Mathematical Sciences: Nonlinear Partial Differential Equations from Hydrodynamics

这个项目要做的工作将集中在流体力学中出现的非线性问题上。将特别强调波在分层介质中的传播。几十年来，对盐度或温度变化引起的密度分层引起的这些波进行了分析研究。已在实验中观察到的许多内波现象仍有待阐明。用伪流函数的非线性偏微分方程组给出了层状波浪运动的数学模型。计算这一点，以及波速，是该项目的主要目标之一。目前的工作集中在具有光滑密度的方程上，其中海洋温跃层中存在密度逐渐变化的方程。在出现突然变化的情况下，工作将继续进行，以确定沿着解决方案分支的流线化程度有多大。其他工作考虑了双流体系统中的浪涌。在以前的文献中还没有对这种类型的内波进行严格的分析研究。现在已经发展了一种动力系统方法来解决相应的椭圆问题，并建立了一个中心流形来模拟小波的行为。涌浪的全局行为问题是一个相当开放的问题，人们将利用全局分支方法来关注这一问题。还将对多种内波结构进行数值研究。对不混溶流体之间的周期性行进重力波进行了初步研究。目前的研究也在分析一种新的悬垂波浪现象。此外，还在继续研究上下固定水平墙的双流体系统中孤立波的极限行为。这个模型模拟了冰盖下的海洋流动。这项研究在理解重要波动现象方面的应用是显而易见的。

Work to be done on this project will concentrate on nonlinear problems arising in hydrodynamics. Particular emphasis will be placed on the propagation of waves in stratified media. These waves, arising from density stratification due to changes in salinity or temperature have been investigated analytically for several decades. Much of the internal wave phenomena which has been observed experimentally, remains to be clarified. Mathematical models for stratified wave motion are given by nonlinear partial differential equations for the pseudostream function. Computing this, together with the wave speed, is one of the primary goals of the project. Current work has focused on equations with smooth densities in which there is a gradual density change in an ocean thermocline. In cases where there are abrupt changes, work will proceed on the question of how much streamlines steepen along solution branches. Other work has considered surges in two-fluid systems. No rigorous analytic studies of this type of internal wave have been presented in the literature before. A dynamical systems approach to the corresponding elliptic problem and a center manifold to model the behavior of small waves has now been developed. The question of global behavior of surges is quite open and attention will be given to this issue by using global bifurcation methods. Numerical studies on the multitude of internal wave structures will also be carried out. Initial work was carried out on periodic progressing gravity waves between immiscible fluids. Present investigations are also analyzing a new phenomenon of overhanging waves. In addition, a study is being continued on the limiting behavior of solitary waves in a two-fluid system with upper and lower fixed horizontal walls. This models ocean flows under ice caps. Applications of this research to the understanding of important wave phenomena are manifest.