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.
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 developed 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. In particular, our findings allow us to predict the parameter dependence of such structures when key features of the underlying system are perturbed. 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. We also constructed and analysed microscopic traffic flow models that reproduce actual traffic density-flow diagrams and that help explain the occurrence of traffic jams that occur without the presence of bottlenecks. The key feature of this model is that it accounts for variable safety distances that drivers choose based on the velocity of the preceding car and the distance to the car head. Finally, we developed algorithms that help assimilate data from floaters and drifters into mathematical fluid-flow models. The goal of data assimilation is to reconcile numerical predictions of the evolving fluid with actual observations; the challenge is that the observations are sparse and give only limited information about the fluid. The algorithm developed here helps overcome challenges that come from the nonlinearity and high-dimensionality of fluid-flow equations.
