COMPLEX DYNAMICS OF LOCALIZED PATTERNS IN REACTION-DIFFUSION SYSTEMS

Abstract

In this thesis we study complex dynamics of the localized patterns that occur in certain partial differential equations. We study three different types of localized patterns: interfaces in one dimension, spots in two and three dimensions, and vortices in two dimensions. In the first part of the thesis, we study the oscillatory motion of multiple interfaces in one dimension for a certain class of reaction-diffusion systems. Within that class, we prove that the eventual fate of the system can be reduced to the study of a single interface. We then study a pattern consists of a single spot within a circular domain in a two-dimensional Schnakenberg model. Depending on parameter regime, such a spot can undergo periodic height oscillations or oscillations in its position. These oscillations are due to the presence of two different Hopf bifurcations. We derive explicit thresholds on the parameters which delineate these two regimes. Beyond the Hopf bifurcation, we also study the motion of a rotating spot and characterise explicitly the radius and frequency of its rotation. In three-dimensional context, we derive the slow dynamics of spot patterns and extend the analysis to the spatially varying feeding rate case. We then study vortex dynamics in the context of Bose-Einstein Condensates (BECs) with a rotating trap, with or without anisotropy. Starting with the Gross-Pitaevskii equation (GPE) , we derive a novel reduced ODE system that governs the slow dynamics and stability of multiple co-rotating vortices. In the limit of many vortices, we derive the effective vortex crystal density and its radius. For an anisotropic potential, we show that a pair of vortices lying on the long (short) axis is linearly stable (unstable), which is in agreement with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density. In each case, extensive full numerical simulations are used to confirm our analytical predictions.