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  Dynamics of Nonlinear Waves on Bounded Domains

Maliborski, M. (2014). Dynamics of Nonlinear Waves on Bounded Domains. PhD Thesis.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-002A-F04F-E Version Permalink: http://hdl.handle.net/11858/00-001M-0000-002C-46B1-8
Genre: Thesis

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1603.00935.pdf (Preprint), 22MB
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 Creators:
Maliborski, Maciej1, Author              
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1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, escidoc:24012              

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Free keywords: 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.

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 Dates: 2016-03-022014
 Publication Status: Published in print
 Pages: PhD thesis, Jagiellonian University, November 2014. 251 pages, 70 figures, 10 tables. Corrected misprints
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 Identifiers: arXiv: 1603.00935
URI: http://arxiv.org/abs/1603.00935
 Degree: PhD

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