Global bifurcation structure and geometric properties for steady periodic water waves with vorticity

TL;DR

This paper analyzes the bifurcation structure and geometric features of steady periodic water waves with vorticity, revealing symmetry, monotonicity, inflection points, and concavity/convexity properties of wave profiles.

Contribution

It introduces two continuous bifurcation curves for water waves with vorticity, demonstrating their geometric properties and behavior under certain conditions.

Findings

Existence of two bifurcation curves meeting once

Wave profiles are symmetric and monotone between crests and troughs

Presence of inflection points and curvature properties at crests and troughs

Abstract

This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work \cite{ConstantinStrauss}, we first obtain two continuous bifurcation curves which meet the laminar flow only one time by using modified analytic bifurcation theorem. They are symmetric waves whose profiles are monotone between each crest and trough. Furthermore, we find that there is at least one inflection point on the wave profile between successive crests and troughs and the free surface is strictly concave at any crest and strictly convex at any trough. In addition, for favorable vorticity, we prove that the vertical displacement of water waves decreases with depth.