Stability of Traveling Waves

本文主要研究行波的稳定性理论，重点研究了连续介质中的锋面传播和可压缩流的动力学理论。这类问题模拟了现实世界的关键现象，如粘性气体或等离子体中的激波，反应气体中的爆炸，以及粘性流体中相边界的传播。研究者通过探索关键结构特征（如对称性和真实耦合）的影响，研究了这些地区行波稳定性的长期开放问题。通过详尽的数值研究，其中包括埃文斯函数计算，他寻求更好地理解行波在这些重要问题领域的稳定性特性，特别是在分析方法目前提供的信息很少的情况下。结合最近与他的合作者的工作，研究者为埃文斯函数计算开发了更好的算法。这项工作的另一个好处是研究者的免费数字埃文斯函数工具箱的持续发展，这也允许系统的探索超出了我们的直接兴趣。此外，研究者使用这个包作为一个调查工具，以学生为中心的本科研究。行波在自然界中无处不在，从生态学中的人口模型到海洋科学中的海啸传播，或者从超音速喷气机的冲击波到光纤电缆中闪烁的光脉冲，无处不在。这些行波的稳定性描述了它们在存在干扰时能够持续的程度。这项工作的重点是行波的数学和发展的计算方法，以探索波的稳定性性质在纯科学和应用科学的无数领域。

HumpherysDMS-0607721 This work focuses on the stability theory of travelingwaves, with an emphasis on front propagation arising in thecontinuum and kinetic theories of compressible flow. This classof problems models key real-world phenomena such as shock wavesin a viscous gas or plasma, detonations in a reactive gas, andthe propagation of phase boundaries in a viscous fluid. Theinvestigator studies long-standing open questions about thestability of traveling waves in these areas by exploring theeffects of key structural features, such as symmetrizability andgenuine coupling. Through an exhaustive numerical study, whichincludes Evans function computation, he seeks a betterunderstanding of the stability properties of traveling waves inthese important problem areas, particularly in regimes whereanalytical methods currently provide little information. Inconnection with recent work with his collaborators, theinvestigator develops better algorithms for Evans functioncomputation. An additional benefit of this work is the continueddevelopment of the investigator's freely available numericalEvans function toolbox, which also allows for the exploration ofsystems beyond our immediate interest. Moreover, theinvestigator uses this package as an investigative tool instudent-focused undergraduate research. Traveling waves are ubiquitous in nature, occurringeverywhere from population models in ecology to the propagationof tsunamis in the oceanic sciences, or from shock waves of asupersonic jet to the flickering pulses of light in a fiber opticcable. The stability properties of these traveling wavesdescribe the degree to which they can persist in the presence ofdisturbances. This work focuses on the mathematics of travelingwaves and develops computational methods for exploring thestability properties of waves in a myriad of areas in the pureand applied sciences.