This chapter is devoted to three additional topics.

The equations we have discussed have been those for which the inverse scattering transform is applicable. However, given any evolution equation, it is natural to ask whether it can be solved by the inverse scattering transform (1ST); in other words, how do we decide if a given equation is completely integrable? This question is still open but a promising conjecture concerns the so-called Painlevé property. We shall describe how the Painlevé equations arise, what they are and the conjecture itself.

If the evolution equation cannot be solved by the 1ST, but is close to one which can be (by virtue of a small parameter), we may adopt the following procedure. The 1ST method is formulated in the conventional way but the time evolution of the scattering data now involves the small parameter. This parameter can be used as the basis for generating an asymptotic solution of the inverse scattering problem, and hence of the original equation. We shall outline the development of this argument.

Finally, if neither of the above methods is applicable, or if a graphical representation of the solution is required, then we may use a numerical solution. Indeed, the original motivation for the 1ST came from a study of numerical solutions of the KdV equation. We shall, in the final section of this chapter, present some numerical methods suited to the solution of the initial-value problem for evolution equations.

The theory of solitons is attractive and exciting; it brings together many branches of mathematics, some of which touch on deep ideas. Several of its aspects are amazing and beautiful; we shall present some of them in this book. The theory is, nevertheless, related to even more areas of mathematics, and has even more applications to the physical sciences, than the number which are included here. It has an interesting history and a promising future. Indeed, the work of Kruskal and his associates which gave us the ‘inverse scattering transform’ – a grand title for soliton theory – is a major achievement of twentieth-century mathematics. Their work was stimulated by a physical problem together with some surprising computational results. This is a classic example of how numerical results lead to the development of new mathematics, just as observational and experimental results have done since the time of Archimedes.

This book has grown out of Solitons written by one of us (PGD). That book originated from lectures given to final-year mathematics honours students at the University of Bristol. Much of the material in this version has also been used as the basis for an introductory course on inverse scattering theory given to MSc students at the University of Newcastle upon Tyne. In both courses the aim was to present the essence of inverse scattering clearly, rather than to develop the theory rigorously and completely. That is also the overall aim of this book.

Strong interactions, local instabilities and turbulence: a postscript

Short waves and long waves

Small-wavelength disturbances may ride on large-amplitude long gravity waves. The orbital velocities of fluid particles due to the long waves provide a variable surface current through which the short waves propagate. When this current is comparable with the propagation velocity of the short waves relative to the long ones, their interaction is no longer weak. Nevertheless, the characteristics of the short waves may still be described, at least in part, by Whitham's theory of slowly-varying wave-trains in an inhomogeneous medium (see §11.3). Phillips (1981b) deduced from wave action conservation that capillary waves are likely to be ‘blocked’ by steep gravity waves. In much the same way, Gargett & Hughes (1972) earlier showed that short gravity waves may be trapped by long internal waves, so leading to caustic formation and local wave breaking. Untrapped modes also undergo amplitude modulations by the straining of the dominant wave field.

Computations of Longuet-Higgins (1978a, b) and McLean et al. (1981), already described in §22.2, display generation of short waves by high-order instability of steep gravity waves. In addition, finite-amplitude wave-trains necessarily contain bound harmonics, which travel with the fundamental Fourier component. Weakly-nonlinear interaction of neighbouring frequency components may also give rise to phase-locking of modes.

For these reasons, and doubtless others besides, measurements of the phase speeds of Fourier components of wave fields often reveal significant departures from the linear dispersion relation, even after allowance is made for wave-induced mean currents.