NSF-CBMS Regional Conference on Water Waves-Theory and Experiment

水波理论有着悠久而杰出的历史，欧拉、斯托克斯、开尔文、扎哈罗夫等人都有重要的贡献。这门学科在历史上一直有实际应用；2004年的海啸，2005年袭击新奥尔良的风暴潮，以及似乎神秘地出现和消失的“巨浪”，都戏剧性地证明了它最近的重要性。在每一种情况下，这些波的大振幅预测得很差或根本没有，造成了巨大的破坏。由于这些原因和其他原因，目前对水波的研究大多侧重于有限振幅的非线性波的动力学。更抽象地说，这门学科是一个动力系统的原型，它展示了有趣的非线性现象：共振相互作用、孤子、波破等。在一系列的十个讲座中，Harvey Segur教授（科罗拉多大学博尔德分校）将发展水波的数学理论，从基本原理开始，在几个方向的研究前沿结束。讲座的目的是让有偏微分方程和经典物理基础知识的人能够理解。该系列讲座有两个目标：(i)确定已被证明对解决本主题重要问题有用的数学方法；（ii）识别重要的开放性问题。讨论的主题包括：-水波的控制方程是什么？斯托克斯的公式（1847年）是对这个问题最普遍接受的陈述，它描述了许多（但不是全部）观察到的现象。从这些方程中可以很容易地预测出哪些物理现象？例如相速度、群速度、频散波、波聚焦、深水与浅水等等。-与会者如何观察这些物理现象？本系列讲座与其他CBMS系列讲座的不同之处在于，将建立一个实验室，在那里Diane Henderson教授（宾夕法尼亚州立大学）将创建一系列关于水波的物理实验。会议将鼓励与会者在实验室中“玩”，并看看讲座中的数学概念如何自然地出现在物理实验中。确定性模型和统计模型都用于预测海洋事件，如风暴、危险的公海和海啸。每种模式的优缺点是什么？——对斯托克斯的公式进行仔细的分析，得出了令人印象深刻的新结果，无论是在适定性方面，还是在永久形式的波浪模式的存在方面。这些结果是什么？我们从水波的近似理论中学到了什么？还有什么值得向他们学习的吗？例如，孤子理论的连锁奇迹首先是在KdV方程中发现的，KdV方程是由Korteweg和de Vries作为浅水中波传播的近似模型推导出来的。什么重要的物理现象没有被斯托克斯的公式所描述？-这门学科有哪些悬而未决的问题？

The theory of water waves has a long and distinguished history, with important contributions from Euler, Stokes, Kelvin, Zakharov and others. The subject has had practical application throughout history; its recent significance was demonstrated dramatically by the tsunami of 2004, the storm surge that battered New Orleans in 2005, and "rogue waves" that seem to appear and disappear mysteriously. In each of these cases, the large amplitudes of these waves were predicted poorly or not at all and caused enormous damage. For these reasons and others, much of the emphasis in current research on water waves is on the dynamics of nonlinear waves, with finite amplitude.More abstractly, this subject is a prototype of a dynamical system that exhibits interesting nonlinear phenomena: resonant interactions, solitons, wave breaking and more.In a series of ten lectures, Professor Harvey Segur (University of Colorado at Boulder) will develop the mathematical theory of water waves, starting from basic principles and ending at forefronts of research in several directions. The lectures are intended to be accessible to someone with a basic knowledge of partial differential equations and of classical physics. The lecture series has two objectives: (i) to identify mathematical methods that have proven useful in solving important problems in this subject; and (ii) to identify important open problems. Some of the topics to be discussed are:-What are the governing equations of water waves? Stokes' formulation (from 1847) is the most commonly accepted statement of the problem, and it describes very many (but not all) of the phenomena observed. -Which physical phenomena can be predicted easily from these equations? Some examples are phase velocity, group velocity, dispersive waves, wave focusing, deep water vs. shallow water, and more. -How can participants at the conference observe these physical phenomena?This lecture series will differ from others in the CBMS series in that a laboratory will be set up, where Professor Diane Henderson (Penn State University) will create a series of physical experiments on water waves. Participants at the conference will be encouraged to 'play' in the laboratory, and to see how mathematical concepts from the lectures appear naturally in physical experiments. -Both deterministic and statistical models are used to predict oceanic events like storms, dangerously high seas and tsunamis. What are the advantages and disadvantages of each kind of model?-Careful analysis, applied to Stokes' formulation, has led to impressive new results, both in terms of well-posedness and in terms of existence of wave patterns of permanent form. What are these results? -What have we learned from approximate theories of water waves? Is there more to learn from them? For example, the interlocking miracles of soliton theory were first discovered in the KdV equation, which was derived by Korteweg and de Vries as an approximate model of wave propagation in shallow water. -What important physical phenomena are not described by Stokes' formulation? -What are some of the open problems in this subject?