Global bifurcation and highest waves on water of finite depth

TL;DR

This paper rigorously proves the existence of extreme steady water waves with vorticity on finite depth, including highest Stokes waves with specific angles, and explores their properties and bifurcation behavior.

Contribution

It provides the first rigorous proof of highest Stokes waves with vorticity on finite depth and introduces new results on wave regularity and bifurcation analysis.

Findings

Existence of highest Stokes waves with 120-degree angles for nonnegative vorticity

Lower bounds on the wavelength of Stokes waves

Elimination of wave breaking for waves with non-negative vorticity

Abstract

We consider the two-dimensional problem for steady water waves with vorticity on water of finite depth. While neglecting the effects of surface tension we construct connected families of large amplitude periodic waves approaching the limiting wave, which is either a solitary wave, the highest solitary wave, the highest Stokes wave or a Stokes wave with a breaking profile. In particular, when the vorticity is nonnegative we prove the existence of highest Stokes waves with an included angle of 120 degrees. In contrast to previous studies we fix the Bernoulli constant and consider the wavelength as a bifurcation parameter, which guarantees that the limiting wave has a finite depth. In fact, this is the first rigorous proof of the existence of extreme Stokes waves with vorticity on water of finite depth. Beside the existence of highest waves we provide a new result about the regularity of Stokes waves of arbitrary amplitude (including extreme waves). Furthermore, we prove several new facts about steady waves, such as a lower bound for the wavelength of Stokes waves, while also eliminate a possibility of the wave breaking for waves with non-negative vorticity.