We construct time-periodic solutions for a system of a self-gravitating massless scalar field, with a negative cosmological constant, in d+1 spacetime dimensions at spherical symmetry, both perturbatively and numerically. We estimate the convergence radius of the formally obtained perturbative series and argue that it is greater then zero. Moreover, this estimate coincides with the boundary of the convergence domain of our numerical method and the threshold for the black-hole formation. Then we confirm our results with a direct numerical evolution. This also gives strong evidence for the nonlinear stability of the constructed time-periodic solutions.

We consider spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant in five dimensions and analyze evolution of small perturbations of anti-de Sitter spacetime using the recently proposed resonant approximation. We show that for typical initial data the solution of the resonant system develops an oscillatory singularity in finite time. This result hints at a possible route to establishing instability of AdS under arbitrarily small perturbations.Introduction. A few years ago two of us gave numerical evidence that anti-de Sitter (AdS) spacetime in four dimensions is unstable against black hole formation for a large class of arbitrarily small perturbations [1]. More precisely, we showed that for a perturbation with amplitude ε a black hole forms on the timescale O(ε −2 ). Using nonlinear perturbation analysis we conjectured that the instability is due to the turbulent cascade of energy from low to high frequencies. This conjecture was extended to higher dimensions in [2].Since the computational cost of numerical simulations rapidly increases with decreasing ε, our conjecture was based on extrapolation of the observed scaling behavior of solutions for small (but not excessively so) amplitudes, which left some room for doubts whether the instability will persist to arbitrarily small values of ε (see e.g.[3]). To resolve these doubts, in this paper we validate and reinforce the above extrapolation with the help of a recently proposed resonant approximation [4][5][6]. The key feature of this approximation is that the underlying infinite dynamical system (hereafter referred to as the resonant system) is scale invariant: if its solution with amplitude 1 does something at time t, then the corresponding solution with amplitude ε does the same thing at time t/ε 2 . Moreover, the latter solution remains close to the true solution (starting with the same initial data) for times ε −2 (provided that the errors due to omission of higher order terms do not pile up too rapidly). Thus, by solving the resonant system we can probe the regime of arbitrarily small perturbations (whose outcome of evolution is beyond the possibility of numerical verification).For concreteness, in this paper we focus our attention on AdS 5 (the most interesting case from the viewpoint of AdS/CFT correspondence); an extension to other dimensions is straightforward and will be presented elsewhere.

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.

We consider a spherically symmetric self-gravitating massless scalar field enclosed inside a timelike worldtube R×S(3) with a perfectly reflecting wall. Numerical evidence is given that arbitrarily small generic initial data evolve into a black hole.