Existence, Stability, and Dynamics of Nonlinear Waves

本研究项目主要研究模拟各种物理现象的数学方程的特殊解的稳定性和行为，这些物理现象包括光通信、浅层或分层流体流动和倾斜薄膜流动。重点研究了具有空间周期结构的非线性波的稳定性，这种结构在许多应用中构成了更复杂的解的基本构件。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)具有非常重要的实际意义，因为不稳定的波在物理应用中不会自然地表现出来，除了可能的瞬变现象。这个项目的目的是为研究人员提供一个数学上严格的理论，通过这个理论，他们可以区分稳定的、因此有可能在现实中表现出来的数学解，以及那些不稳定的数学解。该项目还将包括本科生和研究生参与研究项目。通过这项工作开发的方法和技术将被纳入适合不同科学学科的学生的研讨会、阅读课程和专题课程中。本项目重点研究在数学物理和流体力学中自然产生的模型中的行波解的特殊类别的存在性、稳定性和动力学。其目的是在能量守恒的哈密顿色散偏微分方程组以及能量由于粘性效应而部分耗散的双曲-抛物型守恒定律和平衡律的双曲-抛物型方程中，系统地发展空间周期站立或行进结构的线性和非线性稳定性分析。在色散背景下，该研究项目将考虑各种非局部描述色散的模型方程，并将开发出不仅能够处理许多经典理论领域中的低频现象，而且能够处理解的高频行为的技术和方法，这超出了经典结果的有效性范围。在耗散的背景下，该研究项目将对滚波进行系统的研究，滚波是在许多应用中自然产生的常见的水动力不稳定性，例如管道中的流体流动。将特别注意理解最近发展的这种波的粘性理论和更著名的无粘性理论之间的联系。

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.