Collaborative Research: Nonlinear Water Waves

许多物理系统都承认非线性波。这些系统的耗散通常被认为是可以忽略的或弱的。然而，研究人员最近的工作和最近发表的对海洋膨胀耗散率的观测表明，使用弱耗散的模式不能充分描述非线性表面水波演变的某些方面。因此，本研究的主要重点是建立一个从第一性原理出发的理论，用于非线性表面水波在粘性流体上的传播和演化，而不需要对耗散大小进行近似。该研究涉及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)在粘性流体上传播的表面波和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解决这些问题，以获得对物理现象的基本数学描述。然后，他们将使用来自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层面上理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波浪的影响，随后的海洋膨胀传播，以及浅水和深水中异常浪的发展；根据里氏地震的震级和有关地震地点的一些现成的地质信息，预测危险与无害的海啸；并利用本文研究的新的数学方法，在可变水深的海滩上模拟大振幅波浪。此外，数学结果有望描述几种情况下的非线性波，包括光纤中的光波，薄磁膜中的自旋波等。指导研究生和本科生进行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方面的培训。

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.