Chapter 2 - Nonlinear Water Wave Equations

Abstract

The first successful wave theories are linear and dispersive and solvable by the ordinary, linear Fourier transform. Higher-order theories, such as the Korteweg–de Vries (KdV) and nonlinear Schrödinger (NLS) equations arise from nonlinear singular perturbations of these leading order linear theories using the Euler equations as the natural (nonlinear) starting point. Many of the simpler derived nonlinear partial differential equations have been found to be integrable and are solvable by a relatively new method of mathematical physics known as the inverse scattering transform (IST). IST is a natural nonlinear generalization of the linear Fourier transform. The solutions of these nonlinear wave equations typically include solitons, and the equations and methods of solution are often referred to as “soliton theories.” These theories are the natural generalizations of linear wave theory to nonlinear wave motion––that is, by allowing a suitable nonlinear parameter to become small, the linear dispersive wave theories are naturally recovered. The soliton theories have many kinds of coherent structures––that is, they include solitons, negative solitons (“holes”), shocks, vortices, and unstable “rogue” modes. These structures are typically nonlinear Fourier components in the IST theory.