A plethora of three-dimensional periodic travelling gravity-capillary water waves with multipulse transverse profiles

This article presents a rigorous existence theory for three-dimensional gravity-capillary water waves which are uniformly translating and periodic in one spatial direction x and have the profile of a uni- or multipulse solitary wave in the other z. The waves are detected using a combination of Hamiltonian spatial dynamics and homoclinic Lyapunov-Schmidt theory. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which z is the time-like variable, and a family of points Pk,k+1, k = 1, 2, . . . in its two-dimensional parameter space is identified at which a Hamiltonian 0202 resonance takes place (the zero eigenspace and generalised eigenspace are respectively two and four dimensional). The point Pk,k+1 is precisely that at which a pair of two-dimensional periodic linear travelling waves with frequency ratio k : k+1 simultaneously exist (‘Wilton ripples’). A reduction principle is applied to demonstrate that the problem is locally equivalent to a four-dimensional Hamiltonian system near Pk,k+1. It is shown that a Hamiltonian real semisimple 1 : 1 resonance, where two geometrically double real eigenvalues exist, arises along a critical curve Rk,k+1 emanating from Pk,k+1. Unipulse transverse homoclinic solutions to the reduced Hamiltonian system at points of Rk,k+1 near Pk,k+1 are found by a scaling and perturbation argument, and the homoclinic Lyapunov-Schmidt method is applied to construct an infinite family of multipulse homoclinic solutions which resemble multiple copies of the unipulse solutions.