Asymptotic Methods for Weakly Nonlinear and Other Water Waves

Abstract

The model and equations that describe the classical (inviscid) water-wave problem are introduced and some standard applications listed. The particular interests here are outlined, based on the relevant non-dimensionalisation and scalings; the resulting form of the equations is then interpreted in terms of the various parameters. The main development in this contribution makes use of the ideas and techniques of asymptotic (parameter) expansions, methods which are briefly described. This approach is then used to develop some important, approximate equations that are generated by the water-wave problem, many of which are of completely integrable type i.e. they are ‘soliton’ equations, for which an inverse scattering transform (IST) exists; these important ideas are briefly described. The corresponding problems of modulated waves are also mentioned. Further, the various models and approximations introduced here can be improved in order to represent, more accurately, physically realistic flows (but always in the absence of viscosity). To this end we provide, as an example, the problem of variable depth for weakly nonlinear, dispersive waves. A class of problems, of considerable practical and current mathematical interest, involves the inclusion of vorticity (i.e. the prescription of some background flow on which a wave propagates). In particular, we will present the corresponding problems associated with some of the classical examples introduced earlier, such as the Korteweg-de Vries and Camassa-Holm equations. We conclude this type of problem by discussing the propagation of ring waves on a flow with some prescribe flow (i.e. non-zero vorticity) in a given direction. We finish with an introduction to the way in which we can use asymptotic expansions to extract some details of solutions that exist only in a formal sense. As examples of this, we look at two rather special wave-propagation problems: periodic waves with vorticity and a novel approach to the problem of edge waves.