Nonlinear stability of patterns

翻译后摘要：时空结构和模式已被观察到在许多自然过程。例如，在有髓神经纤维中神经冲动传播的模型中出现了行进脉冲;在颗粒介质、粘土悬浮液和化学反应中观察到了振荡的空间局部结构;嵌入振荡介质中的缺陷在化学反应中以螺旋波的形式出现，在流体容器中以表面波的形式出现。该项目的目标是研究这种模式的出现及其相对于小扰动的稳定性。这项工作的目的是进一步了解特定的模型方程，以及建立更广泛适用的结果。这个项目大致分为三个部分。在第一部分中，研究者分析了在强迫和非强迫反应扩散模型中可能导致混沌子的Hopf分支和Turing-Hopf分支。主要的方法是使用，并扩展，空间动力学技术与几何爆破方法。第二部分研究了离散FitzHugh-Nagumo方程中多快脉冲的存在性和稳定性。该策略是扩展技术从几何奇异摄动理论附近的非双曲慢流形从有限维设置的情况下，不适定的功能微分方程的先进和落后的条款。在第三个项目中，研究人员和合作者研究了接触缺陷的非线性稳定性，源汇对和接触缺陷的相互作用，以及平面螺旋波的线性稳定性。空间动力学技术被用来获得关于缺陷的线性化的预解核的扩展，然后使用拉普拉斯变换将其转换为相关的绿色函数的逐点估计。

Abstract:Spatio-temporal structures and patterns have been observed in many natural processes. For instance, travelling pulses arise in models of the propagation of nerve impulses in myelinated nerve fibers; oscillating, spatially localized structures have been observed in granular media, clay suspensions, and chemical reactions; defects that are embedded in an oscillatory medium occur as spiral waves in chemical reactions and as surface waves in fluid containers. The goal of this project is to investigate the emergence of such patterns and their stability with respect to small perturbations. The work aims to further our understanding of specific model equations as well as to establish results that apply more broadly.This project is divided broadly into three parts. In the first part, the investigator analyses Hopf and Turing Hopf bifurcations, which may lead to oscillons, in forced and unforced reaction-diffusion models. The main approach is to use, and extend, spatial-dynamics techniques together with geometric blow-up methods. The second part is to study the existence and stability of fast multi-pulses in the discrete FitzHugh-Nagumo equation. The strategy is to extend techniques from geometric singular perturbation theory near nonhyperbolic slow manifolds from the finite-dimensional setting to the case of ill-posed functional differential equations with advanced and retarded terms. In the third project, the investigators and collaborators study the nonlinear stability of contact defects, the interaction of source-sink pairs and of contact defects, and the linear stability of planar spiral waves. Spatial-dynamics techniques are used to derive expansions of the resolvent kernel of the linearization about a defect, which are then converted into pointwise estimates of the associated Green's function using Laplace transforms.