On the first bifurcation of Stokes waves

TL;DR

This paper investigates the conditions under which the first bifurcation of Stokes waves occurs, providing explicit criteria for the eigenvalue assumptions involved in bifurcation analysis.

Contribution

It offers explicit conditions for the eigenvalue assumption critical to the bifurcation of Stokes waves, extending previous existence results.

Findings

Explicit eigenvalue conditions identified

Bifurcation criteria clarified

Connection to large amplitude waves established

Abstract

We consider Stokes water waves on the vorticity flow in a two-dimensional channel of finite depth. In the paper "V.Kozlov, On first subharmonic bifurcations in a branch of Stokes waves, JDE, 2024," it was proved existence of subharmonic bifurcations on a branch of Stokes waves. Such bifurcations occur near the first bifurcation in the set of Stokes waves. Moreover it is shown in that paper that the bifurcating solutions build a connected continuum containing large amplitude waves. This fact was proved under a certain assumption concerning the second eigenvalue of the Frechet derivative. In this paper we investigate this assumption and present explicit conditions when it is satisfied.