Self-Organization, Stability, and Defects in Pattern-Forming Systems

自组织是系统在缺乏外部指导的情况下开发模式或规则结构的过程。这种图案形成的现象发生在许多自然和物理系统中，例如，在半干旱的生态系统中，植被组织成一块块的斑块，动物皮毛上出现的条纹和斑点，或拉伸薄膜的涟漪。本项目将通过分析模型偏微分方程（PDEs）在应用中描述模式来推进模式理论。一个特别的焦点将是模式的稳定性，或者它们对扰动或环境变化的弹性，以及模式形成系统中缺陷或不完美的形成和控制。在应用领域，本项目将有助于我们对植被格局和沙漠化现象的认识，并将探索环境条件对格局选择的影响。此外，该项目将推进有关应力材料中起皱模式的结构和控制的知识，以及两亲系统中各种形态的形成和稳定性。本科生将通过暑期REU（本科生研究经验）机会参与该项目。该项目围绕三个主要主题进行组织：(i)描述植被模式的反应-扩散-平流方程中的条纹和斑点；（ii）各向同性Swift-Hohenberg系统中的缺陷和起皱模式；（iii）密度函数模型中两亲结构的存在和稳定性。这些研究课题将告知具体的应用，以及发展和推进平面（和高维）模式的理论；此外，他们将研究各向异性在模式选择、稳定性和缺陷形成方面的影响。本研究通过发展和使用奇异摄动方法，建立了平面和高维远起模式的存在性和稳定性理论；这些方法将广泛适用于反应-扩散-平流系统，以及更普遍的奇摄动偏微分方程。具体来说，将开发工具来分析条纹图案的外观和稳定性，以及植被背景下的径向对称图案，如斑点和环，以及两亲系统中的圆柱形/球形胶束和囊泡。空间动力学和功能分析方法也将被开发来构建远起点径向晶格模式。此外，中心流形理论和动力系统技术将用于分析各向同性系统中各种缺陷的稳定性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Self-organization is the process by which a system develops patterns or regular structure in the absence of external guidance. This phenomenon of pattern formation occurs in many natural and physical systems, for example the organization of vegetation into patches in semiarid ecosystems, stripes and spots which appear on animal coats, or the rippling of stretched thin films. This project will advance the theory of patterns through the analysis of model partial differential equations (PDEs) describing patterns in applications. A particular focus will be the stability of patterns, or their resilience to perturbations or environmental changes, as well as the formation and control of defects, or imperfections, in pattern forming systems. In terms of application areas, this project will contribute to our understanding of vegetation patterns and the phenomenon of desertification and will explore the effect of environmental conditions on pattern selection. Additionally, the project will advance knowledge concerning the structure and control of wrinkling patterns in stressed materials, as well as the formation and stability of various morphologies in amphiphilic systems. Undergraduate students will participate in the project through summer REU (research experience for undergraduates) opportunities.This project is organized around three primary topics: (i) stripes and spots in reaction-diffusion-advection equations describing vegetation patterns, (ii) defects and wrinkling patterns in (an)isotropic Swift–Hohenberg systems, (iii) existence and stability of amphiphilic structures in density functional models. These research topics will inform specific applications as well as develop and advance the theory of planar (and higher dimensional) patterns; further, they will examine the effect of anisotropy in pattern selection, stability, and defect formation in example systems. This research builds on the existence and stability theory of planar and higher-dimensional far-from-onset patterns through the development and use of singular perturbation methods; these methods will have broad applicability in reaction-diffusion-advection systems, and singularly perturbed PDEs more generally. Specifically, tools will be developed to analyze the appearance and stability of striped patterns, as well as radially symmetric patterns such as spots and rings in the context of vegetation, and cylindrical/spherical micelles and vesicles in amphiphilic systems. Spatial dynamics and functional analytic methods will also be developed to construct far-from-onset radial lattice patterns. Additionally, center manifold theory and dynamical systems techniques will be used to analyze the stability of a variety of defects in (an)isotropic systems.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.