Spatial Localization in Several Dimensions

该研究项目将研究在流体、非线性光学、化学催化和其他连续系统中出现的空间局部模式，如锋面、斑点或脉冲。例子包括流体中的涡流、水滴和孤波，光学和化学系统中的斑点，细长结构在压缩下的局部屈曲，沿神经纤维传播的脉冲，以及振动颗粒介质中的局部振荡。这些不同的系统有两个共同点：(i)它们耗散能量，需要外部强迫维持；（ii）它们对这种外部强迫很敏感，微小的变化可能导致截然不同的结果（模式）。本项目力求将现有理论扩展到二维和三维的类似系统，并全面了解当强迫变化时导致模式增长的机制。它将应用结果来提高对旋转对流和熔体晶体生长中的漩涡的理解。该项目将有一名研究生参与研究。空间定域结构在物理建模中出现的许多偏微分方程的解中是常见的。本研究项目研究周期强迫驱动的耗散系统，其中不同的强迫振幅可导致不同的局域状态。该项目旨在将现有理论扩展到更高的维度，并提供对参数变化时观察到的不同类型结构生长背后机制的全面理解。感兴趣的结构不仅包括空间局部化的脉冲态，还包括连接两个不同状态的前沿和其他空间周期性结构中的缺陷或空穴。此外，该项目旨在将结果应用于物理和工程中的几个重要系统，包括旋转对流中的涡结构和过冷液体的结晶。所使用的技术包括可逆和近可逆系统的分岔理论，以及实际系统的数值分支跟踪和直接数值模拟。

This research project will study spatially localized patterns such as fronts, spots, or pulses that arise in fluids, nonlinear optics, chemical catalysis, and other continuum systems. Examples include vortices, drops, and solitary waves in fluids, spots in optical and chemical systems, localized buckling of slender structures under compression, pulses propagating along nerve fibers, and localized oscillations in vibrating granular media. These diverse systems have two things in common: (i) they dissipate energy and need to be sustained by external forcing, and (ii) they are sensitive to this external forcing and small changes may lead to distinctly different outcomes (patterns). This project seeks to extend existing theory to similar systems in two and three dimensions and to provide a comprehensive understanding of the mechanisms responsible for growth of patterns when the forcing is varied. It will apply the results to improved understanding of vortices in rotating convection and crystal growth from a melt. The project will involve a graduate student in the research.Spatially localized structures are common in solutions to many partial differential equations arising in physical modeling. This research project studies dissipative systems driven by periodic forcing in which different forcing amplitudes can lead to distinct localized states. The project seeks to extend existing theory to higher dimensions and to provide a comprehensive understanding of the mechanisms behind the different types of growth of the structures that are observed when parameters are varied. The structures of interest include not only spatially localized pulse-like states, but also fronts connecting two distinct states and defects or holes in otherwise spatially periodic structures. In addition, the project aims to apply the results to several systems of importance in physics and engineering, including vortex structures in rotating convection and crystallization from a supercooled liquid. The techniques used include bifurcation theory for reversible and near-reversible systems, coupled with numerical branch-following and direct numerical simulations of realistic systems.