Modulation Instability (Benjamin-Feir) — NLS Equation

iψₜ + ½ψₓₓ + |ψ|²ψ = 0 — uniform wave train destabilizes into peaks

Amplitude A₀1.00

Perturbation ε0.020

Mod wavenumber k1.00

MI gain: σ = k_mod·√(2A₀²−k_mod²). Unstable when k_mod < √2·A₀. Peregrine soliton is the limiting case (doubly localized rogue wave).