Semianalytic theory of standing waves in deep water, for several dominant modes

We use an appropriate normal-form technique for transforming the classical two-dimensional standing-wave problem. We are then able to prove in a simpl er way the result previously obtained by Amick and Toland on the formal com putation of standing waves with one dominant mode. In addition, we prove in the same way the existence of formal solutions with an arbitrary number, p ossibly infinite, of dominant modes.