Imperfect Patterns-Existence, Stability, and Essential Spectrum

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How do zebras get their stripes? How do cell membranes get their shapes and keep from bursting into pieces? These two seemingly disconnected scientific topics can actually be investigated via a general mathematical framework outlined within this project. Such a study not only applies to the specific topics above, but also to any physical systems arising from similar mathematical models, such as sand patterns in deserts, cloud patterns in the sky, cell fission/fusion processes, cast ionomers in solar cells, and network morphology in soapy water. The aim of this project is to systematically understand imperfections of patterns, which are ubiquitous in nature and pivotal in various scientific settings and technical applications. The investigator studies defects in periodic patterns, with an application to understanding the development of grain boundaries in materials; he also studies defects in bilayers such as arise in cell membranes. Part of the project includes research experience opportunities for undergraduate students.The investigator applies dynamical systems techniques, combined with functional analysis, differential geometry, asymptotic analysis, and large deviation theory, to study pattern formation, with an emphasis on the role of the essential spectrum in the formation and stability of defects of periodic patterns and bilayer interfaces. The most interesting and challenging case occurs when the essential spectrum touches the origin; here he seeks explicit illustrations of formation mechanisms of stripe patterns, amphiphilic structures, and their imperfections. The classical Swift-Hohenberg equation provides a prototype for rigorous studies of periodic patterns. Firstly, the imperfections of periodic patterns to instantaneous, constant, and random perturbations are investigated. Secondly, in the case of instability, the local biases and the rotational symmetry of the system together give rise to various line and point defects, such as grain boundaries, dislocations, and disclinations. The investigator studies grain boundaries, providing a novel functional analytical machinery to construct deformed patterns. He also investigates defects of amphiphilic morphology in the functionalized Cahn-Hilliard FCH) setting, suggesting mechanisms of the formation of lipid rafts, end caps and Y-junctions. Here the essential spectrum of the linearized operator at bilayer interfaces is continuous up to the origin but is not "simple." It serves as a benchmark for understanding the role of essential spectra. Part of the project includes research experience opportunities for undergraduate students.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.