Bifurcation analysis of Stokes waves with piecewise smooth vorticity in deep water

TL;DR

This paper proves the existence of complex water waves with discontinuous vorticity in deep water, using advanced bifurcation techniques to analyze their behavior and potential for large amplitude or stagnation.

Contribution

It introduces a novel bifurcation analysis for Stokes waves with piecewise smooth vorticity, addressing challenges of internal interfaces and unbounded domains.

Findings

Existence of waves with discontinuous vorticity established.

Wave profiles can reach arbitrarily large speeds.

Profiles may approach horizontal stagnation.

Abstract

In this paper, we establish the existence of Stokes waves with piecewise smooth vorticity in a two-dimensional, infinitely deep fluid domain. These waves represent traveling water waves propagating over sheared currents in a semi-infinite cylinder, where the vorticity may exhibit discontinuities. The analysis is carried out by applying a hodograph transformation, which reformulates the original free boundary problem into an abstract elliptic boundary value problem. Compared to previously studied steady water waves, the present setting introduces several novel features: the presence of an internal interface, an unbounded spatial domain, and a non-Fredholm linearized operator. To address these difficulties, we introduce a height function formulation, casting the problem as a transmission problem with suitable transmission conditions. A singular bifurcation approach is then employed, combining global bifurcation theory with Whyburns topological lemma. Along the global bifurcation branch, we show that the resulting wave profiles either attain arbitrarily large wave speed or approach horizontal stagnation.