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

Abstract

This article presents a rigorous existence theory for three-dimensional gravity-capillary water waves which are uniformly translating and periodic in one spatial directionx and have the profile of a uni- or multipulse solitary wave in the otherz. 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 whichz is the timelike variable, and a family of pointsP k,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 pointP k,k+1is precisely that at which a pair of two-dimensional periodic linear travelling waves with frequency ratiok: 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 nearP k,k+1. It is shown that a Hamiltonian real semisimple 1∶1 resonance, where two geometrically double real eigenvalues exist, arises along a critical curveR k,k+1emanating fromP k,k+1.Unipulse transverse homoclinic solutions to the reduced Hamiltonian system at points ofR k,k+1nearP k,k+1are 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.