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Conformal flow on S3 and weak field integrability in AdS4

Bizon, P., Craps, B., Evnin, O., Hunik, D., Luyten, V., & Maliborski, M. (2017). Conformal flow on S3 and weak field integrability in AdS4. Communications in Mathematical Physics, 353(3), 1179-1199. doi:10.1007/s00220-017-2896-8.

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### Creators

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Creators:
Bizon, Piotr, Author
Craps, Ben, Author
Evnin, Oleg, Author
Hunik, Dominika, Author
Luyten, Vincent, Author
Maliborski, Maciej1, Author
Affiliations:
1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24012

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Free keywords: Mathematics, Analysis of PDEs, math.AP,General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP,Nonlinear Sciences, Exactly Solvable and Integrable Systems, nlin.SI
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.

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Dates: 2016-08-252016-08-302017
Publication Status: Published in print
Pages: 22 pages, v2: minor revisions, several references added
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Rev. Type: -
Identifiers: arXiv: 1608.07227
URI: http://arxiv.org/abs/1608.07227
DOI: 10.1007/s00220-017-2896-8
Degree: -

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### Source 1

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Title: Communications in Mathematical Physics
Source Genre: Journal
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Publ. Info: Heidelberg : Springer-Verlag Heidelberg
Pages: - Volume / Issue: 353 (3) Sequence Number: - Start / End Page: 1179 - 1199 Identifier: ISSN: 0010-3616
CoNE: https://pure.mpg.de/cone/journals/resource/954925392313