Exact Theory of Three-Dimensional Water Waves at the Critical Speed

Abstract

The paper concerns three-dimensional (3D) traveling gravity-capillary waves in water of finite depth. It was well known that the waves are determined by two constants: nondimensional wave speed F (called Froude number) and nondimensional surface tension b (called Bond number). For two-dimensional (2D) waves, it was known that $F=1$ is a critical value, and there were many existence results for 2D waves with large or small b. However, there was still no existence of 3D waves when F is near 1 and b is small. In this paper, the existence of 3D waves in this case is discussed. It is shown that the exact Euler equations have a generalized solitary-wave solution (solitary waves with small oscillations at infinity) which is uniformly translating in the propagation direction and periodic in the transverse direction. The first-order approximation for this 3D wave is the solution of a system of coupled Schrödinger–KdV equations.