Three-dimensional travelling gravity-capillary water waves

1.1 The hydrodynamic problem The classical water-wave problem concerns the irrotational flow of a perfect fluid of unit density subject to the forces of gravity and surface tension. The fluid motion is described by the Euler equations in a domain bounded below by a rigid horizontal bottom {y = −h} and above by a free surface which is described as a graph {y = η(x, z, t)}, where the function η depends upon the two horizontal spatial directions x, z and time t. In terms of an Eulerian velocity potential φ(x, y, z, t) the mathematical problem is to solve the equations φxx + φyy + φzz = 0 − h < y < η, (1) φy = 0 on y = −h, (2) φy = ηt + ηxφx + ηzφz on y = η (3)