Long Time Dynamics of Resonant Systems

This thesis studies the long time dynamics of resonant systems in the weakly nonlinear regime. It is divided into two main parts. In the first one, we consider the resonant equation, which captures the energy transfer between normal modes of the system. Different tools to extract analytic information from the resonant equation are developed. After that, we apply them to a large number of resonant models. Some of them consist of a scalar field in different geometries as well as the Gross-Pitaevskii equation. In the second part of this thesis, asymptotically anti-de Sitter geometries subject to time-periodic boundary conditions are studied. The phenomenology allowed by these conditions is explored through the environment of time-periodic geometries. In particular, we construct their phase-space and delimit the regions of linear stability. We also present a protocol to dynamically construct time-periodic geometries

Energy returns in global AdS(4)

Recent studies of the weakly nonlinear dynamics of probe fields in global AdS 4 (and of the nonrelativistic limit of AdS resulting in the Gross-Pitaevskii equation) have revealed a number of cases with exact dynamical returns for two-mode initial data. In this paper, we address the question whether similar exact returns are present in the weakly nonlinear dynamics of gravitationally backreacting perturbations in global AdS 4 . In the literature, approximate returns were first pointed out numerically and with limited precision. We first provide a thorough numerical analysis and discover returns that are so accurate that it would be tantalizing to sign off the small imperfections as an artifact of numerics. To clarify the situation, we introduce a systematic analytic approach by focusing on solutions with spectra localized around one of the two lowest modes. This allows us to demonstrate that in the gravitational case the returns are not exact. Furthermore, our analysis predicts and explains specific integer numbers of direct-reverse cascade sequences that result in particularly accurate energy returns (elaborate hierarchies of more and less precise returns arise if one waits for appropriate longer multiple periods in this manner). In addition, we explain, at least in this regime, the ubiquitous appearance of direct-reverse cascades in the weakly nonlinear dynamics of AdS-like systemsThis research has been supported by CUniverse research promotion project (CUAASC), by FWO-Vlaanderen (projects No. G044016N and No. G006918N), by Vrije Universiteit Brussel through the Strategic Research Program “High-Energy Physics,” by FPA2014-52218-P from Ministerio de Economia y Competitividad, by Xunta de Galicia ED431C 2017/07, by the European Regional Development Fund (FEDER) and by Grant María de Maeztu Unit of Excellence MDM-2016- 0692.This research has benefited from the use computational resources/services provided by the Galician Supercomputing Centre (CESGA). A. B. thanks the Spanish program “ayudas para contratos predoctorales para la formación de doctores 2015” and its mobility program for his stay at Vrije Universiteit Brussel, where part of this project was developedS

Floquet scalar dynamics in global AdS

We study periodically driven scalar fields and the resulting geometries with global AdS asymptotics. These solutions describe the strongly coupled dynamics of dual finite-size quantum systems under a periodic driving which we interpret as Floquet condensates. They span a continuous two-parameter space that extends the linearized solutions on AdS. We map the regions of stability in the solution space. In a significant portion of the unstable subspace, two very different endpoints are reached depending upon the sign of the perturbation. Collapse into a black hole occurs for one sign. For the opposite sign instead one attains a regular solution with periodic modulation. We also construct quenches where the driving frequency and amplitude are continuously varied. Quasistatic quenches can interpolate between pure AdS and sourced solutions with time periodic vev. By suitably choosing the quasistatic path one can obtain boson stars dual to Floquet condensates at zero driving field. We characterize the adiabaticity of the quenching processes. Besides, we speculate on the possible connections of this framework with time crystals.This work of was supported by grants FPA2014-52218-P from Ministerio de Economia y Competitividad, by Xunta de Galicia ED431C 2017/07, by FEDER and by Grant Mar a de Maeztu Unit of Excellence MDM-2016-0692. A.S. is happy to acknowledge support from the International Centre for Theoretical Sciences (ICTS-TIFR), and wants to express his gratitude to the ICTS community, and especially to the String Theory Group, for their warm welcome. D.M. thanks the FRont Of pro-Galician Scientists (FROGS) for unconditional support. A.B. thanks the support of the Spanish program \Ayudas para contratos predoctorales para la formaci on de doctores 2015" associated to FPA2014-52218-P. This research has benefited from the use computational resources/services provided by the Galician Supercomputing Centre (CESGA).S