The work presented in this thesis focuses on the ultra-slow dynamics of localised modes (breathers) for the Discrete Nonlinear Schrödinger Equation (DNLSE), and on the mechanisms which trigger their astronomically slow decays.The thesis starts with a literature review of breathers and the DNLSE, giving information about the theoretical background of the project. It then continues with a description of the numerical methods used throughout the project, and of the general behaviour of the DNLSE under numerical integration.Discrete breathers in Bose-Einstein Condensates in optical lattices or in arrays of optical wave-guides oscillate with frequencies which are much higher than those present in the spectrum of the background. Hence, the interaction between localized breathers and their surroundings is extremely weak, allowing the development of a multiple-time scale perturbation expansion, which is presented in Chapter 4. This analysis will predict a lower bound of the breather drift times and will explain the topological differences between breathers in dimers, trimers and in spatially extended one-dimensional lattices even in the presence of transport from boundary heat baths.These analytical boundaries hold true for lattices of any length, due to the highly localised nature of breathers.We later look closer at the exceedingly slow thermalisation occurring in the DNLSE.We provide evidence that the breather norm is an adiabatic invariant, and this freezes the dynamics of a tall breather. Consequently, relaxation proceeds via rare events,where energy is suddenly released towards the background. A more detailed investigation of these events is provided in Chapter 5.In Chapter 6, a Heterogeneous Multiscale Method is introduced. We take advantage of the fast frequency of the breather oscillation to perform averaging, and to build a numerical integrator which behaves similarly to a predictor-corrector propagator,speeding up the running times by a factor of at least 5.
|Date of Award||10 May 2019|
- University Of Strathclyde
|Sponsors||University of Strathclyde|
|Supervisor||Gian-Luca Oppo (Supervisor) & Gordon Robb (Supervisor)|