Stability of Nonlinear Waves in Dissipative and Dispersive PDE

拟议的研究重点是稳定性和长期动力学的行波解的非线性偏微分方程，出现在许多领域的科学和工程。 这个建议将发展一个系统的分析频谱，线性化，非线性稳定性和长时间动态的空间周期性旅行结构在各种模型。特别是，稳定性的单向和多周期行波相对于单向以及高维横向扰动将被考虑。此外，还将研究这种行波结构对引起全局相移的非定域扰动的稳定性。该项目的主要目的是研究在许多方程中行波的稳定性和行为，例如，作为水波和薄膜的模型，以及在等离子体物理和工程中。 这种解以恒定速度行进而不改变形状，并且通常形成模型方程的更复杂解的基本构建块。研究它们对小扰动的稳定性（即波是否很快恢复到原始波形）具有实际重要性，因为不稳定的波不会自然地在物理情况下表现出来，除非可能是瞬态现象。这项建议旨在为研究人员提供实用和有效的“经验法则”，以确定物理模型中出现的数学解的稳定性。该提案的一个核心组成部分是本科生和研究生的教育和专业培训，他们将通过研讨会，会议和专题课程接受跨学科指导。

The focus of the proposed research is stability and long-time dynamics of travelling wave solutions of nonlinear partial differential equations that arise in many areas of science and engineering. This proposal will develop a systematic analysis of spectral, linearized, and nonlinear stability and long-time dynamics of spatially periodic traveling structures in a variety of models. In particular, stability of uni-directional and multiply periodic traveling waves with respect to uni-lateral as well as higher-dimensional transverse perturbations will be considered. Furthermore, stability of such traveling structures to non-localized perturbations inducing global phase shifts will be investigated.The main thrust of the project is a study of stability and behavior of traveling waves in many equations arising, for example, as models of water waves and thin films, and in plasma physics and engineering. Such solutions travel at constant speed without changing shape, and often form fundamental building blocks for more complicated solutions of the model equations. Studying their stability to small perturbations (i.e. whether the wave is quickly restored to the orginal wave form) is of practical importance, since waves that are unstable do not naturally manifest themselves in physical situations, except possibly as transient phenomena. This proposal aims to provide researchers with practical and efficient "rules of thumb" to ascertain stability of mathematical solutions arising in physical models. A central component of the proposal is the educational and professional training of undergraduate and graduate students, who will receive interdisciplinary instruction through seminars, conferences, and special topics courses.