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General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th,Mathematics, Analysis of PDEs, math.AP
Abstract:
This thesis is concerned with dynamics of conservative nonlinear waves on
bounded domains. In general, there are two scenarios of evolution. Either the
solution behaves in an oscillatory, quasiperiodic manner or the nonlinear
effects cause the energy to concentrate on smaller scales leading to a
turbulent behaviour. Which of these two possibilities occurs depends on a model
and the initial conditions. In the quasiperiodic scenario there exist very
special time-periodic solutions. They result for a delicate balance between
dispersion and nonlinear interaction. The main body of this dissertation is
concerned with construction (by means of perturbative and numerical methods) of
time-periodic solutions for various nonlinear wave equations on bounded
domains. While turbulence is mainly associated with hydrodynamics, recent
research in General Relativity has also revealed turbulent phenomena. Numerical
studies of a self-gravitating massless scalar field in spherical symmetry gave
evidence that anti-de Sitter space is unstable against black hole formation. On
the other hand there appeared many examples of asymptotically anti-de Sitter
solutions which evade turbulent behaviour and appear almost periodic for long
times. We discuss here these two contrasting scenarios putting special
attention to the construction and properties of strictly time-periodic
solutions. We analyze different models where solutions of this type exist.
Moreover, we describe similarities and differences among these models
concerning properties of time-periodic solutions and methods used for their
construction.
