In this talk, we discuss recent studies on the spectral (in-)stability of periodic waves, focusing on two systems from plasma physics and fluid mechanics—the Euler-Poisson system and the water wave system.

First, we show that small-amplitude periodic waves in the electronic Euler-Poisson system are inherently unstable under localized perturbations, with the instability driven by perturbations with finite wavelengths (or high frequencies).

Second, we examine the transverse instability of Stokes waves (one-dimensional periodic water waves) in a two-dimensional gravity water wave system with finite depth.

Our key finding is that when subjected to perturbations with sufficiently small transverse frequencies, all sufficiently small Stokes waves become unstable.