Spatial Localization in Several Dimensions

这项研究项目将研究在流体、非线性光学、化学催化和其他连续介质系统中出现的空间局域图案，如锋面、斑点或脉冲。例如，流体中的涡旋、液滴和孤立波，光学和化学系统中的斑点，压缩下细长结构的局部屈曲，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，需要外部强迫来维持，(Ii)它们对这种外部强迫很敏感，微小的变化可能会导致截然不同的结果(模式)。这个项目试图将现有的理论扩展到二维和三维的类似系统，并提供对当强迫不同时模式增长的机制的全面理解。它将应用这些结果来更好地理解旋转对流中的漩涡和从熔体中生长晶体。该项目将涉及一名研究生参与研究。空间局部化结构在物理建模中出现的许多偏微分方程解中很常见。这项研究项目研究由周期性强迫驱动的耗散系统，其中不同的强迫幅度可以导致不同的局域态。该项目试图将现有的理论扩展到更高的层面，并全面了解在参数变化时观察到的结构不同类型增长背后的机制。感兴趣的结构不仅包括空间局域的脉冲态，而且还包括连接两个不同状态和其他空间周期结构中的缺陷或空穴的前锋。此外，该项目旨在将结果应用于几个在物理和工程上具有重要意义的系统，包括旋转对流中的涡旋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叉理论，以及对真实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.