无限深度流体弹性Stokes波的调制谱
Modulational spectrum of infinite-depth hydroelastic Stokes waves
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中文总结 AI 辅助
该研究分析无限深度流体弹性Stokes波的调制谱，通过解耦特征值推导调制稳定性判据，识别弹性弯曲产生的稳定岛，揭示无限深度问题的奇异性。
中文摘要 AI 辅助
我们确定了无限深度下小振幅周期性流体弹性Stokes波在重力、表面张力和弹性弯曲共同作用下，从原点分岔出的完整局部Bloch谱。在Wilton型共振集之外，我们构造了实解析Stokes波分支，并分析了线性化流体弹性Euler系统有缺陷零特征值产生的四个特征值。利用解析谱扰动理论和Hamiltonian可逆约化，我们将它们解耦为Benjamin-Feir对和长波对：长波对保持纯虚数，具有奇异尺度$\boldsymbol{\textit{O}}(\boldsymbol{\textit{√|μ|}})$；Benjamin-Feir对由显式判别式控制，其主导符号给出了调制稳定与不稳定的精确判据。我们推导了表面张力-弯曲参数平面内的精确非共振相图，识别出由弹性弯曲产生的有界稳定岛。在不稳定区域且远离漂移简并时，Benjamin-Feir分支形成局部八字形曲线。在零弯曲极限下，约化系数恢复了已知的深水重力波和重力-毛细波结果，而从有限深度$\boldsymbol{\textit{O}}(\boldsymbol{\textit{|μ|}})$长波尺度到$\boldsymbol{\textit{O}}(\boldsymbol{\textit{√|μ|}})$的变化表明，无限深度问题具有奇异性。
英文摘要
We determine the complete local Bloch spectrum bifurcating from the origin for small-amplitude periodic hydroelastic Stokes waves in infinite depth, under the combined effects of gravity, surface tension, and elastic bending. Away from the Wilton-type resonance set, we construct a real-analytic Stokes-wave branch and analyze the four eigenvalues emerging from the defective zero eigenvalue of the linearized hydroelastic Euler system. Using analytic spectral perturbation theory and Hamiltonian-reversible reductions, we decouple them into a Benjamin--Feir pair and a long-wave pair. The long-wave pair remains purely imaginary and has the singular scale $\cO(\sqrt{|μ|})$, whereas the Benjamin--Feir pair is governed by an explicit discriminant whose leading sign yields a sharp criterion for modulational stability and instability. We derive the exact non-resonant phase diagram in the surface-tension-bending parameter plane and identify a bounded stability island generated by elastic bending. In the unstable region, and away from a drift degeneracy, the Benjamin--Feir branches form a local figure-eight curve. In the zero-bending limit, the reduced coefficients recover the known deep-water gravity and gravity-capillary results, while the change from the finite-depth $\cO(|μ|)$ long-wave scale to $\cO(\sqrt{|μ|})$ shows that the infinite-depth problem is singular.
