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)提供资金的。波阵面是渐近连接系统的两个平衡态的行波。本研究项目是关于前锋稳定性分析的研究项目。锋面不稳定的机制是由扰动的增长率和它们的输送速率之间的竞争来定义的。在对流系统中，锋面扰动的传播速度快于它们的增长速度，因此逐点衰减。关于波面的非线性稳定性的关键信息包含在系统关于波浪的线性化的谱中，但在许多情况下，谱信息是不确定的。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。然后，标准的分叉理论技术通常会失败。将系统的区域归结到某些加权空间通常在线性水平上可行，但在加权空间中非线性稳定性的证明存在严重的问题。该项目的目标之一是为各类应用问题制定不稳定对流性质的一般标准。由连续谱引起的不稳定性也可能表现为新的局部或全球结构的出现。这个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突然转变的现象的关键。前沿性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，因此它们的稳定性引起了人们的极大兴趣。对于许多模型来说，完全非线性方程中的前沿的稳定性不能简单地从它的线性逼近的性质中推断出来。这个项目的重点是找出在这种情况下不稳定的对流(或瞬变)性质的标准，并研究同一系统内截然不同的制度之间的转变机制，例如从缓慢的过程到快得多的过程的突然转变。分析地捕捉到这一现象将有助于预测何时发生转变，并探索控制它的方法。这一方向的进展将对燃烧理论、生态学和生物数学的应用具有重要意义。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.