Mathematical Sciences: Analysis on Waves in Stratified Fluids of Infinite Depth

小行星9623060 该项目的目的是提出一个数学和 分层流体中孤立波和周期波的数值研究 在重力作用下的无限深度。该提案 由两部分组成。第一部分旨在对 回答一个开放的问题，是否存在孤立的内部 连续分层介质中无穷远代数衰减波 仅由刚性底部限定的流体，具有常数 密度，除了具有连续密度分层的层之外。 拟议中的研究将为 孤立波的存在性来自一个正式的模型方程， 叫做本杰明-小野方程第二部分阐述 两层流体中大振幅周期波的存在性 没有界限将引入问题的新公式， 其将控制方程转换成单个积分方程。 然后对积分方程的解进行数值研究 理论上，对于任何密度比， 解的幅度。在这个项目中，主要的主旨是展示 分层流体的完全非线性控制方程 有有限和大振幅的解决方案，并给出各种 使用数值和理论方法的解决方案的属性。 微分方程与泛函分析理论的相互作用 对于获得严格的理由至关重要，而 数值计算给出了解的一些重要信息。 %%% 密度变化流体中的大振幅重力波， 也称为分层流体，具有很大的深度， 地球物理的兴趣。大振幅内波扰动是 海洋和低层大气的共同特征。在 特别是孤立波，其形式是局部单峰， 和周期波，它总是有相同的形式后，一定的 距离，与各种海洋和大气现象有关。 大振幅孤立波与 龙卷风在大气中的形成和巨大的运输 海洋中的动力和能量。这项工作的重点是获得 孤立波或周期波存在的条件， 预测波的振幅有多大如果它们存在的话。 它还将试图捕捉这些波的定性特征 使用数值计算。从这项研究中获得的结果 可以为这些波的形成提供一些理论解释 让我们对海洋中的某些波动有更多的了解 和大气，以便可以利用或避免这些波 不同的物理应用。 ***

9623060 Sun The purpose of this project is to present a mathematical and numerical study of solitary and periodic waves in stratified fluids of infinite depth subject to the gravitational force. The proposal consists of two parts. The first part intends to give a rigorous answer to an open question whether there exist solitary internal waves with algebraic decay at infinity in a continuously stratified fluid that is bounded only by a rigid bottom and has a constant density except for a layer with continuous density stratification. The proposed research will give a rigorous justification of the existence of solitary waves derived from a formal model equation, called the Benjamin-Ono equation. The second part deals with the existence of periodic waves of large amplitude in a two-layer fluid without boundaries. A new formulation of the problem will be introduced, which transforms the governing equations into a single integral equation. Then the solutions of the integral equation will be studied numerically and theoretically for any density ratio without restrictions on the amplitude of solutions. In this project, the main thrust is to show that the fully nonlinear governing equations for stratified fluids have solutions of finite and large amplitude and give the various properties of the solutions using numerical and theoretical approaches. An interplay of theories in differential equations and functional analysis will be essential to obtaining the rigorous justifications, while numerical computation gives some crucial information on the solutions. %%% The gravity waves of large amplitude in a fluid of density variation, also called a stratified fluid, with great depth are of considerable geophysical interest. Large amplitude internal wave disturbances are common features in the oceans as well as in the lower atmosphere. In particular, a solitary wave, whose form is a localized single hump, and a periodic wave, which always has a same form after a certain distance, are relevant to various oceanic and atmospherical phenomena. The solitary waves of large amplitude have been associated with the formation of tornados in the atmosphere and the enormous transport of momentum and energy within the oceans. This work focuses on obtaining the conditions under which the solitary or periodic waves can exist and predicting how large the amplitude of the waves can be if they exist. It will also try to capture the qualitative features of these waves using numerical computations. The results obtained from this research may provide some theoretical explanation of the formation of these waves and give more understanding of certain wave motions in the oceans and atmosphere so that these waves may be either utilized or avoided in different physical applications. ***