Collaborative Research: Nonlinear Water Waves

许多物理系统承认非线性波。这些系统的耗散通常被认为是可以忽略或很弱的。然而，研究人员最近的工作和最近发表的关于海浪耗散率的观测表明，使用弱耗散的模型不能很好地描述非线性表层水波演化的某些方面。因此，这项研究的一个主要重点是从基本原理出发发展一种关于非线性表面水波在粘性流体中传播和演化的理论，而不是对耗散大小进行近似。这项研究涉及建立在这一概念基础上和/或旨在将数学结果应用于海洋应用的几个项目。这些项目分为两类：A)在粘性流体上传播的面波和B)在浅水或有限深度水中传播的波。研究人员将结合他们在建模和分析、数值模拟和物理实验方面的个人优势来处理这些问题，以获得对物理现象的基本数学描述。然后，他们将使用从海洋观测中获得的公开数据，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在水面上的传播，并将这种理解应用于在实验室和海洋中测量的波的观测。具体应用包括：消散对风产生的海浪、海浪的随后传播以及浅水和深水中无赖海浪的发展的影响；根据对负责地震的震级进行里氏规模测量并提供有关其位置的一些现成资料，预测危险海啸与良性海啸；以及使用此处研究的新的数学方法模拟具有可变水深的海滩上的大振幅海浪。此外，数学结果有望描述几种环境中的非线性波，包括光纤中的光波、薄膜中的自旋波等。研究生和本科生将接受指导，并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.