Communications in Mathematical Physics

, Volume 353, Issue 3, pp 1179–1199 | Cite as

Conformal Flow on S3 and Weak Field Integrability in AdS4

  • Piotr BizońEmail author
  • Ben Craps
  • Oleg Evnin
  • Dominika Hunik
  • Vincent Luyten
  • Maciej Maliborski
Open Access


We consider the conformally invariant cubic wave equation on the Einstein cylinder \({\mathbb{R} \times \mathbb{S}^3}\) for small rotationally symmetric initial data. This simple equation captures many key challenges of nonlinear wave dynamics in confining geometries, while a conformal transformation relates it to a self-interacting conformally coupled scalar in four-dimensional anti-de Sitter spacetime (AdS4) and connects it to various questions of AdS stability. We construct an effective infinite-dimensional time-averaged dynamical system accurately approximating the original equation in the weak field regime. It turns out that this effective system, which we call the conformal flow, exhibits some remarkable features, such as low-dimensional invariant subspaces, a wealth of stationary states (for which energy does not flow between the modes), as well as solutions with nontrivial exactly periodic energy flows. Based on these observations and close parallels to the cubic Szegő equation, which was shown by Gérard and Grellier to be Lax-integrable, it is tempting to conjecture that the conformal flow and the corresponding weak field dynamics in AdS4 are integrable as well.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided 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.

Authors and Affiliations

  • Piotr Bizoń
    • 1
    • 4
    Email author
  • Ben Craps
    • 2
  • Oleg Evnin
    • 2
    • 3
  • Dominika Hunik
    • 1
  • Vincent Luyten
    • 2
  • Maciej Maliborski
    • 4
  1. 1.Institute of PhysicsJagiellonian UniversityKrakówPoland
  2. 2.Theoretische NatuurkundeVrije Universiteit Brussel and The International Solvay InstitutesBrusselsBelgium
  3. 3.Department of Physics, Faculty of ScienceChulalongkorn UniversityBangkokThailand
  4. 4.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutGolmGermany

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