Three-dimensional gravity–capillary waves on water — Small surface tension case

This paper considers three-dimensional gravity–capillary waves on water of finite-depth, which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time-like variable. The existence of the solutions of the system is determined by two non-dimensional constants, the Bond number b and the Froude number F , which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow. Assume that λ = F − 2 , C 1 is the curve in the ( b , λ ) -plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and the intersection point of C 1 with { λ = 1 } is b 1 > 0 . In this paper, the case for 0 < b < b 1 and ( b , λ ) near C 1 is considered. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic (called generalized solitary-wave, thereafter) solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants.