Nonlinear Guyon waves
To study stationary periodic weakly vortical waves on water (the Gouyon waves), the method of the modified Lagrangian variables is suggested. The wave vorticity Ω is specified as a series in the small steepness parameter ε in
the form: Ω =\sum( ε^n · Ω_n (b)), where Ω_n are arbitrary functions of the vertical Lagrangian coordinate b. Earlier Gouyon (1958) studied waves in linear and quadratic approximations. The present paper provides a complete problem solution in a cubic approximation. Explicit expressions are obtained for the coordinates of the liquid particle trajectories and pressure. The quadratic correction to the wave velocity is determined. The comparison with the Stokes and Gerstner waves is made. The class of quadratically vortical Gouyon waves is distinguished, for which Ω_1 = 0. On the basis of the general dispersion theory method for waves of this class, the nonlinear Schrödinger equation is written. The criterion, when the equation is focusing and rogue Gouyon waves can be formed, is indicated.