Critical Phenomena in Coherent Structure Formation

大型复杂动力系统常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在生物体内出现模式和功能的各种过程的中心。首席研究员(PI)研究相干结构在通过选择图案和形状来组织系统动力学方面的作用，以及通过定量预测来验证模型的作用。该项目的第一部分涉及入侵过程，其中一个新的结构在参数突然变化后在系统中传播。人们对这种入侵过程知之甚少，在计算上也颇具挑战性。PI将开发新的分析和计算工具，以系统地分析、预测并验证模型。该项目的第二部分涉及软物质中出现的点缺陷。点缺陷不仅组织了系统动力学，而且也对大系统的宏观、同质化描述提出了挑战。这项拟议的工作进一步发展了分析和计算方法，旨在精细地描述点缺陷的核心形状及其对远离核心的介质的影响。研究生和本科生将从事该项目的研究。PI将分析入侵前沿和点缺陷。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的传播进入不稳定状态。通过边际逐点稳定性分析，通常可以从不稳定状态的线性分析中很好地预测入侵过程之后产生的传播速度和新状态。PI将首先为相关的线性问题开发新的计算工具，并辅之以严格的收敛分析。在第二步中，PI将解决这些线性预测在非线性系统中的有效性，特别是与边际稳定性猜想相关的问题，该猜想假设在入侵过程中只选择关键的、边缘稳定的前沿。该提案的第二部分研究了条状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理解缺陷的核心在介质的大范围变形中的作用。CORE和FARFIELD之间的相互作用是使用分析、微扰工具和计算方法进行分析的，这些方法依赖于无限大小的有限域的严格近似。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.