Growth and patterns: existence, stability, and dynamics

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This project is focused on how spatially-periodic patterns arise in natural systems and how they are selected and controlled by domain growth, geometry, and more generally, spatio-temporal heterogeneities. A pattern typically refers to a regular, repeating geometric structure, and examples abound in many different scientific domains, such as striped convection rolls in fluids, spiral-seed arrangements on a sunflower, and differentiation in mammalian embryos in early stage gastrulation. In both natural and man-made settings, spatial growth processes have proven to be powerful tools for organizing and harnessing such pattern-forming mechanisms, while at the same time suppressing the formation of imperfections, commonly known as defects. This project seeks to rigorously understand the interaction of growth and patterns in mathematical models arising in various areas such as chemistry, material science, fluid dynamics, and biology, and used to study phenomena such as skeletal patterning in animals, quenching of eutectic metal alloys, evaporative chemical deposition, and defect suppression in elastic surface crystals. Such an understanding could also aid in the fabrication of novel and functional materials at macro-, micro, and nano-meter length scales, giving a cookbook for creating a desired structure. A facet of this project incorporates research experience opportunities for undergraduates.This project aims to develop and implement new techniques in infinite-dimensional dynamical systems theory and functional analysis to rigorously study growth in prototypical partial differential equation models for pattern-formation. It will also explore and illuminate complex phenomena by developing novel numerical approaches to approximate and continue spatially patterned structures in bounded and unbounded domains. To characterize the effect of growth speed and boundary curvature on pattern formation, the project will first focus on quenching inhomogeneities, which allow patterns in a sub-domain and suppress them in the complement. In simple planar quenching, the project will study existence, wavenumber selection, and stability of pattern-forming fronts in multi-dimensional spatial domains. As standard spatial dynamics methods apply here in only limited settings, functional analytic approaches will be developed to deal with the presence of continuous spectrum in bifurcation, perturbation, or continuation schemes for such patterned solutions. With the goal of studying patterns in non-planar quenching geometries, this project seeks to extend the tools of spatial dynamical systems to domains with more than one unbounded spatial direction. The project will also investigate pattern selection and defect formation for other types of spatio-temporal heterogeneity of physical interest, such as slowly-varying parameter ramps, temporally slow but spatially homogeneous quenches, and spatially localized source terms. In addition to considering existence, stability, and wavenumber selection of patterns, this project will also use modulational techniques to approximate the interactions and defects of selected patterns in terms of the dynamics of simpler, more tractable partial differential equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.