Stability of Traveling Waves

HUPHERERS DMS-0607721这项工作的重点是行波稳定性理论，重点是可压缩流动的连续波传播理论和动力学理论。这类问题模拟了一些关键的真实世界现象，如粘性气体或等离子体中的冲击波，反应气体中的爆炸，以及粘性流体中相边界的传播。研究人员通过探索对称性和真耦合等关键结构特征的影响，研究了关于这些地区行波稳定性的长期悬而未决的问题。通过详尽的数值研究，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，特别是在目前分析方法提供的信息很少的情况下。在与他的合作者最近的工作中，这位研究人员开发了更好的埃文斯函数计算算法。这项工作的另一个好处是继续开发研究人员免费提供的数字埃文斯函数工具箱，它还允许探索我们直接感兴趣的系统。此外，研究人员在以学生为中心的本科生研究中使用这个包作为调查工具。行波在自然界中无处不在，从生态学中的种群模型到海洋科学中的海啸传播，从超音速喷流的冲击波到光纤电缆中闪烁的光脉冲，随处可见。这些行波的稳定性属性描述了它们在存在扰动时能坚持到什么程度。这项工作专注于行波的数学，并开发了在纯科学和应用科学的众多领域中探索波的稳定性特性的计算方法。

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.