  The goal for this paper is twofold. Our first main objective is to develop
Bahouri-Gerard type profile decompositions for waves on hyperbolic space.
Recently, such profile decompositions have proved to be a versatile tool in the
study of the asymptotic dynamics of solutions to nonlinear wave equations with
large energy. With an eye towards further applications, we develop this theory
in a fairly general framework, which includes the case of waves on hyperbolic
space perturbed by a time-independent potential. Our second objective is to use
the profile decomposition to address a specific nonlinear problem, namely the
question of global well-posedness and scattering for the defocusing, energy
critical, semi-linear wave equation on three-dimensional hyperbolic space,
possibly perturbed by a repulsive time-independent potential. Using the
concentration compactness/rigidity method introduced by Kenig and Merle, we
prove that all finite energy initial data lead to a global evolution that
scatters to linear waves in infinite time. This proof will serve as a blueprint
for the arguments in a forthcoming work, where we study the asymptotic behavior
of large energy equivariant wave maps on the hyperbolic plane.
