Topics in Pattern Formation Far From Threshold

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NSF Award Abstract - DMS-0073087Mathematical Sciences: Topics in Pattern Formation Far From ThresholdAbstract0073087 ErcolaniThis research examines various aspects of partial differential equations that model pattern formation in physical systems when they are stressed well above threshold. Particular emphasis is placed on understanding the structure of defects in these systems, especially in the singular limit as some regularization is removed. For this project the particular model under consideration is the regularized Cross-Newell phase diffusion equation. This equation has a free energy similar to two-dimensional Ginzburg-Landau free energy except that variation is restricted to the domain of gradient vector fields. Associated defect structures are typically one-dimensional rather than being point defects. This project explores four research problems related to this interesting model. The first concerns generalizing the model to incorporate variations over director fields (unoriented analogues of vector fields). This extension may have important consequences for understanding the emergence of labyrinthine patterns far from threshold. The second project involves refining the technique of self-dual reduction (a novel method to determine natural candidates for asymptotic minimizers of the free energy) to incorporate general boundary conditions as well as general geometries. The third project will extend the model to three or more dimensions. Potential applications would be to modeling filamentary collapse, seen in some recent experimental studies of bacterial colonies. The final project will be to undertake a study of the validity of the phase equation within a model pattern-forming microscopic system. Patterns with almost periodic structure are ubiquitous. One sees them in nature as sand ripples, as tiger stripes, as fingerprints and in atmospheric and geological formations. In the laboratory, they are seen in experiments on optical beams, on convection, on flame fronts, as labyrinths on magnetic films, as textured Faraday waves. The striking similarity between pattern textures arising in very different microscopic contexts, not only in planform (striped, hexagonal) but also in defect structures, suggests that patterns are macroscopic objects with universal features depending only on common symmetries shared by different microscopic situations. Finding macroscopic descriptions which unify and simplify our understanding of pattern behavior wherever it occurs is the principal objective motivating this research.