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
= {(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 =
(the Froude number is
) 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)
= 0 on Σ, (1.2)
u · ∇Xϕ +
+ 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
generating a lattice
Γ = {K = n1K1 + n2K2 : nj Z},
and a dual lattice Γ of periods in
such that
Γ = = m1λ1 + m2λ2 : mj Z, λj · Kl = 2πδjl}.
The Fourier expansions of η and ϕ are in terms of
, where K Γ
and K · λ = 2nπ, n Z, for λ Γ. The situation we consider in the
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