On the first bifurcation of Stokes waves

Stokes water waves on the vorticity flow in a two-dimensional channel of finite depth are treated. In author’s paper published in JDE, 2024, the existence of subharmonic bifurcations on a branch of Stokes waves was proved. Such bifurcations occur near the first bifurcation in the set of Stokes waves. Moreover it was 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 Fréchet derivative. In this paper this assumption is investigated and explicit conditions ensuring it are presented.