Mechanisms of localised pattern formation in systems governed by partial differential equations.

Abstract

This thesis studies the time-independent solutions of systems of partial differential equations given by reaction-diffusion models. The study is motivated by two different contexts: non-linear optics and cellular biology. In the context of optics, we study the Eckhaus instability observed in the Lugiato-Lefever model. Computing the amplitude equation, we predict the nascence and existence of solutions with different wave-numbers. The predictions are compared with the values obtained through numerical continuation. The agreement between theory and numerics is quantitatively good near the threshold and qualitatively good far from it. In the context of biology we investigate two models. The first is a generalised model for cell polarisation with two regimes, where the amount of protein is either conserved or not conserved. In the non-conservative regime, several equilibrium solutions are observed, studied and charac- terised. In the conservative regime, a novel description for the so-called wave-pinning mechanism is provided using an energetic approach. The last part of this problem is the transition from the non-conservative to the conservative regime. Numerical continuation reveals that only spike solutions persist, becoming a front solution in the conservative case. The problem is also analysed using matched asymptotics. The second problem is the modelling of the shapes observed at the interface between adjacent cells in the surface of plant leaves. We propose a generalisation of the curve-shortening flow problem coupled with a model for the concentration of ROP proteins. This model exhibits qualitatively the same shapes observed in the so-called pavement cells on the surface of plant leaves and allows some analytical calculations. The final problem is purely mathematical and it corresponds to the transition, observed in several systems, where the homoclinic snaking scenario is annihilated after the primary homoclinic orbit undergoes a Belyakov-Devaney transition. We study this problem using a generalisation of the Shilnikov-type analysis for homoclinic orbits.