Three-Dimensional Steady Capillary-Gravity Waves

Abstract

Three-dimensional steady capillary-gravity water-waves are studied in this paper. Potential flow of an ideal fluid in a layer with finite depth and upper free surface is considered. The existence of these waves is derived through bifurcation processes from the state of rest. The waves are assumed to be periodic in the direction of propagation and just bounded in the transverse direction (modulated periodic travelling waves — MPTW). Restricting the analysis to small amplitude waves, one can reduce the problem to a finite-dimensional reversible and reflectionally symmetric dynamical system. Existence and full information about the geometry of the shape of possible crests then follows via normal form analysis and persistence.