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

自组织是一个系统在没有外部指导的情况下发展模式或规则结构的过程。这种模式形成的现象发生在许多自然和物理系统中，例如半干旱生态系统中的植被组织成斑块，动物皮毛上出现的条纹和斑点，或拉伸薄膜的波纹。本项目将通过分析在应用中描述模式的模型偏微分方程（PDE）来推进模式理论。一个特别的重点将是图案的稳定性，或它们对扰动或环境变化的弹性，以及图案形成系统中缺陷或缺陷的形成和控制。就应用领域而言，该项目将有助于我们了解植被格局和荒漠化现象，并将探讨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。本科生将通过夏季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系统中的缺陷和扭曲模式，（iii）密度泛函模型中两亲结构的存在性和稳定性。这些研究主题将为具体应用提供信息，并发展和推进平面（和更高维）图案的理论;此外，他们将研究各向异性对示例系统中图案选择，稳定性和缺陷形成的影响。这项研究建立在平面和高维远离发病模式的存在性和稳定性理论，通过发展和使用奇异摄动方法，这些方法将具有广泛的适用性，在反应扩散平流系统，奇异摄动偏微分方程更普遍。具体而言，将开发工具来分析条纹图案的外观和稳定性，以及径向对称的图案，如在植被的背景下的斑点和环，以及两亲性系统中的圆柱形/球形胶束和囊泡。空间动力学和功能分析方法也将被开发，以构建远离发病放射状晶格模式。此外，中心流形理论和动力系统技术将用于分析（一个）各向同性系统中的各种缺陷的稳定性。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.