Long time dynamics of resonant systems

Abstract

This thesis studies the long-time dynamics of resonant systems. In particular, we focus our effort on the weakly nonlinear limit models. The thesis is divided into two main branches. In the first part we study the long-time evolution of conservative systems; namely, models with conserved quantities associated with the energy, number of particles, angular momentum, etc. For this purpose, we construct an effective equation that captures the behavior of this kind of systems at long-timescales. We consider this equation with a high degree of generality as our starting point. Specifically, we will construct families of systems that admit an analytic resolution of this equation, as well as, a particular limit where we can extract useful information from the system. Once we develop the general tools, we will apply them to a large number of resonant models. Among them, we can find the Gross-Pitaevskii equation and nonlinear wave equations in anti-de Sitter spacetime. In the second part of this thesis, we consider asymptotically anti-de Sitter geometries subject to time-periodic boundary conditions. Specifically we study two models, a scalar field in anti-de Sitter in global coordinates and purely gravitational fluctuations of the anti-de Sitter soliton. For both models time-periodic geometries perfectly synchronized with the boundary conditions will be constructed. We will also construct the phase-space of these objects, delimiting the regions of linear stability and thereafter the end states of the present instabilities will be inspected. We will find that in addition to the collapse into a black hole, there are other end states which remain regular with a long-time modulation. After that, we will show that the time-periodic geometries can be dynamically constructed from the static geometry by quenching the boundary conditions.