Nonlinear interfacial waves with a free surface

流体在不同密度层中的分层是自然界中的一种众所周知的现象。波可以在两个叠加层之间的界面上传播。例如，由于盐度的变化，在密度不同的两层水之间的尖锐界面上出现的界面波。在合成孔径雷达(SAR)图像中，有对海洋中所谓的“死水”效应和舰船后面窄V形尾迹的实验观测，这可以用界面波与自由表面相互作用的存在来解释。界面和自由表面都是未知的，必须作为解决方案的一部分来找到。寻找这些界面的问题属于一类自由边界问题，它存在于许多物理和生物应用中。界面波系或自由面波系由面波模式和内波模式决定。在研究界面波时，通常忽略自由面的存在。自由表面和界面之间的非线性耦合使问题变得更加困难。我们将研究当自由面不被忽略时可能存在的不同类型的非线性波。还将考虑表面张力和界面张力的影响，并将特别从二维和三维角度研究不同类型孤立波的存在。将开发完全非线性的代码来解决这些问题。计划开发与时间相关的算法，以探索在二维和三维计算的不同类型定常波的稳定性。

The stratification of fluids in layers of different densities is a well-known phenomenon in nature. Waves can propagate at the interface between two superposed layers. Examples are the interfacial waves which occur at a sharp interface between two layers of water of different densities due to the variation in salinity. There are experimental observations of the so-called ``dead-water'' effect in oceans and of narrow V-wakes behind the ships in synthetic-aperture radar (SAR) images, which may be explained by presence of the interfacial waves interacting with the free surface.Both the interface and the free surface are unknown, and have to be found as part of the solution. The problem of finding these interfaces belong to the class of free boundary problems, which occur in many physical and biological applications. The wave system at the interface or at the free surface is determined by a surface-wave mode and an internal-wave mode. The presence of the free surface was usually neglected when the interfacial waves were investigated. The nonlinear coupling between the free surface and interface makes the problem much more difficult. We will study different types of nonlinear waves which may exist when the free surface is not neglected. The effect of the surface tension and interfacial tension will also be considered and the existence of different types of solitary waves will be in particular investigated in two and three dimensions. Fully nonlinear codes will be developed to tackle these problems. Time-dependent algorithms are planned to be developed to explore the stability of different types of steady waves computed in two and three dimensions.