Pattern Selection: Growth, Fronts, and Defects

该研究项目旨在通过直接建模和分析相干结构，以数学方式描述物理和生物生长过程。特定的模型示例创建规则的图案，如条纹或六边形点阵。本研究的目标是找到模式的数学特征，以解释自然界中普遍存在的这种规则模式，同时为微观结构的自组织制造提供配方。该项目包括培训研究生和本科生的研究活动。该项目侧重于摄动方法，适用于没有技术上可行的空间动力学解释的系统，包括非局部和多维问题。本文将探讨区域增长与模式形成的相互作用以及行进锋面速度选择的机制。特别是，本研究旨在开发利用严格的核心/远场分解来研究存在基本谱的无限维动力系统中的微扰和分岔问题的技术。除了局部扰动分析，这些策略将在具有先验和后验误差界的数值连续算法中实现。提出的特定目标是增长域中的波数和模式预测，模式形成系统中入侵速度的表征，以及异质介质中的渐近性。

This research project aims to mathematically describe physical and biological growth processes through direct modeling and analysis of coherent structures. Particular model examples create regular patterns such as stripes or hexagonal lattices of spots. The goal of this research is to find mathematical characterizations of patterns that explain the ubiquity of such regular patterns across nature and, at the same time, to provide recipes for the self-organized manufacturing of microstructure. The project includes research activities that will train graduate and undergraduate students. The project focuses on perturbation methods that are applicable to systems without a technically viable spatial dynamics interpretation, including nonlocal and multi-dimensional problems. It will explore the interaction of the domain growth and pattern-formation and mechanisms of the speed-selection for traveling fronts. In particular, the research aims to develop techniques for the study of perturbation and bifurcation problems in infinite dimensional dynamical systems in the presence of essential spectrum using rigorous core/far-field decompositions. Beyond local perturbation analysis, these strategies will be implemented in numerical continuation algorithms, with a priori and a posteriori error bounds. Particular goals set forth are wavenumber and pattern predictions in growing domains, characterization of invasion speeds in pattern-forming systems, and depinning asymptotics in heterogeneous media.