Nonlinear stability of patterns

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Abstract:Spatio-temporal structures and patterns have been observed in many natural processes. For instance, travelling pulses arise in models of the propagation of nerve impulses in myelinated nerve fibers; oscillating, spatially localized structures have been observed in granular media, clay suspensions, and chemical reactions; defects that are embedded in an oscillatory medium occur as spiral waves in chemical reactions and as surface waves in fluid containers. The goal of this project is to investigate the emergence of such patterns and their stability with respect to small perturbations. The work aims to further our understanding of specific model equations as well as to establish results that apply more broadly.This project is divided broadly into three parts. In the first part, the investigator analyses Hopf and Turing Hopf bifurcations, which may lead to oscillons, in forced and unforced reaction-diffusion models. The main approach is to use, and extend, spatial-dynamics techniques together with geometric blow-up methods. The second part is to study the existence and stability of fast multi-pulses in the discrete FitzHugh-Nagumo equation. The strategy is to extend techniques from geometric singular perturbation theory near nonhyperbolic slow manifolds from the finite-dimensional setting to the case of ill-posed functional differential equations with advanced and retarded terms. In the third project, the investigators and collaborators study the nonlinear stability of contact defects, the interaction of source-sink pairs and of contact defects, and the linear stability of planar spiral waves. Spatial-dynamics techniques are used to derive expansions of the resolvent kernel of the linearization about a defect, which are then converted into pointwise estimates of the associated Green's function using Laplace transforms.