Critical Phenomena in Coherent Structure Formation

大型复杂动力系统经常自组织成简单的连贯行为。描述、分析和计算这些自组织结构的动力学是理解从结晶、自组装或材料硬化到生物体中模式和功能的出现等过程的核心。首席研究员（PI）研究连贯结构的作用，既通过选择模式和形状来组织系统的动力学，也通过定量预测来验证模型。项目的第一部分涉及入侵过程，在参数突然改变后，系统中会传播新的结构。这种入侵过程很难理解，并且在计算上具有挑战性。PI将开发新的分析和计算工具来系统地分析、预测并验证模型。项目的第二部分是关于在软物质中产生的点缺陷。点缺陷既组织了系统动力学，也对大系统的宏观、均质化描述提出了挑战。提出的工作进一步发展了分析和计算方法，旨在精确描述点缺陷的核心形状及其对远离核心的介质的影响。研究生和本科生将参与该项目的研究。PI将分析入侵前线和指出缺陷。入侵前沿描述了从材料科学到生态学和流行病学等许多科学领域进入不稳定状态的传播。传播速度和在入侵过程后产生的新状态通常可以通过边缘点稳定性分析从不稳定状态的线性分析中很好地预测出来。PI将首先为相关的线性问题开发新的计算工具，并辅以严格的收敛分析。在第二步，PI将解决非线性系统中这些线性预测的有效性，特别是与边缘稳定性猜想相关的问题，该猜想假设在入侵过程中选择了关键的，只有边缘稳定的锋面。该建议的第二部分研究了条纹相中的位错和弹性介质中的折痕。在这两种情况下，分析这些点缺陷的存在将使我们更深入地了解缺陷核心在介质大规模变形中的作用。核心和远场之间的相互作用分析使用解析，微扰工具和计算方法，依赖于严格的近似无界的有限大小的域。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。

Large complex dynamical systems often self-organize into simple coherent behavior. Describing, analyzing, and computing the dynamics of these self-organized structures is at the center of understanding processes ranging from crystallization, self-assembly, or material hardening, to the emergence of patterns and function in living organisms. The principal investigator (PI) studies the role of coherent structures both in organizing the dynamics of systems through the selection of patterns and shape, but also in their role for validating models through quantitative predictions. The first part of the project is concerned with invasion processes, where a new structure propagates in the system after a sudden change in parameters. This invasion process is poorly understood and computationally challenging. The PI will develop new analytical and computational tools to systematically analyze, predict, and thus validate models. The second part of the project is concerned with point defects that arise in soft matter. Point defects both organize the system dynamics, but also present challenges to macroscopic, homogenized descriptions of large systems. The proposed work further develops analytical and computational methods that aim at a fine description of the shape of the core of point defects and their effect on the medium far away from the core. Graduate and undergraduate students will be engaged in the research of the project.The PI will analyze invasion fronts and point defects. Invasion fronts describe the propagation into an unstable state across many scientific areas, from material science to ecology and epidemiology. Propagation speeds and the novel state created in the wake of the invasion process can often be well predicted from linear analysis at the unstable state, through a marginal pointwise stability analysis. The PI will first develop novel computational tools for the associated linear questions, complemented by a rigorous convergence analysis. In a second step, the PI will addresse the validity of these linear predictions in nonlinear systems, in particular questions associated with the marginal stability conjecture that postulates that critical, only marginally stable fronts are selected in the invasion process. A second part of the proposal studies dislocations in striped phases and creases in elastic media. In both situations, analysis of the existence of these point defects will yield a deeper understanding of the role of the core of the defect in the large-scale deformation of the medium. The interaction between core and farfield is analyzed using analytical, perturbative tools and computational methods that rely on rigorous approximations of unbounded by finite-size domains.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.