Introduction

It has long been recognized that linear theories cannot capture even the most basic features of waves observed on water surfaces: solitary waves being a particularly famous example where linear equations will not suffice. Hydrodynamic waves are inherently nonlinear. However the basic Euler equations which govern the behaviour of inviscid incompressible fluid flows are notoriously difficult mathematically and are certainly not well understood even at the beginning of the 21st century. When a constant–pressure nonlinear free–boundary condition is added to represent a water wave they become more awkward still, and more nonlinear.