Collaborative Research: Nonlinear Water Waves

许多物理系统都存在非线性波。这些系统通常表现出的耗散被认为是可以忽略不计的或弱的。然而，最近的研究工作的调查人员和最近发表的观测耗散率的海洋涌浪，表明，模型使用弱耗散提供了一个非线性表面水波的演变的某些方面的描述不足。因此，本研究的一个主要重点是开发一个理论，从第一原理的非线性表面水波的传播和演化的粘性流体，没有近似的大小耗散。该研究解决了几个基于这一概念和/或旨在将数学结果应用于海洋应用的项目。这些项目分为两类：A）表面波在粘性流体上的传播和B）浅水或有限深水中的波。研究人员将通过结合他们在建模和分析，数值模拟和物理实验中的个人优势来解决这些问题，以获得物理现象的基本数学描述。然后，他们将利用海洋观测的公开数据，将他们的结果应用于海洋环境。这项研究的主要目标是在基本的数学水平上了解波浪在水面上的传播，并将这种理解应用于实验室和海洋中测量的波浪观测。具体应用包括耗散对风产生的波浪的影响、海洋涌浪的随后传播以及浅水和深水中的流氓波浪的发展;根据对责任地震震级的里氏震级测量以及关于其所在地地质的一些现成信息，预测危险海啸和良性海啸;和模拟大振幅波的海滩上的可变水深，使用新的数学方法研究这里。此外，数学结果预计将描述几种设置中的非线性波，包括光纤中的光波，薄磁膜中的自旋波等。研究生和本科生将被指导和参与实验研究和理论分析;通过这个项目，他们将接受跨学科研究的培训。

Many physical systems admit nonlinear waves. These systems often exhibit dissipation that is considered to be negligible or weak. However, recent work by the investigators and recently published observations of dissipation rates of ocean swell, show that the models using weak dissipation provide an inadequate description of some aspects of the evolution of nonlinear surface water waves. Thus, a main emphasis of this research is to develop a theory from first principles for the propagation and evolution of nonlinear surface water waves on a viscous fluid, with no approximation on the size of dissipation. The research addresses several projects that build on this notion and/or are intended to apply the mathematical results to ocean applications. The projects fall into two categories: A) surface waves propagating on a viscous fluid and B) waves in shallow or finite-depth water. The investigators will approach these problems by combining their individual strengths in modeling and analysis, numerical simulations, and physical experiments to obtain a fundamental mathematical description of the physical phenomena. They will then use publicly available data from ocean observations to apply their results to ocean settings.The primary goal of this research is to understand at a fundamental, mathematical level the propagation of waves on a surface of water, and to apply this understanding to observations of waves measured in the laboratory and in the oceans. Particular applications include the effects of dissipation on the generation of waves by wind, the subsequent propagation of ocean swell, and the development of rogue waves in shallow and deep water; predicting dangerous versus benign tsunamis given the Richter-scale measure of the magnitude of the responsible earthquake and some readily available information about the geology of its location; and modeling large-amplitude waves on beaches with variable bathymetry using the new mathematical approaches investigated here. In addition, the mathematical results are expected to describe nonlinear waves in several settings, including light waves in an optical fiber, spin waves in a thin magnetic film and others. Graduate and undergraduate students will be mentored and involved in experimental studies and theoretical analysis; through this project they will receive training in interdisciplinary research.