Stability of Nonlinear Waves in Dissipative and Dispersive PDE

耗散和色散偏微分方程中非线性波的稳定性

• 批准号: 1211183
• 负责人: Mathew Johnson
• 金额: $ 10万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211183&HistoricalAwards=false
• 关键词: Stability; Nonlinear; Waves; Dissipative; Dispersive

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.

拟议研究的重点是科学和工程许多领域中出现的非线性偏微分方程行波解的稳定性和长期动力学。 该提案将对各种模型中空间周期性移动结构的光谱、线性和非线性稳定性以及长期动力学进行系统分析。特别是，将考虑单向和多周期行波相对于单向以及高维横向扰动的稳定性。此外，还将研究这种行进结构对引起全局相移的非局部扰动的稳定性。该项目的主要目的是研究许多方程中行波的稳定性和行为，例如，作为水波和模型的模型。薄膜，以及等离子体物理和工程。 此类解以恒定速度行进而不改变形状，并且通常构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（即波是否快速恢复到原始波形）具有实际意义，因为不稳定的波不会自然地在物理情况下表现出来，除非可能作为瞬态现象。该提案旨在为研究人员提供实用且有效的“经验法则”，以确定物理模型中出现的数学解决方案的稳定性。该提案的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。