Concatenated Traveling Waves

在耗散偏微分方程解中最简单的是行波，即在以恒定速度运动的同时保持其形状的解。在随波速度移动的坐标系中，行波成为静止的解。这一事实使得行波的稳定性可以通过线性化来研究。由于德雷克塞尔大学的道格拉斯·赖特和比勒费尔德大学的萨布丽娜·塞尔尔最近的工作，现在可以证明某些“串联波”解的稳定性--这种解在左边看起来像一个行波，在右边看起来像另一个速度更快的行波。然而，他们的工作将串联波解视为波的总和，这种方法似乎不适用于某些重要的情况。例如，如果第一波的共同右状态和第二波的左状态仅是边缘稳定，则Wright-Selle方法将这种边缘稳定性扩展到两端，该边缘稳定性应该只是级联波解的中心的问题。在这个研究项目中，这样的解被视为级联波而不是波的和。这里，稳定性分析从一个近似解开始，该近似解由左侧的一个波、右侧的另一个波和一个中间恒定区组成，其中近似解是第一个波的公共右状态和第二个波的左状态。该项目利用拉普拉斯变换研究了这类对象的线性化，其目的是将这一示意图转化为非线性稳定性的严格证明，并将该方法扩展到一些重要的退化情况。单个燃烧锋是行波的一个例子：因为它以恒定的速度移动，如果使用以相同速度移动的坐标系，则该波是静止的。如果行波的微小扰动几乎没有任何影响，则称行波为稳定波；行波很快恢复其形状，并以相同的速度继续。在数学上，行波稳定性的研究是通过使用行波是静止的坐标系来实现的。关于行波的稳定性有一套完善的数学理论，它使人们能够预先预测什么条件下才能产生稳定的波。然而，当波以两种不同的速度传播时(在引信示例中，一个波的速度为负，另一个波的速度为正)，没有方便的坐标系使用，因此稳定性的数学理论发展得很慢。在这个项目中，研究人员将开发一种严格的方法来证明串联行波的非线性稳定性，这种方法比目前可用的理论应用得更广泛。这项工作对研究采油方法中出现的行波现象和井下污染治理具有潜在的应用价值。

Among the simplest solutions of dissipative partial differential equations are traveling waves, that is, solutions that preserve their shape while moving at a constant velocity. In a coordinate system that moves with the velocity of the wave, a traveling wave becomes a stationary solution. This fact allows the stability of traveling waves to be studied by linearization. Thanks to recent work of Douglas Wright of Drexel University and Sabrina Selle of the University of Bielefeld, the stability of certain "concatenated wave" solutions -- solutions that look like one traveling wave at the left and another, with greater velocity, at the right -- can now be proved. However, their work views a concatenated wave solution as a sum of waves, an approach that does not seem to generalize to certain important situations. For example, if the common right state of the first wave and left state of the second wave is only marginally stable, the Wright-Selle approach expands this marginal stability, which should only be an issue in the center of the concatenated wave solution, to both ends. In this research project, such solutions are treated as concatenated waves rather than as sums of waves. Here, analysis of stability starts with an approximate solution that consists of one wave at the left, another at the right, and an intermediate constant region where the approximate solution is the common right state of the first wave and left state of the second. This project studies linearization at such an object using Laplace transforms, and the work aims to convert this sketch into a rigorous proof of nonlinear stability and to extend the approach to some important degenerate situations.The proposed work is motivated in part by a familiar situation: if one lights a fuse in the middle, combustion fronts travel in both directions. A single combustion front is an example of a traveling wave: since it moves with a constant velocity, if one uses a coordinate system that moves with the same velocity, the wave is stationary. A traveling wave is called stable if a small perturbation of it has almost no effect; the wave quickly recovers its shape and continues with the same velocity. Mathematically, the study of the stability of traveling waves is facilitated by using a coordinate system in which the wave is stationary. There is a well-developed mathematical theory of the stability of traveling waves, which allows one to predict in advance what conditions will allow stable waves. However, when waves traveling with two different velocities are present (in the fuse example, one wave has negative velocity, the other has positive velocity), there is no convenient coordinate system to use, and so the mathematical theory of stability is much less developed. In this project, the investigators will develop a rigorous approach to proving nonlinear stability of concatenated traveling waves that applies more broadly than the currently available theory. The work has potential application to study of traveling waves that occur in oil recovery methods and underground pollution cleanup.