Nonlinear Wave Motion

提案编号DMS-9731097 PI：Harvey Segur标题：非线性波动 Kadomtsev-Petviashvili方程是一个二维空间加时间的非线性偏微分方程。 之一 非常少的非线性方程：（i）是完全可积的，并允许 （ii）涉及两个空间维度加上时间;以及 (iii)在物理学中自然产生。 具体来说，该方程描述了 近似于浅水中海浪的运动。 （海浪） 需要两个空间维度，因为它们发生在水面上， 二维曲面）。 对于这个应用程序，最有趣的是 解是精确的或近似的周期性的，就像典型的 海浪是近似周期性的。 根据最近的发现， 这个方程的数学结构，我们现在可以解决这个问题， 一个初始值问题，初始数据要么是精确的，要么是 在空间中近似周期性的。 本研究拟开展的工作 将利用这种结构建立一个精确的浅水波模型， 水 模型中的波将和 海洋通常是：稳定或不稳定，精确或近似 时间或空间或两者的周期性，具有任意大的振幅， 并且具有完全二维的空间模式。 无其他功能 水波模型同时包括所有这些特征，因此， 这个模型将大大提高我们预测浅水波浪的能力。 当系统的扰动触发与扰动不成比例的响应时，就会发生非线性现象。 许多 自然界中的戏剧性事件都是非线性的，包括宇宙大爆炸。 宇宙学、空气动力学中的音爆、飓风和海浪 在物理学中。 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情况下解决它们。 一般意义 一组模型，承认“孤子”， “完全可积”是这个规则的例外：通常这些模型 可以完全解决的细节，他们提供了很大的洞察力， 非线性波动现象的本质。 在这方面要开展的工作 研究将利用最近发现的其中一个数学结构， 完全可集成的模型，以建立准确和现实的模型， 在浅水中的海浪。 更准确的实际影响 浅水波浪的预测包括更好地控制海滩侵蚀， 减少人为废物散布造成的海岸线污染问题， 以及更好的海上结构设计，比如石油平台。

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.