Dynamics near coherent structures

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Many physical processes exhibit interesting patterns with a nontrivial spatio-temporal structure. Stationary and time-periodic localized spots and rings, for instance, occur in autocatalytic chemical reactions, ferro-fluids, cavity lasers, and as vegetation patches in deserts. Sources are more complex patterns that can be thought of as defects that actively organize a spatially periodic background medium: they arise as planar spiral waves and as one-dimensional flip-flops in chemical reactions and as surface waves in fluids. This project aims at developing techniques to analyze the existence, nonlinear stability, and parameter dependence of coherent structures for general reaction-diffusion systems and for systems posed on lattices. Example projects that we will carry out are geometric singular perturbation theory for functional differential equations of mixed type, the analysis of planar and three-dimensional localized structures near Turing and Hopf bifurcations, and the nonlinear stability of sources in reaction-diffusion systems using pointwise Green's function estimates.Coherent structures and nonlinear waves organize the dynamics of many biological, chemical and physical processes. Examples are pulsating combustion fronts, vegetation patches, nerve impulses, spiral waves in cardiac tissue, and localized convection rolls in fluid experiments. We will develop analytical and numerical tools to study under which conditions such structures appear and to determine their spatial extent and their stability properties as functions of system parameters. Among the potential technological applications are semiconductor lasers, which can exhibit localized hexagonal patches that may be used in all-optical storage devices, and fiber lasers whose optimization depends crucially on understanding the effect of design parameters on the power of the generated light wave.