Existence of Multimodal Standing Gravity Waves

We consider two-dimensional standing gravity waves on the surface of an infinitely deep perfect fluid, the flow being potential. It is known that the linearized problem is completely resonant. Following the method described in [4], we prove the existence of an infinity of multimodal standing gravity waves, corresponding to any choice of asymptotic expansion in powers of the amplitude indicated in [2] and [3]. Each one of these solutions exist for a set of values of being dense in 0.