Nonlinear Dynamics of Pattern Forming Invasion Fronts

这个项目关注复杂系统中周期结构的自发形成。这类系统的例子包括滩涂上的贻贝带、细菌聚集模式以及对流问题中的六边形或条纹模式。 空间模式通常是由于简单状态的不稳定性而产生的。 例如，在生态或化学应用中，平衡状态可能由于引入先前不存在的物种而不稳定。 位于这个平衡态附近的初始态将朝着一个新的轮廓演化，人们希望预测这个新的状态将是什么。 对于空间扩展的问题，这个过程涉及到移动界面的形成，称为入侵锋，以固定的速度移动，并用于将不稳定状态转换为其他一些配置。 我们的目标是揭示一个非线性系统的属性，导致选择一个特定的入侵前，因此模式化的状态，是由该前创建。 从不稳定到稳定状态的转变发生在许多应用环境中，这项研究将有助于科学家制定关于这些系统中复杂时空模式形成的预测，甚至控制这些模式。 这项工作具有潜在的应用，通过反应扩散方程的作用，在纳米技术和材料科学的小尺度结构的生成建模。该项目的数学重点是研究反应扩散方程系统中的模式形成，其中周期性结构是由入侵前沿创建的。 几个具体的抛物型偏微分方程系统将被认为是与提取调解波速选择的一般原则的目的。 项目领域包括研究生态和化学应用中产生的反应扩散平流方程中的模式之间的竞争，Keller-Segel类型生物聚集模型中的粗化模式，以及描述超临界图灵分叉附近模式状态的出现和竞争的耦合振幅方程中的前沿存在。 该项目将需要各种工具，包括线性分析，几何奇异摄动理论，爆破技术和比较原则。在描述方程组中的模式形成前沿方面有几个固有的挑战，这需要扩展分析技术。 例如，方程组可以支持入侵前沿，其中不同的组件以不同的速度入侵，对于这些速度，没有单个固定帧可以捕获相关的动态。

This project is concerned with the spontaneous formation of periodic structures in complex systems. Examples of such systems include bands of mussels in tidal flats, bacterial aggregation patterns, and hexagonal or stripe patterns in convection problems. Spatial patterns often arise due to the instability of a simpler state. For example, in ecological or chemical applications an equilibrium state may be destabilized due to the introduction of a previously absent species. Initial states lying near this equilibrium state will evolve towards a new profile and one would like to predict what this new state will be. For spatially extended problems, this process involves the formation of moving interfaces, known as invasion fronts, that move at a fixed speed and serve to transition the unstable state to some other configuration. The objective is to uncover the properties of a nonlinear system that lead to selection of a particular invasion front and consequently the patterned state that is created by that front. Transitions from unstable to stable states occur in many applied contexts and this research will help inform scientists to formulate predictions regarding, or even control over, the formation of complex spatiotemporal patterns in those systems. The work has potential application via the role of reaction-diffusion equations in modeling the generation of small scale structures in nanotechnology and materials science. The mathematical focus of the project is the study of pattern formation in systems of reaction-diffusion equations in which periodic structures are created by invasion fronts. Several specific systems of parabolic partial differential equations will be considered with the aim of extracting general principles mediating wavespeed selection. The project areas include the study of competition amongst patterns in reaction-diffusion-advection equations arising in ecological and chemical applications, coarsening modes in biological aggregation models of Keller-Segel type, and the existence of fronts in coupled amplitude equations describing the emergence of and competition between patterned states near a supercritical Turing bifurcation. The project will require a variety of tools including linear analysis, geometric singular perturbation theory, blow-up techniques, and comparison principles. There are several innate challenges in regards to describing pattern-forming fronts in systems of equations that will require extensions of analytical techniques. For example, systems of equations can support invasion fronts wherein different components invade at different speeds for which no single fixed frame can capture the relevant dynamics.