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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 selfinteracting conformally coupled scalar in fourdimensional antide Sitter spacetime (AdS$_4$) and connects it to various questions of AdS stability. We construct an effective infinitedimensional timeaveraged 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 lowdimensional 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 Laxintegrable, 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 

Pages (fromto)  11791199 
Journal  Communications in Mathematical Physics 
Volume  353 
Issue number  3 
DOIs  
Publication status  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 CMPKeywords
 math.AP
 grqc
 hepth
 mathph
 math.MP
 nlin.SI
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 1 Active

SRP8: Strategic Research Programme: HighEnergy Physics at the VUB
D'Hondt, J., Van Eijndhoven, N., Craps, B. & Buitink, S.
1/11/12 → 31/10/24
Project: Fundamental