Nonlinear interfacial waves with a free surface

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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.