There and Back Again: on (quasi-)periodic solutions in GR

This project proposes to study closed conservative systems described by nonlinear time- depended partial differential equations. The behaviour of conservative dynamical systems highly depends on the domain on which they are defined. For open systems, initial excitations can be radiated away, leaving behind a stationary configuration (e.g. a ringing bell). However, for closed systems, the excess energy can not escape. What happens with the excitation if we wait long enough? There is no clear and easy answer if the system is nonlinear, in which case there are many possibilities. For example, the solution may develop a singularity (e.g. black hole or blowup). Alternatively, the solution may oscillate irregularly or, in special cases, be time-periodic (like an undamped, eternally oscillating string). What exactly happens in the evolution depends on the specifics of the system but also on the initial perturbation. Studies of closed systems are in their infancy, and the current project aims to further develop this field. We want to explore quasi- and time-periodic solutions of the theory of General Relativity and also in simple model equations. For the Einstein equations to resemble a closed system, one needs a negative cosmological constant. In such a case, there is a `boundary` from which the energy and matter can be reflected. A fundamental issue is, whether such closed spacetimes are stable. To study this question we look at small deviations from such solutions. It is known that such spacetimes are generically tend to collapse to black holes. However, there is enough evidence that there exist quasi- or time-periodic configurations as well. Because of the intrinsic complexity of Einstein equations, we want to explore also simpler model equations (which find applications, e.g., in the study of trapped ultracold gases). For toy models, the analysis might be done using rigorous approaches. The knowledge gained here will provide a starting point for studying more complex systems. In our studies, we will use a combination of analytical and numerical methods. The intrinsic complexity of the problem makes computer assistance necessary. Besides, numerical techniques will be required for exploring regimes where approximation techniques are not accurate enough or simply fail (e.g. for studies of large solutions). Moreover, we will use computer-generated data as guidance for further rigorous developments. The finalisation of this research will considerably improve our understanding of the nonlinear dynamics of conservative partial differential equations on closed domains and the systems they model.