Topics in Pattern Formation Far From Threshold

远离阈值的模式形成主题

• 批准号: 0073087
• 负责人: Nicholas Ercolani
• 金额: $ 12.6万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0073087&HistoricalAwards=false
• 关键词: Topics; Pattern; Formation; Far; Threshold

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.

NSF奖摘要-DMS -0073087数学科学：模式形成的主题远离阈值theSoldabstract0073087 Ercolanithis Research研究了部分微分方程的各个方面，这些方程在物理系统中模型形成在物理系统中时，当它们压力很好地高于阈值。 特别重点是理解这些系统中缺陷的结构，尤其是在去除某些正则化时在奇异极限中。 对于该项目，所考虑的特定模型是正规化跨纽维尔相扩散方程。 该方程的自由能类似于二维金茨堡 - 兰道自由能，只是变化仅限于梯度矢量场的域。 相关的缺陷结构通常是一维的，而不是点缺陷。该项目探讨了与这个有趣模型有关的四个研究问题。 第一个涉及概括该模型以将变体与导演字段（向量场的无定向类似物）结合在一起。 该扩展可能会对理解迷宫模式的出现远非阈值产生重要的后果。 第二个项目涉及完善自偶会减少的技术（一种确定自由能的自然候选者的新方法），以结合一般的边界条件以及一般的几何形状。 第三个项目将将模型扩展到三个或多个维度。潜在的应用将是对细菌菌落的一些实验研究中观察到的丝状塌陷。 最终项目将是对模型图案形成微观系统中相位方程的有效性进行研究。 几乎周期性结构的模式无处不在。 人们在自然界中将它们视为沙纹，是老虎条纹，指纹以及大气和地质地层。 在实验室中，在光束，对流，火焰正面，磁性膜上迷宫的实验中可以看到它们，如质感的法拉第波。 在非常不同的显微镜环境中产生的模式纹理之间的惊人相似性，不仅在平面形式（条纹，六角形）中，而且在缺陷结构中，都表明模式是宏观对象具有通用特征的宏观对象，仅取决于不同微观情况所共享的常见对称性。 寻找宏观描述，以统一和简化我们对模式行为的理解是激励这项研究的主要目标。