深水中Stokes波的分支切割。第一部分：数值解和Padé近似
Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Padé approximation
摘要：
分析了具有自由表面和无限深的理想不可压缩流体二维势流Stokes波的复解析结构。Stokes波是以恒定速度传播的完全非线性周期性重力波。为了高精度地寻找Stokes波，并研究Stokes波在波峰处以$2π/3$弧度角逼近其极限形式，进行了四倍精度和变精度的数值模拟。采用保角映射，将Stokes波的自由流体表面映射到实数线上，流体域映射到下复半平面上。上复半平面中的复奇异性是Stokes波的充分特征。这些奇异性通过复平面中斯托克斯波的有理（Padé）插值来解决。Padé近似对复极点密度的收敛性随着数值精度的提高和逼近极点数目的增加而增加，表明Stokes波的唯一奇点是由分支割连接的分支点。收敛密度是跨越分支切口的跳跃。Stokes波的每个水平空间周期$λ$有一个分支切口。每个分支切口严格垂直延伸到斯托克斯波的相应波峰上方，直到复无穷大。分支割的下端是位于与共形变量中的流体表面对应的实线相距$v_c$处的平方根分支点。当奇点到达流体表面时，极限斯托克斯波出现。给出了不同高度Stokes波的Padé近似表。这些表允许以至少$10^{-26}$的相对精度恢复斯托克斯波。该表适用于$v_c/λ\sim 10^{-6}.$的高度非线性Stokes波的几个极点到近百个极点
英文摘要：
Complex analytical structure of Stokes wave for two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave propagating with the constant velocity. Simulations with the quadruple and variable precisions are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with $2π/3$ radians angle on the crest. A conformal map is used which maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half-plane. These singularities are addressed by rational (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé approximation to the density of complex poles with the increase of the numerical precision and subsequent increase of the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one branch cut per horizontal spatial period $λ$ of Stokes wave. Each branch cut extends strictly vertically above the corresponding crest of Stokes wave up to complex infinity. The lower end of branch cut is the square-root branch point located at the distance $v_c$ from the real line corresponding to the fluid surface in conformal variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Padé approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least $10^{-26}$. The tables use from several poles to about hundred poles for highly nonlinear Stokes wave with $v_c/λ\sim 10^{-6}.$
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