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arXiv · 2511.03973

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

Changfeng Gui·Jun Wang·Wen Yang·Yong Zhang
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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 have jump discontinuities across internal streamlines. 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 Whyburn's topological lemma. The singular limit is taken in a natural deep-water topology based on the surface trace and the derivatives of the height function. An exact mean-flux identity controls the closing zero Fourier mode uniformly as the regularization is removed. This yields a connected global continuum of exact waves together with explicit norm, ellipticity, surface-obliqueness, and parameter-boundary alternatives.

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Changfeng Gui, Jun Wang, Wen Yang, Yong Zhang. 2026-08-13. Bifurcation analysis of Stokes waves with piecewise smooth vorticity in deep water. https://arxiv.org/abs/2511.03973

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