Fully dispersive water wave equations

Abstract

The PhD project concerns the surface water wave theory. Liquid is presented as a three or two dimensional layer bounded from below by a rigid horizontal bottom. Above it can have either a free surface or an elastic layer such as an ice cover, for example. The fluid flow is considered to be inviscid, incompressible and irrotational. The flow is de- scribed by the Euler equations with some appropriate boundary conditions. Following Lagrange we assume that at the bottom liquid has zero vertical velocity component and at the free surface there is a constant atmospheric pressure. The first condition is very natural meaning that water cannot penetrate through the bottom and is always attached to it making no cavities. The second condition is based on the fact that the air fluctua- tions are negligible and the pressure is defined by the weight of the atmosphere. Note that it also assumed that the pressure is changing continuously in the air-water media. The latter can be modified in different ways. For example, assuming a strong capillar- ity effect one can assume that the liquid pressure at the top is proportional to the surface curvature. However, it turns out that this effect is more important in water tanks than in the ocean. Another more relevant modification for the real world is the modelling of ice cover. In such situation the surface tension is caused by deformation forces in the ice. The latter is assumed to be elastic. Solving the Euler equations with the mentioned boundary conditions provides with the complete description of the flow dynamics. However, in many situations it is more important to know only the surface time evolution, whereas solving the full problem may be too demanding. Different nonlinear dispersive wave equations, such as the Korteweg-de Vries equation, allow to approximate the free surface dynamics without providing with the complete description of the fluid motion below the surface. There are several ways of testing the validity of these models. The two most important of them are tests on well-posedness and travelling wave existence. A relevant model should at least reflect adequately these two properties. In this work we are mainly concerned with the fully dispersive bi-directional exten- sions of the mentioned KdV equation. Still being simple toy models they are believed to be better approximations to the full Euler equations. Moreover, they allow two di- rectional water motion for the two dimensional liquid layer, whereas the scalar KdV equation describes the waves traveling in one direction. A recent interest in such mod- els was caused by discovery of certain phenomena in the Whitham equation that is a direct fully dispersive one-directional extension of the KdV equation. Among them are solitary waves, the existence of a wave of greatest height predicted by Stokes, the ex- istence of shocks and modulational instability of steady periodic waves. A natural step forward is to try to find an adequate two-directional extension of the scalar Whitham equation displaying the same phenomena or some other water wave properties. In this project we have an overview of several such models recently put forward. We pay a special attention to one particular system introduced by the author. We anal- yse situations demanding the use of such models, testing them with different surface boundary conditions, as for example, putting ice cover atop. Consideration is given to the initial value problem and solitary wave existence. We simulate periodic travelling waves for these models and also for one exact model the so-called Babenko equation, that is also an example of a fully dispersive model, in particular. The thesis is based on ten papers submitted during the work on the project, with eight of them being included in the text.