Skip to main content
SpringerLink
Log in

Solvable Cubic Resonant Systems

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross–Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross–Pitaevskii equation, but not for any other cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balasubramanian V., Buchel A., Green S.R., Lehner L., Liebling S.L.: Holographic thermalization, stability of anti-de Sitter space, and the Fermi–Pasta–Ulam paradox. Phys. Rev. Lett. 113, 071601 (2014) arXiv:1403.6471 [hep-th]

    Article  ADS  Google Scholar 

  2. Craps B., Evnin O., Vanhoof J.: Renormalization group, secular term resummation and AdS (in)stability. JHEP 1410, 48 (2014) arXiv:1407.6273 [gr-qc]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Craps B., Evnin O., Vanhoof J.: Renormalization, averaging, conservation laws and AdS (in)stability. JHEP 1501, 108 (2015) arXiv:1412.3249 [gr-qc]

    Article  ADS  MATH  Google Scholar 

  4. Germain P., Hani Z., Thomann L.: On the continuous resonant equation for NLS: I. Deterministic analysis. J. Math. Pures Appl. 105, 131 (2016) arXiv:1501.03760 [math.AP]

    Article  MathSciNet  MATH  Google Scholar 

  5. Bizoń P., Maliborski M., Rostworowski A.: Resonant dynamics and the instability of anti-de Sitter spacetime. Phys. Rev. Lett. 115, 081103 (2015) arXiv:1506.03519 [gr-qc]

    Article  ADS  Google Scholar 

  6. Germain P., Thomann L.: On the high frequency limit of the LLL equation. Q. Appl. Math. 74, 633 (2016) arXiv:1509.09080 [math.AP]

    Article  MathSciNet  MATH  Google Scholar 

  7. Bizoń P., Craps B., Evnin O., Hunik D., Luyten V., Maliborski M.: Conformal flow on S 3 and weak field integrability in AdS4. Commun. Math. Phys. 353, 1179 (2017) arXiv:1608.07227 [math.AP]

    Article  ADS  MATH  Google Scholar 

  8. Biasi A.F., Mas J., Paredes A.: Delayed collapses of BECs in relation to AdS gravity. Phys. Rev. E 95, 032216 (2017) arXiv:1610.04866 [nlin.PS]

    Article  ADS  Google Scholar 

  9. Biasi A., Bizoń P., Craps B., Evnin O.: Exact lowest-Landau-level solutions for vortex precession in Bose–Einstein condensates. Phys. Rev. A 96, 053615 (2017) arXiv:1705.00867 [cond-mat.quant-gas]

    Article  ADS  Google Scholar 

  10. Bizoń, P., Hunik-Kostyra, D., Pelinovsky, D.: Ground state of the conformal flow on \({\mathbb{S}^3}\). arXiv:1706.07726 [math.AP]

  11. Craps B., Evnin O., Luyten V.: Maximally rotating waves in AdS and on spheres. JHEP 1709, 059 (2017) arXiv:1707.08501 [hep-th]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Gérard, P., Germain, P., Thomann, L.: On the cubic lowest Landau level equation. arXiv:1709.04276 [math.AP]

  13. Biasi, A., Bizoń, P., Craps, B., Evnin, O.: Two infinite families of resonant solutions for the Gross–Pitaevskii equation. arXiv:1805.01775 [cond-mat.quant-gas]

  14. Biasi, A., Craps, B., Evnin, O.: Energy returns in global AdS4. arXiv:1810.04753 [hep-th]

  15. Bizoń, P., Hunik-Kostyra, D., Pelinovsky, D.: Stationary states of the cubic conformal flow on \({\mathbb{S}^3}\). arXiv:1807.00426 [math-ph]

  16. Bizoń P., Rostworowski A.: On weakly turbulent instability of anti-de Sitter space. Phys. Rev. Lett. 107, 031102 (2011) arXiv:1104.3702 [gr-qc]

    Article  ADS  Google Scholar 

  17. Craps B., Evnin O.: AdS (in)stability: an analytic approach. Fortschr. Phys. 64, 336 (2016) arXiv:1510.07836 [gr-qc]

    Article  MathSciNet  MATH  Google Scholar 

  18. Murdock, J.A.: Perturbations: Theory and Methods. SIAM, Philadelphia (1987)

  19. Kuksin, S., Maiocchi, A.: The effective equation method. In: New Approaches to Nonlinear Waves. Springer (2016) arXiv:1501.04175 [math-ph]

  20. Gérard P., Grellier S.: The cubic Szegő equation. Ann. Scient. Éc. Norm. Sup 43, 761 (2010) arXiv:0906.4540 [math.CV]

    Article  MATH  Google Scholar 

  21. Gérard P., Grellier S.: Effective integrable dynamics for a certain nonlinear wave equation. Anal. PDE 5, 1139 (2012) arXiv:1110.5719 [math.AP]

    Article  MathSciNet  MATH  Google Scholar 

  22. Gérard P., Grellier S.: An explicit formula for the cubic Szegő equation. Trans. Am. Math. Soc. 367, 2979 (2015) arXiv:1304.2619 [math.AP]

    Article  MATH  Google Scholar 

  23. Gérard, P., Grellier, S.: The cubic Szegő equation and Hankel operators. arXiv:1508.06814 [math.AP]

Download references

Acknowledgements

We thank Ben Craps, Javier Mas, and Alexandre Serantes for discussions. This research has been supported by FPA2014-52218-P from Ministerio de Economia y Competitividad, by Xunta de Galicia ED431C 2017/07, by European Regional Development Fund (FEDER), by Grant María de Maetzu Unit of Excellence MDM-2016-0692, by Polish National Science Centre Grant Number 2017/26/A/ST2/00530 and by CUniverse research promotion project by Chulalongkorn University (Grant CUAASC). A.B. thanks the Spanish program “ayudas para contratos predoctorales para la formación de doctores 2015” and its mobility program for his stay at Jagiellonian University, where part of this project was developed.

Author information

Authors and Affiliations

  1. Departamento de Física de Partículas, Universidade de Santiago de Compostela and Instituto Galego de Física de Altas Enerxías (IGFAE), Santiago de Compostela, Spain

    Anxo Biasi

  2. Institute of Physics, Jagiellonian University, Kraków, Poland

    Piotr Bizoń

  3. Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, Thailand

    Oleg Evnin

  4. Theoretische Natuurkunde, Vrije Universiteit Brussel and The International Solvay Institutes, Brussels, Belgium

    Oleg Evnin

Authors
  1. Anxo Biasi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Piotr Bizoń
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oleg Evnin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anxo Biasi.

Additional information

Communicated by P. Chrusciel

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biasi, A., Bizoń, P. & Evnin, O. Solvable Cubic Resonant Systems. Commun. Math. Phys. 369, 433–456 (2019). https://doi.org/10.1007/s00220-019-03365-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03365-z

Search

Navigation