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Recent developments in the mathematical theory of water waves

Recent developments in the mathematical theory of water waves (Royal Society Philosophical Transactions A)

The last decade has seen vigorous activity in mathematical theory for the motion of water waves by several independent international research groups, and in 2001 a workshop on mathematical problems of nonlinear hydrodynamic waves was held at the conference centre at Oberwolfach, Southern Germany. The aim of this workshop was to bring these groups together and to focus attention upon certain famous and still outstanding open problems in water waves. These aspects pose considerable challenges and are of importance far beyond the hydrodynamic application in which they emerge. This themed issue contains seven original scientific papers which originated frompresentations or discussions at the workshop.

The mathematical equations which describe the motion of water waves were written down in the nineteenth century by the English mathematician George Stokes. However these basic equations are notoriously difficult mathematically and certainly not well understood even now at the beginning of the twenty-first century. Therefore, at least since the time of Boussinesq and Korteweg and deVries in the late nineteenth century, approximations, derived from assumptions such as `small amplitude`, `small depth-to-wavelength ratio` or `uni-directionality`, have been used to facilitate a simplification of Stokes`s equations leading to simpler differential equations which model water waves.

Typically these model equations, although still nonlinear, have features, such as exact solutions which can be explicitly written down, which make them attractive to a scientist seeking to explain observations. Indeed, Scott Russell explained his observation of the `solitary wave` on the Edinburgh canal in terms of a solution of the Korteweg-deVries equation. However, from a mathematical standpoint there is a considerable difference between `model` theories and the so-called `exact` water-wave equations. An important goal of the mathematical scientist is to show that the approximations used in deriving a model can be justified and to examine rigorously the extent to which solutions of a model equation corresponds to a solution of the exact equations The `exact` water-wave problem is mathematically interesting in another respect: it is a paradigm for most modern methods in nonlinear mathematics, its study calling upon many different approaches from a wide range of applications.

The present collection of papers was compiled to reflect the recent progress on the `exact` water-wave equations reported in the Oberwolfach meeting. Much of the work concerned the existence of small-amplitude permanent waves (waves which move in a fixed direction with a fixed speed). In particular, we see the `spatial dynamics` method brought to a level of refinement so that within its scope now lie deep and difficult fully nonlinear problems in the theory of three-dimensional water waves and multi-layered stratified flows that would have seemed impossible only a few years ago.

With such achievements in small-amplitude two-dimensional water-wave theory, from the viewpoint of existence theory, we also see evidence of the developing interest in the physically more realistic, but mathematically more subtle and demanding, mathematical theory for fully three-dimensional wave phenomena. Two early results in this area include a nonexistence result for `fully localised` gravity waves and a discussion of the transition from two-dimensional `line` solitary waves to three-dimensional `periodically modulated` solitary waves and raises the question of `fully localised` solitary waves for large surface-tension forces.

A result for nonlinear gravity-capillary waves is presented which confirms famous predictions about the stability of solitary waves made upon the basis of the Korteweg-de Vries equation. Numerical computations and mathematical results about the structure of the water-wave equations are also included.

The Royal Society. September 2002.

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