Stability of nonlinear dispersive waves and wave collapse phenomena

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目的目的是研究流体、等离子体中发生的非线性色散波的数学模型，以及许多物理现象中遇到的非线性色散波。将研究非线性色散方程中孤立波的稳定性和不稳定性;还将研究此类方程的波破碎现象。 特别是，Degasperis-Procesi（DP）方程，Ostrovsky（OS）方程和Boussinesq（BQ）方程将被考虑。这个研究领域的核心是非线性波的相互作用。当可能以不同速度和不同方向运动的波相交时，就会发生这种情况。如果相互作用足够强，那么它可能会产生新的波浪，甚至导致波浪破碎。在这些研究中将采用各种数学分析方法。该项目将对OS方程中孤立波的稳定性、BQ方程中孤立波的强不稳定性、DP方程的整体弱解、激波和波破碎现象以及爆破结构进行具体分析和数值模拟。 在开展这一项目中开发的技术，预计将是有用的其他非线性色散波方程。波普遍存在于许多不同的物理环境中，例如，在海洋、大气、声学、电信等中。最复杂的波是非线性波，其演化在不进行非常复杂的计算和建模的情况下很难预测。特别是，众所周知，一些非线性波是不稳定的，也就是说，一个小的变化将产生一个大的扰动。 另一方面，波浪破碎（波浪在短时间内的纵倾或纵荡）是另一种至关重要的物理现象，尤其是对于水波。非线性波的稳定性研究在聚变能研究和光纤通信等应用中是必不可少的。 该项目的目标是发展一个更好的数学理解，哪些波是稳定的，哪些是不稳定的，以及如何以及何时这些波的形成和打破。研究将集中在某些类型的非线性波，很好地描述了数学模型，但很难解决。这些方程的特解的分析对海洋波浪和海流的基本认识具有潜在的影响。 虽然方程是本研究的主要焦点是与水波和海洋动力学，分析特殊波和波破碎的新方法可以扩展到其他领域的流体力学，等离子体物理和其他应用。在应用分析或数值微分方程方面具有新兴专业知识的研究生将参与研究并通过该项目进行培训。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The purpose of this project is to study mathematical models of nonlinear dispersive waves that occur in fluids, plasmas, and are encounters in many physical phenomena. Research will be conducted on the stability and instability of solitary waves in nonlinear dispersive equations; wave breaking phenomena for such equations will also be investigated. In particular, the Degasperis-Procesi (DP) equation, the Ostrovsky (OS) equation and the Boussinesq (BQ) equations will be considered. At the heart of this field of inquiry are nonlinear wave interactions. These occur when waves, moving possibly with different speeds and in different directions, intersect. If the interaction is strong enough, then it may create new waves or even lead to wave breaking. Various methods of mathematical analysis will be employed in these investigations. The project will provide specific analysis and numerical simulations of the stability of solitary waves in the OS equations, the strong instability of solitary wave in the BQ equations, and global weak solutions, shock waves, and wave breaking phenomena as well as blow-up structures of the DP equation. The techniques developed in carrying out this project are expected to be useful for other nonlinear dispersive wave equations. Waves are ubiquitous in many different physical contexts, for example, in the ocean, the atmosphere, acoustics, telecommunications, and so on. The most complicated ones are nonlinear waves whose evolution is difficult to predict without performing very complex computations and modeling. In particular, it is known that some of nonlinear waves are not stable; that is, a small change will produce a large disturbance. On the other hand, wave breaking (the wave plunging or surging in short time) is another physical phenomenon of critical importance, especially for water waves. The study of stability of nonlinear waves is necessary for applications such as fusion energy research and fiber-optic communications. The objectives of this project are to develop a better mathematical understanding of which waves are stable and which are unstable, as well as how and when these waves form and break. Research will be focused on certain kinds of nonlinear waves that are well described by mathematical models, but quite difficult to solve. The analysis of these special solutions to the equations has potential impact on the fundamental understanding of ocean waves and currents. Although the equations which are the primary focus of this research are related to water waves and ocean dynamics, the new methods of analysis of special waves and wave breaking developed here could be extended to other areas of fluid mechanics, plasma physics, and other applications. Graduate students with emerging expertise in applied analysis or numerical differential equations will be involved into research and trained through this project.