Derivation and numerical solution of fully nonlinear and fully dispersive water wave model equations

Abstract

The water wave theory is a classical part of Fluid Mechanics. It has a long scientific history with a great number of mathematical results [10, 32]. The first studies in this field were done by Stokes in 1847 [34]. He developed some approximations to periodic waves and proposed conjectures about their behavior on deep water. Today these waves are known as Stokes waves. The problem of water waves concerns the two-dimensional flow of an inviscid, incompressible fluid, bounded above by a free surface and below by a rigid horizontal bottom. In this situation, the flow is described by Euler equations with appropriate boundary conditions [13]. By solving these equations, one obtains a complete understanding of the flow dynamics. However, for some applications the dynamics of the free surface is of particular interest. Nonlinear dispersive wave equations, such as the Korteweg–de Vries equation [22], allow to approximate the description of the free surface evolution without having to provide a complete solution of the fluid flow below the surface. Different questions related to these equations are actively researched. The existence of traveling and solitary waves solutions [15], well posedness of these equations [21, 24, 25] are two examples. In this work, nonlinear dispersive water wave equations are analyzed from a number of points: their accuracy in approximating the solutions of Euler equations, derivation from Hamiltonian formulation of the water wave problem, numerical solution of these equations, stability of their solutions and investigation of their bifurcation curves. The Part I of this thesis consists of three chapters. Chapter 1 is devoted to mathematical formulation of the surface water-wave problem. We revise the Eulerian formulation of the problem, the Linear water-wave theory and the boundary conditions applied. The second chapter describes a method for deriving nonlinear dispersive water-wave equations. The main focus here is on the Hamiltonian formulation of the problem and different scaling regimes. A numerical method for solving the water-wave equations is also presented in Chapter 2. Chapter 3 gives a brief summary of the research results obtained in the course of the doctoral studies. Part II contains the research papers, published and submitted for publication, which were written for this PhD project.