Existence, Stability, and Dynamics of Nonlinear Waves

非线性波的存在性、稳定性和动力学

• 批准号: 1614785
• 负责人: Mathew Johnson
• 金额: $ 17.5万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1614785&HistoricalAwards=false
• 关键词: Existence; Stability; Dynamics; Nonlinear; Waves

This research project focuses on stability and behavior of special classes of solutions of mathematical equations modeling a variety of physical phenomena including optical communication, shallow or stratified fluid flow, and inclined thin film flow. Emphasis is placed on the stability of nonlinear waves exhibiting a spatially periodic structure, which form fundamental building blocks for more complicated solutions in many applications. The stability of such structures (i.e. their ability to retain their form when disturbed) is of great practical importance, as waves that are not stable do not naturally manifest themselves in physical applications, except possibly for transient phenomena. The aim of this project is to provide researchers a mathematically rigorous theory by which they may distinguish between mathematical solutions that are stable, and hence have a possibility of being manifested in reality, and those that are not. This project will also include undergraduate and graduate students in research projects. The methodologies and techniques developed through this work will be incorporated into seminars, reading courses, and special topics courses appropriate for students from a variety of scientific disciplines.This project focuses on the existence, stability, and dynamics of special classes of traveling wave solutions in models arising naturally in mathematical physics and fluid mechanics. The aim is to systematically develop a linear and nonlinear stability analysis of spatially periodic standing or traveling structures in both Hamiltonian dispersive partial differential equations, in which energy is conserved, as well as hyperbolic-parabolic systems of conservation and balance laws, where energy is partially dissipated due to, for example, viscous effects. In the dispersive context, the research project will consider a variety of model equations with nonlocal descriptions of dispersion and will develop techniques and methodologies capable of treating not only low-frequency phenomena, which is in the realm of many classical theories, but also high-frequency behaviors of solutions, which are beyond the regime of validity of classical results. In the dissipative context, the research project will develop a systematic investigation of roll-waves, which are commonly-observed hydrodynamic instabilities arising naturally in many applications, such as fluid flow in conduits. Particular attention will be placed on understanding connections between the recently-developed viscous theories of such waves and more well-known inviscid theories.

该研究项目的重点是数学方程的特殊类的解决方案的稳定性和行为建模的各种物理现象，包括光通信，浅或分层的流体流动，倾斜的薄膜流。 重点放在表现出空间周期性结构的非线性波的稳定性，在许多应用中形成更复杂的解决方案的基本积木。 这种结构的稳定性（即它们在受到干扰时保持其形状的能力）具有很大的实际重要性，因为不稳定的波在物理应用中不会自然地表现出来，除了可能的瞬态现象。 该项目的目的是为研究人员提供一个数学上严格的理论，使他们可以区分稳定的数学解决方案，因此有可能在现实中表现出来，而那些不是。 该项目还将包括本科生和研究生的研究项目。 通过这项工作开发的方法和技术将被纳入研讨会，阅读课程，以及适合学生从各种科学学科的专题课程。本项目侧重于存在性，稳定性和动态的特殊类的行波解的模型自然产生的数学物理和流体力学。 其目的是系统地开发一个线性和非线性稳定性分析的空间周期性的站立或旅行结构在两个哈密顿色散偏微分方程，其中能量守恒，以及双曲抛物系统的守恒和平衡定律，其中能量部分耗散，例如，粘性效应。 在色散的情况下，该研究项目将考虑各种模型方程与非局部描述的色散，并将开发技术和方法，不仅能够治疗低频现象，这是在许多经典理论的领域，但也解决方案的高频行为，这是超出了经典结果的有效性制度。 在耗散方面，该研究项目将对滚波进行系统研究，滚波是在许多应用中自然产生的常见流体动力学不稳定性，例如管道中的流体流动。 特别注意将放在理解最近发展的粘性理论，这种波和更知名的无粘理论之间的联系。