Large-Amplitude Steady Solitary Water Waves with General Vorticity

Jifeng Chu; Zihao Wang; Yong Zhang

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arXiv · 2603.19699

Large-Amplitude Steady Solitary Water Waves with General Vorticity

Jifeng Chu·Zihao Wang·Yong Zhang

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Abstract

We study two-dimensional steady solitary gravity water waves with general vorticity, allowing for overhanging free-surface profiles. The main challenges arise from the free boundary, the unbounded fluid domain, and the inherent complexity of the general vorticity setting. To address these, we introduce a conformal reformulation that reduces the problem to an equivalent system on a fixed strip, consisting of an overdetermined elliptic problem coupled with an elliptic boundary-value problem. This framework enables a local analysis without imposing restrictive assumptions on the vorticity. Using the center-manifold construction of Chen, Walsh and Wheeler \cite{chennonlinearity}, we first establish the existence of small-amplitude solitary waves for such non-trivial vorticity distributions. Subsequently, via a global analytic bifurcation argument, we prove the existence of continuous branches of large-amplitude solitary waves for analytic vorticity functions. Along these global solution curves, the free surfaces may develop overhanging profiles and need not remain graphs over the horizontal coordinate. Our results provide a new constructive framework for large-amplitude solitary waves with general vorticity, extending the existing theory beyond the classical settings of constant vorticity \cite{susannaarma} and non-overhanging profiles \cite{milesjma}.

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Jifeng Chu, Zihao Wang, Yong Zhang. 2026-09-23. Large-Amplitude Steady Solitary Water Waves with General Vorticity. https://arxiv.org/abs/2603.19699

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