CAREER: Pattern formation in singularly perturbed partial differential equations

职业：奇异摄动偏微分方程中的模式形成

• 批准号: 2238127
• 负责人: Paul Carter
• 金额: $ 49.67万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2028-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2238127&HistoricalAwards=false
• 关键词: CAREER; Pattern; formation; singularly; perturbed

Spatiotemporal pattern formation appears throughout the natural world; examples include the dynamics of vegetation patterns in dryland ecosystems, phase separation in mixtures, and patterns in chemical reactions. Fundamental theoretical questions concern identifying universal pattern-forming mechanisms and predicting the nature and dynamics of the patterns which appear. This project aims to address these questions through the study of pattern-forming instabilities in the setting of partial differential equations (PDEs) arising in models from biology, ecology, and chemical reactions. In particular, the project focuses on the development of techniques to tackle pattern formation in PDEs which are singularly perturbed; such equations arise naturally in the study of systems where components operate on widely differing length or time scales. Application areas include desertification fronts in dryland ecosystems, as well as wave phenomena in mathematical biology and neuroscience. Included in the project are activities which integrate research and education through the supervision of graduate students as well as outreach programs and summer research opportunities for undergraduate students.This project focuses on three primary research areas, with the goal of providing insight into complex spatiotemporal pattern formation phenomena in singularly perturbed reaction diffusion PDEs. Motivated by the phenomenon of desertification fronts in dryland ecosystems, the first research area concerns pattern-forming instabilities of bistable planar interfaces in multi-component reaction diffusion systems. The second area aims to build a rigorous theory of temporal pulse replication, by which a single traveling pulse (or localized wave) self-replicates, sequentially nucleating additional pulses in a manner resembling a spatiotemporal canard explosion. The third area is concerned with the nature and properties of far-from-onset patterns formed in the wake of pattern-forming invasion processes, whereby a pattern is nucleated when a disturbance invades an unstable steady state. These research areas will grow and advance the theory of far-from-onset pattern-forming mechanisms through the development of geometric singular perturbation methods and spatial dynamical systems tools to be used in the existence, stability, and bifurcation analysis of waves and patterns. Several of the phenomena explored in this project occur in regions where standard singular perturbation methods break down, requiring the development of novel techniques. Graduate students and undergraduate students will participate in various aspects of the project.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.

时空模式形成出现在整个自然世界中。例子包括旱地生态系统中植被模式的动力学，混合物中的相分离以及化学反应中的模式。基本的理论问题涉及确定普遍模式形成机制并预测出现模式的性质和动态。该项目旨在通过研究模型，生物学，生态和化学反应的模型中引起的部分微分方程（PDE）来解决这些问题。 特别是，该项目的重点是针对奇异扰动的PDE中的模式形成的技术的发展；这种方程自然出现在对组件以差异差异或时间尺度差异的系统的研究中。应用领域包括旱地生态系统中的荒漠化阵线，以及数学生物学和神经科学中的波浪现象。该项目包括通过研究生的监督以及针对本科生的宣传计划和夏季研究机会整合研究和教育的活动。该项目侧重于三个主要研究领域，目的是洞悉复杂的时空模式形成形成现象，用于奇异扰动的受扰反应反应扩散PDES。由旱地生态系统中荒漠化阵线现象的激励，第一个研究领域涉及多组分反应扩散系统中可动的平面界面的模式形成不稳定性。第二个区域旨在建立一个严格的时间脉冲复制理论，通过该理论，单个行进脉冲（或局部波）自我复制，以类似于时空的canard爆炸的方式将其他脉冲依次成核。第三个区域涉及在模式形成的入侵过程中形成的远距离发作模式的性质和特性，当干扰侵入不稳定的稳态时，将模式成核。这些研究领域将通过开发几何奇异扰动方法和空间动力学系统工具来发展和推进远离模式形成机制的理论。该项目中探索的几种现象发生在标准奇异扰动方法分解并需要开发新技术的地区。研究生和本科生将参与该项目的各个方面。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。