Self-Organization, Stability, and Defects in Pattern-Forming Systems

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Self-organization is the process by which a system develops patterns or regular structure in the absence of external guidance. This phenomenon of pattern formation occurs in many natural and physical systems, for example the organization of vegetation into patches in semiarid ecosystems, stripes and spots which appear on animal coats, or the rippling of stretched thin films. This project will advance the theory of patterns through the analysis of model partial differential equations (PDEs) describing patterns in applications. A particular focus will be the stability of patterns, or their resilience to perturbations or environmental changes, as well as the formation and control of defects, or imperfections, in pattern forming systems. In terms of application areas, this project will contribute to our understanding of vegetation patterns and the phenomenon of desertification and will explore the effect of environmental conditions on pattern selection. Additionally, the project will advance knowledge concerning the structure and control of wrinkling patterns in stressed materials, as well as the formation and stability of various morphologies in amphiphilic systems. Undergraduate students will participate in the project through summer REU (research experience for undergraduates) opportunities.This project is organized around three primary topics: (i) stripes and spots in reaction-diffusion-advection equations describing vegetation patterns, (ii) defects and wrinkling patterns in (an)isotropic Swift–Hohenberg systems, (iii) existence and stability of amphiphilic structures in density functional models. These research topics will inform specific applications as well as develop and advance the theory of planar (and higher dimensional) patterns; further, they will examine the effect of anisotropy in pattern selection, stability, and defect formation in example systems. This research builds on the existence and stability theory of planar and higher-dimensional far-from-onset patterns through the development and use of singular perturbation methods; these methods will have broad applicability in reaction-diffusion-advection systems, and singularly perturbed PDEs more generally. Specifically, tools will be developed to analyze the appearance and stability of striped patterns, as well as radially symmetric patterns such as spots and rings in the context of vegetation, and cylindrical/spherical micelles and vesicles in amphiphilic systems. Spatial dynamics and functional analytic methods will also be developed to construct far-from-onset radial lattice patterns. Additionally, center manifold theory and dynamical systems techniques will be used to analyze the stability of a variety of defects in (an)isotropic systems.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.