Asymmetrical Three-Dimensional Travelling Gravity Waves

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 $${\varepsilon_{1},\varepsilon_{2}}$$ along the basis of wave vectors which satisfy the dispersion relation, we first give the formal asymptotic expansion of bifurcating solutions in powers of $${\varepsilon_{1},\varepsilon_{2}.}$$ Then, introducing an additional equation for the unknown diffeomorphism of the torus associated with an irrational rotation number, which allows us 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 $${(\varepsilon_{1}^{2},\varepsilon_{2}^{2})}$$ in a subset of the first quadrant of the plane with asymptotic full measure at the origin.