Dynamics near coherent structures

许多物理过程都表现出有趣的时空结构模式。例如，在自动催化化学反应、铁流体、腔激光和沙漠中的植被斑块中，存在固定的和时间周期的局部斑点和环。源是更复杂的图案，可以被认为是活跃地组织空间周期性背景介质的缺陷：它们以平面螺旋波的形式出现，在化学反应中作为一维触发器出现，在流体中作为表面波出现。这个项目的目的是开发技术来分析一般反应扩散系统和格子上系统的相干结构的存在性、非线性稳定性和参数依赖性。我们将开展的例子项目有混合型泛函微分方程的几何奇异摄动理论，图灵和Hopf分支附近的平面和三维局域结构的分析，以及反应扩散系统中源的非线性稳定性的逐点格林函数估计。相干结构和非线性波组织了许多生物、化学和物理过程的动力学。例如脉动燃烧锋面、植被斑块、神经冲动、心脏组织中的螺旋波，以及流体实验中的局部对流滚动。我们将开发分析和数值工具来研究这种结构在什么条件下出现，并确定它们的空间范围及其稳定性作为系统参数的函数。其中潜在的技术应用包括半导体激光器，它可以显示出可用于全光存储设备的局部六角形贴片，以及光纤激光器，其优化关键取决于了解设计参数对产生的光波功率的影响。

Many physical processes exhibit interesting patterns with a nontrivial spatio-temporal structure. Stationary and time-periodic localized spots and rings, for instance, occur in autocatalytic chemical reactions, ferro-fluids, cavity lasers, and as vegetation patches in deserts. Sources are more complex patterns that can be thought of as defects that actively organize a spatially periodic background medium: they arise as planar spiral waves and as one-dimensional flip-flops in chemical reactions and as surface waves in fluids. This project aims at developing techniques to analyze the existence, nonlinear stability, and parameter dependence of coherent structures for general reaction-diffusion systems and for systems posed on lattices. Example projects that we will carry out are geometric singular perturbation theory for functional differential equations of mixed type, the analysis of planar and three-dimensional localized structures near Turing and Hopf bifurcations, and the nonlinear stability of sources in reaction-diffusion systems using pointwise Green's function estimates.Coherent structures and nonlinear waves organize the dynamics of many biological, chemical and physical processes. Examples are pulsating combustion fronts, vegetation patches, nerve impulses, spiral waves in cardiac tissue, and localized convection rolls in fluid experiments. We will develop analytical and numerical tools to study under which conditions such structures appear and to determine their spatial extent and their stability properties as functions of system parameters. Among the potential technological applications are semiconductor lasers, which can exhibit localized hexagonal patches that may be used in all-optical storage devices, and fiber lasers whose optimization depends crucially on understanding the effect of design parameters on the power of the generated light wave.