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Small divisor problem in the theory of three-dimensional water gravity waves
We consider doubly-periodic travelling waves at the surface of an infinitely
deep perfect fluid, only subjected to gravity and resulting from the
nonlinear interaction of two simply periodic travelling waves making an angle
between them. \newline Denoting by the dimensionless
bifurcation parameter ( is the wave length along the direction of the
travelling wave and is the velocity of the wave), bifurcation occurs for
. For non-resonant cases, we first give a large family of
formal three-dimensional gravity travelling waves, in the form of an expansion
in powers of the amplitudes of two basic travelling waves. "Diamond waves" are
a particular case of such waves, when they are symmetric with respect to the
direction of propagation.\newline \emph{The main object of the paper is the
proof of existence} of such symmetric waves having the above mentioned
asymptotic expansion. Due to the \emph{occurence of small divisors}, the main
difficulty is the inversion of the linearized operator at a non trivial point,
for applying the Nash Moser theorem. This operator is the sum of a second order
differentiation along a certain direction, and an integro-differential operator
of first order, both depending periodically of coordinates. It is shown that
for almost all angles , the 3-dimensional travelling waves bifurcate
for a set of "good" values of the bifurcation parameter having asymptotically a
full measure near the bifurcation curve in the parameter plane Comment: 119
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