# Conformal Flow on S-3 and Weak Field Integrability in AdS(4)

Piotr Bizoń, Ben Craps, Oleg Evnin, Dominika Hunik, Vincent Luyten, Maciej Maliborski

Research output: Contribution to journalArticlepeer-review

23 Citations (Scopus)

## Abstract

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 (AdS$_4$) 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 Szego equation, which was shown by Gerard and Grellier to be Lax-integrable, it is tempting to conjecture that the conformal flow and the corresponding weak field dynamics in AdS$_4$ are integrable as well.
Original language English 1179-1199 Communications in Mathematical Physics 353 3 https://doi.org/10.1007/s00220-017-2896-8 Published - Aug 2017

### Bibliographical note

22 pages, v2: minor revisions, several references added, v3: typos corrected, v4: typos corrected, one reference added, matches version accepted by CMP

• math.AP
• gr-qc
• hep-th
• math-ph
• math.MP
• nlin.SI

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