4 1. INTRODUCTION

allows to use the center manifold reduction method. In fact if we restrict

the study to periodic solutions as here, the center manifold reduction is not

necessary, and the infinite depth case might be considered in using an exten-

sion of the proof of Lyapunov-Devaney center theorem in the spirit of [I], in

this case where 0 belongs to the continuous spectrum. However, it appears

that the number of imaginary eigenvalues becomes infinite when the surface

tension cancels, which prevents the use of center manifold reduction in the

limiting case we are considering in the present paper, not only because of

the infinite depth.

1.2. Formulation of the Problem

Since we are looking for waves travelling with velocity c, let us consider

the system in the moving frame where the waves look steady. Let us denote

by ϕ the potential defined by

ϕ = φ − c · X,

where φ is the usual velocity potential, X = (x1, x2) is the 2-dim horizontal

coordinate, x3 is the vertical coordinate, and the fluid region is

Ω = {(X, x3) : −∞ x3 η(X)},

which is bounded by the free surface Σ defined by

Σ = {(X, x3) : x3 = η(X)}.

We also make a scaling in choosing |c| for the velocity scale, and L for

a length scale (to be chosen later), and we still denote by (X, x3) the new

coordinates, and by ϕ, η the unknown functions. Now defining the parameter

µ =

gL

c2

(the Froude number is

√c

gL

) where g denotes the acceleration of

gravity, and u the unit vector in the direction of c, the system reads

∆ϕ = 0 in Ω, (1.1)

∇Xη · (u + ∇Xϕ) −

∂ϕ

∂x3

= 0 on Σ, (1.2)

u · ∇Xϕ +

(∇ϕ)2

2

+ µη = 0 on Σ, (1.3)

∇ϕ → 0 as x3 → −∞.

Hilbert spaces of periodic functions. We specialize our study to

spatially periodic 3-dimensional travelling waves, i.e. the solutions η and

ϕ are bi-periodic in X. This means that there are two independent wave

vectors K1, K2 ∈

R2

generating a lattice

Γ = {K = n1K1 + n2K2 : nj ∈ Z},

and a dual lattice Γ of periods in

R2

such that

Γ = {λ = m1λ1 + m2λ2 : mj ∈ Z, λj · Kl = 2πδjl}.

The Fourier expansions of η and ϕ are in terms of

eiK·X

, where K ∈ Γ

and K · λ = 2nπ, n ∈ Z, for λ ∈ Γ. The situation we consider in the