Concatenated Traveling Waves

耗散偏微分方程最简单的解之一是行波，即以恒定速度运动时保持其形状的解。在以波的速度运动的坐标系中，行波变成了一个定解。这一事实使行波的稳定性可以用线性化来研究。由于德雷塞尔大学的道格拉斯·赖特和比勒菲尔德大学的萨布丽娜·塞勒最近的工作，某些“串联波”解的稳定性现在可以被证明了——这种解看起来像左边的一个行波，右边的另一个行波速度更快。然而，他们的工作将串联波解视为波的总和，这种方法似乎不能推广到某些重要的情况。例如，如果第一波的共同右状态和第二波的共同左状态只是边缘稳定，那么Wright-Selle方法将这种边缘稳定性扩展到两端，这应该只是连接波解的中心问题。在本研究项目中，这些解被视为串联波而不是波的总和。在这里，稳定性分析从一个近似解开始，这个近似解由一个波在左边，另一个波在右边，以及一个中间常数区域组成，其中近似解是第一波的共同右状态和第二波的共同左状态。本项目利用拉普拉斯变换研究这种对象的线性化，并将此草图转化为非线性稳定性的严格证明，并将该方法推广到一些重要的退化情况。这项工作的动机部分来自于一个熟悉的情况：如果在中间点燃保险丝，燃烧前沿会向两个方向移动。单个燃烧锋面是行波的一个例子：由于它以恒定的速度运动，如果使用以相同速度运动的坐标系，那么波是静止的。如果一个行波受到很小的扰动几乎没有影响，那么它就被称为稳定行波；波浪很快恢复形状，并以相同的速度继续前进。在数学上，用静止的坐标系来研究行波的稳定性是很方便的。关于行波的稳定性，有一个发展得很好的数学理论，它允许人们提前预测什么条件会允许稳定的波。然而，当波以两种不同的速度传播时（在保险丝的例子中，一个波的速度是负的，另一个波的速度是正的），没有方便的坐标系可以使用，因此稳定性的数学理论就不那么发达了。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。该工作在石油开采方法和地下污染清理中的行波研究中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.