Evnin, AdS (in)stability: an analytic approach, Fortsch. Phys. 64 (2016) 336 [arXiv:1510.07836] [INSPIRE].
I. Bloch, J. Dalibard and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80 (2008) 885 [arXiv:0704.3011] [INSPIRE].
N.R. Cooper, Rapidly rotating atomic gases, Adv. Phys. 57 (2008) 539 [arXiv:0810.4398].
A.L. Fetter, Rotating trapped Bose-Einstein condensates, Rev. Mod. Phys. 81 (2009) 647 [Laser Phys. 18 (2008) 1] [arXiv:0801.2952] [INSPIRE].
J.A. Murdock, Perturbations: theory and methods, SIAM, U.S.A., (1987).
V. Balasubramanian, A. Buchel, S.R. Green, L. Lehner and S.L. Liebling, Holographic thermalization, stability of anti-de Sitter space and the Fermi-Pasta-Ulam paradox, Phys. Rev. Lett. 113 (2014) 071601 [arXiv:1403.6471] [INSPIRE].
B. Craps, O. Evnin and J. Vanhoof, Renormalization group, secular term resummation and AdS (in)stability, JHEP 10 (2014) 048 [arXiv:1407.6273] [INSPIRE].
B. Craps, O. Evnin and J. Vanhoof, Renormalization, averaging, conservation laws and AdS (in)stability, JHEP 01 (2015) 108 [arXiv:1412.3249] [INSPIRE].
P. Basu, C. Krishnan and A. Saurabh, A stochasticity threshold in holography and the instability of AdS, Int. J. Mod. Phys. A 30 (2015) 1550128 [arXiv:1408.0624] [INSPIRE].
P. Germain, Z. Hani and L. Thomann, On the continuous resonant equation for NLS: I. Deterministic analysis, J. Math. Pur. App. 105 (2016) 131 [arXiv:1501.03760].
P. Germain and L. Thomann, On the high frequency limit of the LLL equation, Quart. Appl. Math. 74 (2016) 633 [arXiv:1509.09080].
P. Bizoń, B. Craps, O. Evnin, D. Hunik, V. Luyten and M. Maliborski, Conformal flow on S 3 and weak field integrability in AdS 4, Commun. Math. Phys. 353 (2017) 1179 [arXiv:1608.07227] [INSPIRE].
A. Biasi, P. Bizon, B. Craps and O. Evnin, Exact LLL solutions for BEC vortex precession, arXiv:1705.00867 [INSPIRE].
P. Gérard and S. Grellier, The cubic Szegö equation, Ann. Scient. École Norm. Sup. 43 (2010) 761 [arXiv:0906.4540].
A. Rostworowski, Higher order perturbations of anti-de Sitter space and time-periodic solutions of vacuum Einstein equations, Phys. Rev. D 95 (2017) 124043 [arXiv:1701.07804] [INSPIRE].
G. Martinon, G. Fodor, P. Grandclément and P. Forgàcs, Gravitational geons in asymptotically anti-de Sitter spacetimes, Class. Quant. Grav. 34 (2017) 125012 [arXiv:1701.09100] [INSPIRE].
O.J.C. Dias and J.E. Santos, AdS nonlinear instability: breaking spherical and axial symmetries, arXiv:1705.03065 [INSPIRE].
M.W. Choptuik, O.J.C. Dias, J.E. Santos and B. Way, Collapse and nonlinear instability of AdS with angular momenta, arXiv:1706.06101 [INSPIRE].
A. Karch and E. Katz, Adding flavor to AdS/CFT, JHEP 06 (2002) 043 [hep-th/0205236] [INSPIRE].
T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog. Theor. Phys. 113 (2005) 843 [hep-th/0412141] [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a holographic superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic superconductor/insulator transition at zero temperature, JHEP 03 (2010) 131 [arXiv:0911.0962] [INSPIRE].
P. Basu and S.R. Das, Quantum quench across a holographic critical point, JHEP 01 (2012) 103 [arXiv:1109.3909] [INSPIRE].
P. Basu, D. Das, S.R. Das and T. Nishioka, Quantum quench across a zero temperature holographic superfluid transition, JHEP 03 (2013) 146 [arXiv:1211.7076] [INSPIRE].
I.-S. Yang, Missing top of the AdS resonance structure, Phys. Rev. D 91 (2015) 065011 [arXiv:1501.00998] [INSPIRE].
O. Evnin and C. Krishnan, A hidden symmetry of AdS resonances, Phys. Rev. D 91 (2015) 126010 [arXiv:1502.03749] [INSPIRE].
O. Evnin and R. Nivesvivat, AdS perturbations, isometries, selection rules and the Higgs oscillator, JHEP 01 (2016) 151 [arXiv:1512.00349] [INSPIRE].
K. Ohashi, T. Fujimori and M. Nitta, Conformal symmetry of trapped Bose-Einstein condensates and massive Nambu-Goldstone modes, arXiv:1705.09118 [INSPIRE].
U. Niederer, The maximal kinematical invariance group of the harmonic oscillator, Helv. Phys. Acta 46 (1973) 191 [INSPIRE].
A.F. Biasi, J. Mas and A. Paredes, Delayed collapses of Bose-Einstein condensates in relation to anti-de Sitter gravity, Phys. Rev. E 95 (2017) 032216 [arXiv:1610.04866] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1707.08501
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Craps, B., Evnin, O. & Luyten, V. Maximally rotating waves in AdS and on spheres. J. High Energ. Phys. 2017, 59 (2017). https://doi.org/10.1007/JHEP09(2017)059
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2017)059