Nonlinear Wave Motion

提案#DMS-9731097 PI：Harvey Segur标题：非线性波动Kadomtsev-Petviashvili方程是一个二维加上时间的非线性偏微分方程。它是极少数的几个非线性方程之一：(I)是完全可积的，并且允许“孤子”解；(Ii)涉及两个空间维度加上时间；(Iii)在物理中自然产生。具体地说，该方程近似描述了海浪在浅水中的运动。(海浪需要两个空间维度，因为它们发生在水的二维表面上。)对于这种应用，最有趣的解要么是精确周期的，要么是近似周期的，就像典型的海浪是近似周期的一样。基于对这个方程数学结构的最新发现，我们现在可以将这个问题作为空间中精确或近似周期的初始数据的初值问题来解决。这项研究将要开展的工作将利用这种结构来建立一个准确的浅水海浪模型。模型中的波浪将像海洋中的波浪一样复杂：稳定或不稳定，在时间或空间上具有精确或近似的周期，或两者兼而有之，具有任意大的振幅，并具有完全二维的空间模式。没有其他的水波功能模型同时包括所有这些特征，所以这个模型应该会极大地提高我们预测浅水海浪的能力。当对系统的摄动触发与摄动不成比例的响应时，就会出现非线性现象。自然界中的许多戏剧性事件都是非线性的，包括宇宙学中的大爆炸、空气动力学中的音爆、飓风和地球物理学中的破碎海浪。用来描述这些现象的数学模型本质上是非线性的，通常我们无法在任何一般意义上解决它们。允许“孤子”和“完全可积”的一组模型是这一规则的例外：通常这些模型可以完全详细地求解，并且它们提供了对非线性波动现象的本质的极大洞察。这项研究将利用最近发现的这些完全可积模型之一的数学结构来建立一个准确和现实的浅水海浪模型。对浅水波进行更准确预测的实际意义包括更好地控制海滩侵蚀，减少人为废物扩散造成的海岸线污染问题，以及更好地设计石油平台等近海结构。

Proposal # DMS-9731097 PI: Harvey Segur Title: Nonlinear Wave Motion The Kadomtsev-Petviashvili equation is a nonlinear partial differential equation in two spatial dimensions plus time. It is one of very few nonlinear equations that: (i) is completely integrable and admits "soliton" solutions; (ii) involves two spatial dimensions plus time; and (iii) arises naturally in physics. Specifically, the equation describes approximately the motion of ocean waves in shallow water. (Ocean waves require two spatial dimensions because they occur on the water's two-dimensional surface.) For this application, the most interesting solutions are either exactly or approximately periodic, just as typical ocean waves are approximately periodic. Based on recent discoveries of the mathematical structure of this equation, we can now solve this problem as an initial-value problem for initial data that are either exactly or approximately periodic in space. The work to be carried out in this study will exploit this structure to build an accurate model of waves in shallow water. The waves in the model will be as complicated as waves in the ocean usually are: steady or unsteady, with either exact or approximate periodicity in time or in space or both, with arbitrarily large amplitudes, and with fully two-dimensional spatial patterns. No other functioning model of water waves includes all of these features simultaneously, so this model should vastly improve our ability to predict waves in shallow water. Nonlinear phenomena occur when a perturbation to a system triggers a response that is not proportional to the perturbation. Many of the dramatic events in nature are nonlinear, including the "big bang" in cosmology, sonic booms in aerodynamics, hurricanes and breaking ocean waves in geophysics. The mathematical models used to describe these phenomena are intrinsically nonlinear, and typically we cannot solve them in any general sense. The set of models that admit "solitons" and are "completely integrable" are exceptions to this rule: usually these models can be solved in complete detail, and they have provided great insight into the nature of nonlinear wave phenomena. The work to be carried out in this study will exploit recently a discovered mathematical structure of one of these completely integrable models to build an accurate and realistic model of ocean waves in shallow water. Practical implications of more accurate predictions of shallow water waves include better control of beach erosion, fewer problems with shoreline pollution from dispersal of man-made waste, and better design of offshore structures like oil platforms.