In this thesis we study the chaotic behavior of multidimensional Hamiltonian systems in the presence of nonlinearity and disorder. It is known that any localized initial excitation in a large enough linear disordered system spreads for a finite amount of time and then halts forever. This phenomenon is called Anderson localization (AL). What happens to AL when nonlinearity is introduced is an interesting question which has been considered in several studies over the past decades. Recent works focussing on two widely–applicable systems, namely the disordered Klein-Gordon (DKG) lattice of anharmonic oscillators and the disordered discrete nonlinear Schr¨odinger (DDNLS) equation, mainly in one spatial dimension suggest that nonlinearity eventually destroys AL. This leads to an infinite diffusive spreading of initially localized wave packets whose extent (measured for instance through the wave packet's second moment m2) grows in time t as t αm with 0 < αm < 1. However, the characteristics and the asymptotic fate of such evolutions still remain an issue of intense debate due to their computational difficulty, especially in systems of more than one spatial dimension. Two different spreading regimes, the so-called weak and strong chaos regimes, have been theoretically predicted and numerically identified. As the spreading of initially localized wave packets is a non-equilibrium thermalization process related to the ergodic and chaotic properties of the system, in our work we investigate the properties of chaos studying the behavior of observables related to the system's tangent dynamics. In particular, we consider the DDNLS model of one (1D) and two (2D) spatial dimensions and develop robust, efficient and fast numerical integration schemes for the long-time evolution of the phase space and tangent dynamics of these systems. Implementing these integrators, we perform extensive numerical simulations for various sets of parameter values. We present, to the best of our knowledge for the first time, detailed computations of the time evolution of the system's maximum Lyapunov exponent (MLE–Λ) i.e. the most commonly used chaos indicator, and the related deviation vector distribution (DVD). We find that although the systems' MLE decreases in time following a power law t αΛ with αΛ < 0 for both the weak and strong chaos cases, no crossover to the behavior Λ ∝ t −1 (which is indicative of regular motion) is observed. By investigating a large number of weak and strong chaos cases, we determine the different αΛ values for the 1D and 2D systems. In addition, the analysis of the DVDs reveals the existence of random fluctuations of chaotic hotspots with increasing amplitudes inside the excited part of the wave packet, which assist in homogenizing chaos and contribute to the thermalization of more lattice sites. Furthermore, we show the existence of a dimension-free relation between the wave packet spreading and its degree of chaoticity between the 1D and 2D DDNLS systems. The generality of our findings is confirmed, as similar behaviors to the ones observed for the DDNLS systems are also present in the case of DKG models.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uct/oai:localhost:11427/33780 |
| Date | 17 August 2021 |
| Creators | Many, Manda Bertin |
| Contributors | Skokos, Charalampos |
| Publisher | Faculty of Science, Department of Mathematics and Applied Mathematics |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Doctoral Thesis, Doctoral, PhD |
| Format | application/pdf |
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