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
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