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Speaker: Ed McDugald, Applied Mathematics
Title: Natural Stripe Patterns: Extracting Order Parameters and Locating Defects in Elliptical Domains
Abstract: Natural stripe patterns, common in systems such as convection, nonlinear optics, sand ripples, and epidermal patterns, are often modeled through the Swift–Hohenberg equation. Reduced descriptions in terms of order parameters are motivated by the fact that such patterns are locally plane-wave-like. This observation makes it possible to introduce slow variables—such as phase and amplitude—that vary smoothly except at special points known as defects, where the regular stripe pattern breaks down or reorganizes. These coarse-grained models capture the large-scale, averaged dynamics of the pattern, describing the evolution of features like orientation and wavelength without tracking every microscopic detail. Classical reductions such as the Cross–Newell equation capture phase diffusion in defect-free regions but fall short in describing defect dynamics. I will present progress toward two related problems. First, we are developing algorithms to reliably extract order parameters from simulation data, with the goal of using data-driven discovery methods (e.g., weak SINDy) to identify PDE models that extend the Cross–Newell description to defect-rich regimes. Second, we study pattern formation in restricted geometries, focusing on convection in ellipses. Numerical evidence suggests a geometric rule predicting defect locations, which informs the formulation of a variational problem we are working to solve. Together, these advances provide new insight into both the structure and possible modeling approaches for natural stripe patterns.