On loops in water wave branches and monotonicity of water waves

TL;DR

This paper presents a straightforward method to prove the absence of loops in bifurcation branches of rotational water waves, including cases with stagnation points, overhanging surfaces, and critical layers, and discusses monotonicity properties.

Contribution

It introduces a simple approach to establish the absence of loops in water wave bifurcation branches with arbitrary vorticity and provides criteria based on wave period and velocity components.

Findings

No loops in bifurcation branches under certain conditions.

Monotonicity properties of the free surface are established.

A simple criterion for absence of loops when bifurcation parameter is wave period.

Abstract

We give a quite simple approach how to prove the absence of loops in bifurcation branches of water waves in rotational case with arbitrary vorticity distribution. The supporting flow may contain stagnation points and critical layers and water surface is allowed to be overhanging. Monotonicity properties of the free surface are presented. Especially simple criterium of absence of loops is given for bifurcation branches when the bifurcation parameter is the water wave period. We show that there are no loops if you start from a water wave with a positive/negative vertical component of velocity on the positive half period.