Nonlinear Collapse in the Semilinear Wave Equation in AdS

TL;DR

This paper investigates the collapse behavior of the semilinear wave equation in Anti-de Sitter space, revealing that even small initial data can lead to collapse due to boundary-induced dispersion, contrasting with Minkowski space results.

Contribution

It demonstrates that in AdS space, small initial data can cause collapse, highlighting the role of boundary conditions and dispersion in nonlinear wave dynamics.

Findings

Small initial data in AdS can lead to collapse due to boundary effects.

A threshold exists below which initial data does not collapse.

Boundary conditions induce dispersion that affects collapse behavior.

Abstract

Previous studies of the semilinear wave equation in Minkowski space have shown a type of critical behavior in which large initial data collapse to singularity formation due to nonlinearities while small initial data does not. Numerical solutions in spherically symmetric Anti-de Sitter (AdS) are presented here which suggest that, in contrast, even small initial data collapse eventually. Such behavior appears analogous to the recent result of Ref. [1] that found that even weak, scalar initial data collapse gravitationally to black hole formation via a weakly turbulent instability. Furthermore, the imposition of a reflecting boundary condition in the bulk introduces a cut-off, below which initial data fails to collapse. This threshold appears to arise because of the dispersion introduced by the boundary condition.