Asymmetrical three-dimensional travelling gravity waves

Gérard Iooss (1, 2) , Pavel Plotnikov (3)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 IUF - Institut universitaire de France
3 LIH - Lavrentyev Institute of Hydrodynamics

We consider periodic travelling gravity waves at the surface of an infinitely deep perfect fluid. The pattern is non symmetric with respect to the propagation direction of the waves and we consider a general non resonant situation. Defining a couple of amplitudes ε 1 , ε 2 along the basis of wave vectors which satisfy the dispersion relation, first we give the formal asymptotic expansion of bifurcating solutions in powers of ε 1 , ε 2. Then, introducing an additional equation for the unknown diffeomorphism of the torus, associated with an irrational rotation number, which allows to transform the differential at the successive points of the Newton iteration method, into a differential equation with two constant main coefficients, we are able to use a descent method leading to an invertible differential. Then by using an adapted Nash Moser theorem, we prove the existence of solutions with the above asymptotic expansion, for values of the couple (ε1^2 , ε2^2) in a subset of the first quadrant of the plane, with asymptotic full measure at the origin.

Mots clés: nonlinear water waves; small divisors; bifurcation theory; pseudodifferential operators; travelling gravity waves; asymmetric 3D waves; rotation number AMS: 76B15; 47J15; 35S15; 76B07

Domaines: Equations aux dérivées partielles [math.AP]; Mécanique des fluides [physics.class-ph]; Formation de Structures et Solitons [nlin.PS]; Systèmes dynamiques [math.DS]

Gérard Iooss, Pavel Plotnikov. Asymmetrical three-dimensional travelling gravity waves. Archive for Rational Mechanics and Analysis, 2011, 200 (3), pp.92. ⟨hal-01265172⟩