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 bb and the Froude number FF, 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−2λ=F−2, C1C1 is the curve in the (b,λ)(b,λ)-plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and the intersection point of C1C1 with {λ=1}{λ=1} is b1>0b1>0. In this paper, the case for 0