Stability of Nonlinear Waves in Dissipative and Dispersive PDE

本文的研究重点是非线性偏微分方程行波解的稳定性和长期动力学问题，这些问题广泛存在于科学和工程领域。这一建议将在各种模型中对空间周期运动结构的谱、线性化和非线性稳定性以及长时间动力学进行系统的分析。特别地，将考虑单向和多周期行波相对于单侧和高维横向扰动的稳定性。此外，还将研究这种行波结构对引起全球相移的非局域扰动的稳定性。该项目的主要目的是研究许多方程中的行波的稳定性和行为，例如水波和薄膜的模型，以及等离子体物理和工程中的行波。这样的解在不改变形状的情况下以恒定的速度移动，并且通常构成模型方程的更复杂解的基本构件。研究它们对小扰动的稳定性(即波是否迅速恢复到原始波形)具有重要的实际意义，因为不稳定的波在物理情况下不会自然地表现出来，除非可能是瞬变现象。这一建议旨在为研究人员提供实用而有效的“经验法则”，以确定物理模型中数学解的稳定性。该提案的核心部分是对本科生和研究生进行教育和专业培训，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.