Pelwan, Chad2021-04-262021-04-262020https://hdl.handle.net/10539/31004A thesis submitted to the Faulty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. September 2020In Part I, we consider a quantum quench from the strongly correlated ground state of the Kondo model to a Fermi sea. We calculate the overlap between the ground states before and after the quench, as well as the Loschmidt echo, that is, the transition amplitude between the initial state and the evolved state at a time t after the quench. The overlap is known to determine the dynamics of the echo at large times. We show, in addition, that the overlap depends algebraically on the emergent Kondo length, with a power-law exponent that is the difference of long-and short-time contributions that appear in the echo. Our result suggests that in general, there may be more information contained in the overlap than previously. In Part II, we study the effects of increasing modulation instability and random fluctuations on the onset times of rogue waves in a waveguide array as described by the discrete unstable nonlinear Schrödinger equation (UNLSE). We analytically determine regions of instability, where rogue waves are likely to occur in the UNLSE and then we use numerical techniques to study the time evolution of these systems. We find that the effects of the fluctuations are prominent on the onset times of rogue waves only for small modulation instability, but otherwise large modulation instability dominated the onset time behaviourenThesis