Nonlinear modulational instabililty of the Stokes waves in 2d full water waves

Modulational Instability in the Whitham Equation for Water Waves

We show that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations,are spectrally unstable to long wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin-Feir instability of Stokes waves; they are spectrally stable to square integrable perturbations otherwise. The proof involves a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and the small amplitude…

An alternative proof of modulation instability of Stokes waves in deep water

We generalize the periodic Evans function approach recently used to study the spectral stability of Stokes wave and gravity-capillary (including Wilton ripples) in water of finite depth to study spectral stability of Stokes waves in water of infinite depth. We prove waves of sufficiently small amplitude are always low-frequency unstable regardless of the wave number and gravity, giving an alternative proof for the Benjamin-Feir modulational instability in the infinite depth case. Here, the first proof for the infinite depth case is recently obtained by Nguyen and Strauss. We also study the spe…

Modulational instability in the full-dispersion Camassa-Holm equation

We determine the stability and instability of a sufficiently small and periodic traveling wave to long wavelength perturbations, for a nonlinear dispersive equation which extends a Camassa-Holm equation to include all the dispersion of water waves and the Whitham equation to include nonlinearities of medium amplitude waves. In the absence of the effects of surface tension, the result qualitatively agrees with the Benjamin-Feir instability of a Stokes wave. In the presence of the effects of surface tension, it qualitatively agrees with those from formal asymptotic expansions of the physical pro…

Modulational instability in a full-dispersion shallow water model

We propose a shallow water model which combines the dispersion relation of water waves and the Boussinesq equations, and which extends the Whitham equation to permit bidirectional propagation. We establish that its sufficiently small, periodic wave train is spectrally unstable to long wavelength perturbations, provided that the wave number is greater than a critical value, like the Benjamin-Feir instability of a Stokes wave. We verify that the asso- ciated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability away from t…

深水重力-毛细 Stokes 波的本征结构

Spectral structure of the Benjamin-Feir instability in deep-water gravity-capillary Stokes waves

本文研究了深水重力-毛细 Stokes 波的 Benjamin-Feir 不稳定性，通过 Euler 方程的严格 Bloch-Floquet 谱分析，揭示了多特征值在原点的分裂，识别出非零实部的特征值对，从而获得表面张力参数下的稳定性区域。

空间周期水波的长期动力学

Long time dynamics of space periodic water waves

本文研究了空间周期水波的长期动力学，探讨了准周期解的分支、长期良好定义性结果以及斯托克斯波的调制不稳定性。

McLean共振与Stokes波的$3d$谱不稳定性

McLean resonances and $3d$ spectral instability of Stokes waves

通过Kato扰动分析、KAM型解耦和解析延拓，首次严格描述了深水小振幅重力Stokes波在McLean共振曲线邻域内的三维不稳定谱，揭示了Benjamin-Feir不稳定性与高频孤立的共同根源。

A Hamiltonian Dysthe equation for deep-water gravity waves with constant vorticity

This paper is a study of the water wave problem in a two-dimensional domain of infinite depth in the presence of nonzero constant vorticity. A goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear quasi-monochromatic surface gravity waves. Starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we derive a Hamiltonian Dysthe equation for the time evolution of the wave envelope. Consistent with previous studies, we observe that the uniform shear flow tends to enhance or weaken the modulational i…

修正Camassa-Holm方程中的谱稳定性

Spectral stability in the modified Camassa-Holm equation

研究修正Camassa-Holm方程小振幅周期行波解的谱稳定性，利用Kato扰动理论完整描述线性化算子原点附近谱，证明波数k²≤3时谱稳定，k²>3时出现不稳定性。

稳定与不稳定的毛细重力波

Stable and unstable capillary-gravity waves

本文通过周期性Evans函数方法研究非共振毛细重力波和小振幅Wilton涟漪的稳定性，提出新的分析框架，证明了非共振频率附近稳定性及共振频率附近稳定性与不稳定性特征。