CHAPTER 1

Introduction

1.1. Presentation and History of the Problem

We consider small-amplitude three-dimensional doubly periodic travel-

ling gravity waves on the free surface of a perfect fluid. These unforced

waves appear in literature as steady 3-dimensional water waves, since they

are steady in a suitable moving frame. The fluid layer is supposed to be

infinitely deep, and the flow is irrotational only subjected to gravity. The

bifurcation parameter is the horizontal phase velocity, the infinite depth case

being not essentially different from the finite depth case, except for very de-

generate situations that we do not consider here. The essential diﬃculty

here, with respect to the existing literature is that we assume the absence of

surface tension. Indeed the surface tension plays a major role in all existing

proofs for three-dimensional travelling gravity-capillary waves, and when the

surface tension is very small, which is the case in many usual situations, this

implies a reduced domain of validity of these results.

In 1847 Stokes [Sto] gave a nonlinear theory of two-dimensional trav-

elling gravity waves, computing the flow up to the cubic order of the am-

plitude of the waves, and the first mathematical proofs for such periodic

two-dimensional waves are due to Nekrasov [N], Levi-Civita [Le] and Struik

[Str] about 80 years ago. Mathematical progress on the study of three-

dimensional doubly periodic water waves came much later. In particular, to

our knowledge, first formal expansions in powers of the amplitude of three-

dimensional travelling waves can be found in papers [Fu] and [Sr]. One can

find many references and results of research on this subject in the review

paper of Dias and Kharif [DiK] (see section 6). The work of Reeder and

Shinbrot (1981)[ReSh] represents a big step forward. These authors con-

sider symmetric diamond patterns, resulting from (horizontal) wave vectors

belonging to a lattice Γ (dual to the spatial lattice Γ of the doubly periodic

pattern) spanned by two wave vectors K1 and K2 with the same length, the

velocity of the wave being in the direction of the bissectrix of these two wave

vectors, taken as the x1 horizontal axis. We give in Figure 1 two examples

of patterns for these waves (see the detailed comment about these pictures

at the end of subsection 2.4). These waves also appear in the literature

as ”short crested waves” (see Roberts and Schwartz [RoSc], Bridges, Dias,

Menasce [BDM] for an extensive discussion on various situations and nu-

merical computations). If we denote by θ the angle between K1 and the x1−

1