Topics in Pattern Formation Far From Threshold

数学科学：远离阈值的模式形成主题[摘要]ercolani本研究考察了当物理系统的应力远高于阈值时，模拟模式形成的偏微分方程的各个方面。特别强调的是理解这些系统中缺陷的结构，特别是在奇异极限中，因为一些正则化被删除了。对于这个项目，所考虑的特定模型是正则化的Cross-Newell相扩散方程。该方程的自由能与二维金兹堡-朗道自由能相似，只是变化仅限于梯度向量场的域。相关的缺陷结构通常是一维的，而不是点缺陷。这个项目探讨了与这个有趣的模型相关的四个研究问题。第一个问题是将模型一般化，以包含方向场（矢量场的无方向类似物）的变化。这种扩展可能对理解远离阈值的迷宫模式的出现产生重要影响。第二个项目涉及改进自对偶约简技术（一种确定自由能渐近极小值的自然候选者的新方法），以结合一般边界条件和一般几何形状。第三个项目将把模型扩展到三个或更多的维度。潜在的应用将是模拟纤维的崩溃，在最近的一些细菌菌落的实验研究中看到。最后的项目将是研究相方程在模型模式形成微观系统中的有效性。几乎具有周期性结构的模式是普遍存在的。人们在自然界中看到它们，就像沙子的涟漪，老虎的条纹，指纹，在大气和地质构造中。在实验室里，它们在光束实验、对流实验、火焰前沿实验、磁膜实验、法拉第波实验中都能看到。在非常不同的微观环境中产生的图案纹理之间惊人的相似性，不仅在平面（条纹，六边形）中，而且在缺陷结构中，表明图案是具有普遍特征的宏观物体，仅依赖于不同微观情况共享的共同对称性。寻找宏观的描述，统一和简化我们对模式行为的理解，无论它发生在哪里，这是推动本研究的主要目标。

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.