Babenko’s equation describes traveling water waves in holomorphic coordinates. It has been used in the past to obtain properties of Stokes waves with smooth profiles analytically and numerically. We show in the deep-water limit that properties of Stokes waves with peaked profiles can also be recovered from the same Babenko’s equation. In order to develop the local analysis of singularities, we rewrite Babenko’s equation as a fixed-point problem near the maximal elevation level. As a by-product, our results rule out a corner point singularity in the holomorphic coordinates, which has been obtained in a local version of Babenko’s equation.

中文翻译：

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峰值斯托克斯波作为 Babenko 方程的解


Babenko 方程描述了在全态坐标中行进的水波。过去，它已被用于通过分析和数值获得具有平滑轮廓的斯托克斯波的特性。我们在深水极限中表明，具有峰值剖面的斯托克斯波的特性也可以从相同的巴边科方程中恢复。为了发展奇点的局部分析，我们将 Babenko 方程改写为接近最大高程水平的定点问题。作为副产品，我们的结果排除了全态坐标中的角点奇点，这是在 Babenko 方程的局部版本中获得的。