Traveling Fronts with Unstable Continuous Spectrum: Geometric Structure and Nonlinear Stability Properties

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。波前是渐近地连接系统的两个平衡状态的行波。本研究课题是关于锋面稳定性的分析。锋面不稳定的机制是由扰动增长率和扰动输送率之间的竞争所决定的。 在对流状态下，对锋面的扰动被输送走的速度比它们的增长速度快，因此逐点衰减。关于波前非线性稳定性的关键信息包含在关于波的系统线性化的谱中，但在许多情况下谱信息不是确定的。例如，当线性化算子的连续谱穿过虚轴时，就会发生这种情况。标准的分叉理论技术通常会失败。将系统的域缩减到一些加权空间通常在线性水平上起作用，但是在加权空间中存在与非线性稳定性的证明相关的严重问题。这个项目的目标之一是为应用问题的类别的不稳定性的对流性质制定一般准则。由连续光谱引起的不稳定性也可以表现为新的局部或全局结构的出现。该项目的另一个目标是研究连续谱引起的不稳定性是否可能是解释从一个相干结构到另一个相干结构的突然转变所表征的现象的关键点。前沿出现在来自不同领域的各种应用问题中：光通信，燃烧理论，生物数学（组织中的钙波、神经传导、种群动力学）、化学、生态学等等，因此它们的稳定性是非常感兴趣的。对于许多模型，完全非线性方程中波前的稳定性不能简单地从其线性近似的性质推断出来。该项目的重点是寻找在这种情况下不稳定性的对流（或瞬态）性质的标准，并研究同一系统内完全不同的制度之间的过渡机制，例如从缓慢过程突然过渡到更快的过程。分析捕捉这一现象将有助于预测转变何时发生，并探索控制它的方法。这一方向的进展将在燃烧理论，生态学和生物数学的应用中具有重要意义。分析技术将基于波浪的几何结构与其稳定性之间的关系。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Fronts are traveling waves that asymptotically connect two equilibrium states of a system. This research project is concerned with the analysis of the stability of fronts. The mechanism of the instability of fronts is defined by the competition between the rate of growth of perturbations and the rate of their transport. In the convective regime perturbations to the front are transported away faster than they grow and thus decay pointwise. Critical information about nonlinear stability of the front is contained in the spectrum of the linearization of the system about the wave, but in many cases the spectral information is not definitive. That happens, for example, when the continuous spectrum of the linearized operator crosses the imaginary axis. Standard bifurcation theory techniques then typically fail. Reducing the domain for the system to some weighted spaces often works on the linear level, but there are serious issues related to the proof of nonlinear stability in the weighted spaces. One of the goals of this project is to develop general criteria for the convective nature of instability for classes of applied problems. An instability caused by the continuous spectrum can also manifest itself in the appearance of new local or global structures. Another goal of this project is to investigate whether the instability caused by the continuous spectrum may be the key point in the explanation of phenomena that are characterized by a sudden transition from one coherent structure to another.Fronts arise in a variety of applied problems from different fields: optical communication, combustion theory, biomathematics (calcium waves in tissue, nerve conduction, population dynamics), chemistry, ecology, to name a few, therefore their stability is of a great interest. For many models the stability of a front in a full nonlinear equation cannot be simply inferred from the properties of its linear approximation. This project is focused on finding criteria for the convective (or transient) nature of the instability in such cases and investigating the mechanism of the transition between drastically different regimes within the same system, such as a sudden transition from a slow process to a much faster one. Capturing this phenomenon analytically will assist in predicting when the transition happens and exploring ways to control it. Progress in this direction will be of importance for applications in combustion theory, ecology, and biomathematics. The techniques of the analysis will be based on the relation between the geometric structure of the wave and its stability.