Pattern formation with a conservation law
The evolution of many pattern-forming systems is strongly influenced by the presence of a conserved quantity. Diverse physical phenomena such as magnetoconvection, rotating fluid convection, binary fluid convection, vibrated granular and fluid layers, filament dynamics and sandbank formation, all possess a conservation law which plays a central role in their nonlinear dynamics. In this thesis, this influence of a conserved quantity is examined through analyses of three distinct pattern-formation problems. Firstly, the consequences of conservation of mass are investigated in a phenomenological model of a vibrated granular layer. A new weakly nonlinear analysis is performed that reveals the existence of modulational instabilities for patterns composed of either stripes and squares. The nonlinear evolution of these instabilities is numerically studied and a plethora of patterns and localised arrangements are exhibited. The second component of this work concerns an oscillatory bifurcation in the presence of a conserved quantity. Building upon existing theory for the corresponding stationary bifurcation, universal amplitude equations are constructed through symmetry and asymptotic considerations. Subsequently, the stability properties of travelling and standing waves are found to be significantly altered and new modulational instabilities are uncovered. Numerical simulations reveal that, in the presence of a conserved quantity, travelling and standing waves lose stability to spatially localised patterns, either coherent, time-periodic or chaotic. Finally, wave-like behaviour of large-scale modes is examined through an analysis of a model for Faraday waves, that has been modified to account for flnite fluid depth. Several approaches to the weakly nonlinear analysis are considered and two sets of amplitude equations are derived that account for the unusual wave-like behaviour of large-scale modes. Numerical simulations reveal amplitude-modulated and localised patterns away from the small-amplitude, weak-viscosity limit.