Comparison of modulational instabilities in full-dispersion shallow water models

浅水波局部模型中周期行波的调制稳定性

Modulational stability of the periodic traveling wave in a local model for shallow water waves

本文研究浅水波局部模型中周期行波的调制稳定性，通过Floquet-Bloch理论、摄动理论和谱分析建立小振幅周期行波的稳定性，并分析具有立方非线性的扩展Hunter-Saxton方程的调制不稳定性。

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…

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…

Modulational Instability in the Whitham Equation with Surface Tension and Vorticity

We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant vorticity. When the surface tension coefficient is large, we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin-Feir instability of gravity waves. In the case of weak surface tension, we find inter…

KdV-like方程中Benjamin-Feir不稳定性研究

The Benjamin-Feir instability in KdV-like equations with general dispersion and monomial nonlinearity

本文研究了具有通用色散和多项式非线性的KdV-like方程的Benjamin-Feir不稳定性，通过线性化周期行波，完整刻画了线性算子在原点附近的谱结构，证明了当模化不稳定性存在时，谱结构呈现闭合的八字形图案。

Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law

In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow down an incline. We begin by introducing a natural set of spectral stability assumptions, motivated by considerations from the Whitham averaged equations, and outline the recent proof yielding nonlinear stability under these conditions. We then turn to an analytical and numerical investigation of the verification of these spectral stability assumptions. While …