Non-linear periodic waves on the Einstein cylinder

Abstract

Motivated by the study of small amplitudes non-linear waves in the Anti-de-Sitter spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically-symmetric Yang-Mills equations on the Einstein cylinder R × S 3 . For the conformal cubic wave equation, we consider both spherically-symmetric solutions and complexed-valued aspherical solutions with an ansatz relying on the Hopf fibration of the 3-sphere. In all three cases, the equations reduce to 1+1 semi-linear wave equations. Our proof relies on a theorem of Bambusi-Paleari for which the main assumption is the existence of a seed solution, given by a non-degenerate zero of a non-linear operator associated with the resonant system. For the problems that we consider, such seed solutions are simply given by the mode solutions of the linearized equations. Provided that the Fourier coefficients of the systems can be computed, the non-degeneracy conditions then amount to solving infinite dimensional linear systems. Since the eigenfunctions for all three cases studied are given by Jacobi polynomials, we derive the different Fourier and resonant systems using linearization and connection formulas as well as integral transformation of Jacobi polynomials. In the Yang-Mills case, the original version of the theorem of Bambusi-Paleari is not applicable because the non-linearity of smallest degree is nonresonant. The resonant terms are then provided by the next order non-linear terms with an extra correction due to backreaction terms of the smallest degree non-linearity and we prove an analogous theorem in this setting. Finally, we emphasize that, in view of the KAM type method used, the time periodic solutions that we construct exist only when the small perturbative parameter belongs to a Cantor-like set.

Mode stability results for the Teukolsky equations on Kerr–anti-de Sitter spacetimes

“…8 The (in)stability of the anti-de Sitter limiting space M = a = 0 with Dirichlet-type conditions is still an open question. See [BR11] for numerical simulations revealing a turbulent instability mechanism in the context of the Einsteinscalar field system, [Mos18,Mos20] for recent breakthrough results establishing the instability of anti-de Sitter space in the context of the spherically symmetric Einstein-Vlasov system, and [CS22] for the construction of periodic solutions to non-linear wave equations on anti-de Sitter space.…”