Concatenated Traveling Waves

其中最简单的解决方案耗散偏微分方程是行波，即解决方案，保持其形状，而以恒定的速度移动。 在一个随波的速度移动的坐标系中，行波成为一个定态解。 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塞布丽娜塞尔最近的工作，某些“级联波”解的稳定性--看起来像左边的一个行波和右边的另一个速度更大的行波--现在可以被证明。然而，他们的工作将级联波解视为波的总和，这种方法似乎并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共同左状态仅是边缘稳定的，则赖特-塞尔方法将这种边缘稳定性扩展到两端，这应该仅是级联波解的中心的问题。在这个研究项目中，这样的解决方案被视为级联波，而不是作为波的总和。 在这里，稳定性分析从一个近似解开始，该近似解由左侧的一个波、右侧的另一个波和中间恒定区域组成，其中近似解是第一波的公共右侧状态和第二波的公共左侧状态。 这个项目研究线性化在这样一个对象使用拉普拉斯变换，和工作的目的是将这个草图转换成一个严格的证明非线性稳定性和扩展的方法，一些重要的退化situations.The拟议的工作是由一个熟悉的情况的部分动机：如果一个点燃的引信在中间，燃烧前锋在两个方向上的旅行。一个单一的燃烧前缘是行波的一个例子：因为它以恒定的速度移动，如果使用一个以相同速度移动的坐标系，波是静止的。一个行波被称为稳定的，如果它的一个小扰动几乎没有影响;波很快恢复其形状，并继续以相同的速度。 在数学上，行波的稳定性的研究是通过使用一个坐标系，其中波是固定的。 有一个关于行波稳定性的成熟的数学理论，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（在引信的例子中，一个波具有负速度，另一个波具有正速度），没有方便的坐标系可以使用，因此稳定性的数学理论发展得更少。 在这个项目中，研究人员将开发一种严格的方法来证明级联行波的非线性稳定性，这种方法比目前可用的理论适用得更广泛。这项工作对研究采油方法和地下污染清除中发生的行波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.