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Abstract

We study the dynamics of unstable Reissner-Nordstr\"om anti-de Sitter black
holes under charged scalar field perturbations in spherical symmetry. We
unravel their general behavior and approach to the final equlibrium state. In
the first part of this work, we present a numerical analysis of massive charged
scalar field quasinormal modes. We identify the known mode families -
superradiant modes, zero-damped modes, AdS modes, and the near-horizon mode -
and we track their migration under variation of the black hole and field
parameters. We show that the zero-damped modes become superradiantly unstable
for large RNAdS with large gauge coupling; the leading unstable mode is
identified with the near-horizon condensation instability. In the second part,
we present results of numerical simulations of perturbed large RNAdS, showing
the nonlinear development of these unstable modes. For generic initial
conditions, charge and mass are ransferred from the black hole to the scalar
field, until an equilibrium solution with a scalar condensate is reached. We
use results from the linear analysis, however, to select special initial data
corresponding to an unstable overtone mode. We find that these data evolve to
produce a new equilibrium state - an excited hairy black hole with the scalar
condensate in an overtone configuration. This state is, however, unstable, and
the black hole eventually decays to the generic end state. Nevertheless, this
demonstrates the potential relevance of overtone modes as transients in black
hole dynamics.