Stability of Traveling Waves

HumpherysDMS-0607721 本文着重于行波的稳定性理论，着重于连续体中的波前传播和可压缩流的动力学理论。 这类问题模拟了关键的现实世界现象，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。 研究人员通过探索关键结构特征（如对称性和真正的耦合）的影响，研究了这些领域中关于行波稳定性的长期未决问题。 通过详尽的数值研究，其中包括埃文斯函数计算，他寻求更好地了解行波在这些重要问题领域的稳定性，特别是在政权的分析方法目前提供的信息很少。 在最近的工作与他的合作者，调查员开发更好的算法埃文斯函数计算。 这项工作的另一个好处是继续发展的调查人员的免费提供numericalEvans函数工具箱，这也允许探索的系统超出了我们的直接利益。 此外，研究者使用这个包作为一个调查工具，在学生为中心的本科研究。 行波在自然界中无处不在，从生态学中的种群模型到海洋科学中海啸的传播，或者从超音速射流的冲击波到光纤电缆中闪烁的光脉冲，到处都有行波的存在。 这些行波的稳定性描述了它们在干扰存在下能够持续的程度。 这项工作的重点是数学的行波和开发计算方法，探索稳定性的波在无数领域的纯科学和应用科学。

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.