The theory of water waves has been a source of intriguing and often difficult mathematical problems for at least 150 years. Virtually every classical mathematical technique appears somewhere within its confines. Beginning with the introduction of the appropriate equations of fluid mechanics, the opening chapters of this text consider the classical problems in linear and non-linear water-wave theory. This sets the ground for a study of more modern aspects, problems that give rise to soliton-type equations. The book closes with an introduction to the effects of viscosity. All the mathematical developments are presented in the most straightforward manner, with worked examples and simple cases carefully explained. Exercises, further reading, and historical notes on some of the important characters in the field round off the book and help to make this an ideal text for a beginning graduate course on water waves.

In this paper we first describe the current method for obtaining the Camassa–Holm equation in the context of water waves; this requires a detour via the Green–Naghdi model equations, although the important connection with classical (Korteweg–de Vries) results is included. The assumptions underlying this derivation are described and their roles analysed. (The critical assumptions are, (i) the simplified structure through the depth of the water leading to the Green–Naghdi equations, and, (ii) the choice of submanifold in the Hamiltonian representation of the Green–Naghdi equations. The first of these turns out to be unimportant because the Green–Naghdi equations can be obtained directly from the full equations, if quantities averaged over the depth are considered. However, starting from the Green–Naghdi equations precludes, from the outset, any role for the variation of the flow properties with depth; we shall show that this variation is significant. The second assumption is inconsistent with the governing equations.)Returning to the full equations for the water-wave problem, we retain both parameters (amplitude, ε, and shallowness, δ) and then seek a solution as an asymptotic expansion valid for, ε → 0, δ → 0, independently. Retaining terms O(ε), O(δ2) and O(εδ2), the resulting equation for the horizontal velocity component, evaluated at a specific depth, is a Camassa–Holm equation. Some properties of this equation, and how these relate to the surface wave, are described; the role of this special depth is discussed. The validity of the equation is also addressed; it is shown that the Camassa–Holm equation may not be uniformly valid: on suitably short length scales (measured by δ) other terms become important (resulting in a higher-order Korteweg–de Vries equation, for example). Finally, we indicate how our derivation can be extended to other scenarios; in particular, as an example, we produce a two-dimensional Camassa–Holm equation for water waves.

The steady-state solution of the non-linear equation \[ h_t + hh_x + h_{xxx} = \delta h_{xx} \] with both damping and dispersion is examined in the phase plane. For small damping an averaging technique is used to obtain an oscillatory asymptotic solution. This solution becomes invalid as the period of the oscillation approaches infinity, and is matched to a straightforward expansion solution. The results obtained are compared with a numerical integration of the equation.

The techniques that have been developed for the application of the inverse-scattering transform method to the solution of the Camassa-Holm equation, principally by Constantin, are implemented. We use this approach, first, to represent the known solitary-wave solution in a simple parametric form, and then, second, to obtain the general two-and three-soliton solutions. (These latter two solutions require rather extensive use of mathematical packages, Mathematica and Maple, in order to complete the construction of solutions.) A number of examples are presented; the phase shifts, evident after an interaction, are found and the special limit that recovers the peakon solutions is discussed.