Conformal Flow on S3 and Weak Field Integrability in AdS4

Abstract

We consider the conformally invariant cubic wave equation on the Einstein cylinder R×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 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 AdS 4 are integrable as well.

Nonsingular spacetimes with a negative cosmological constant: Stationary solutions with matter fields

“…The question of stability of the solutions with V ′′ (0) < 0 is unclear (compare [36], and there are in fact hints that some solutions with V ′′ (0) < 0 might be stable [9]), but this is irrelevant from the point of view of the question existence of static or stationary solutions, which is our main interest in this work.…”

Breathing Modes, Quartic Nonlinearities and Effective Resonant Systems

“…In the case of Bose-Einstein condensates, the existence of breathing modes is well-known [1][2][3], though their relation to the weakly nonlinear solutions within the resonant approximation has not been duly appreciated in the literature. For the case of [13], it is the solution of the resonant approximation that came first, and our present analysis will supply the corresponding breathing modes responsible for these solutions. The general type of resonant systems relevant for us here, where the presence of an extra conserved quantity imposes relations between mode couplings and generates some explicit analytic solutions, has been constructed in [17].…”

“…The structures outlined above have been observed in the recent literature for a number of special cases, motivated by rather disparate topics in physics. Thus, the analysis of [10][11][12] is rooted in the Gross-Pitaevskii equation and the physics of Bose-Einstein condensates, while the analysis of [13][14][15][16] originates in studies of nonlinear dynamics in Anti-de Sitter spacetimes, which is of interest for mathematical general relativity and high-energy physics. In the case of Bose-Einstein condensates, the existence of breathing modes is well-known [1][2][3], though their relation to the weakly nonlinear solutions within the resonant approximation has not been duly appreciated in the literature.…”

Ground State of the Conformal Flow on 𝕊3

“…This system, called the conformal flow on S 3 , has been introduced and studied in [3]. A major mathematical challenge in this context is to describe the energy transfer between eigenmodes of the corresponding linearized flow near the zero equilibrium.…”

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We consider the conformal flow model derived in Bizoń et al (2017) as a normal form for the conformally invariant cubic wave equation on S 3 . We prove that the energy attains a global constrained maximum at a family of particular stationary solutions that we call the ground state family.

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