NLS Modulation Instability (Benjamin-Feir)

iψ_t + ½ψ_xx + |ψ|²ψ = 0 (focusing NLS). A uniform wave A·e^(iA²t) is unstable to sidebands with wavenumber k<A√2 (Benjamin-Feir instability). Breathers (Akhmediev, Peregrine) emerge from this instability.

Amplitude A: 1.0 Perturbation ε: 0.05 Wavenumber k: 0.8 Speed: 1x

|ψ(x,t)| — envelope

Spacetime diagram |ψ(x,t)| (x horizontal, t vertical, scroll downward)