A spatial dynamics approach to three-dimensional gravity-capillary steady water waves

This paper contains a rigorous existence theory for three-dimensional stead y gravity-capillary finite-depth water waves which are uniformly translatin g in one horizontal spatial direction x and periodic in the transverse dire ction z. Physically motivated arguments are used to find a formulation of t he problem as an infinite-dimensional Hamiltonian system in which x is the time-like variable, and a centre-manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite-dimensional Hamiltonian system. General statements concerning the existence of waves w hich are periodic or quasiperiodic in x (and periodic in z) are made by app lying standard tools in Hamiltonian-systems theory to the reduced equations . A critical curve in Bond number-Froude number parameter space is identified which is associated with bifurcations of generalized solitary waves. These waves are three dimensional but decay to two-dimensional periodic waves (s mall-amplitude Stokes waves) far upstream and downstream. Their existence a s solutions of the water-wave problem confirms previous predictions made on the basis of model equations.