NLS sideband growth, rogue wave formation, Peregrine soliton, ocean wave focusing
NLS: i∂ψ/∂T + ∂²ψ/∂X² + 2|ψ|²ψ = 0; ψ = A(X,T)e^(iX)
BF growth rate: σ = ΔK√(2ε²−ΔK²); unstable if ΔK < √2·ε (Benjamin-Feir 1967)
Benjamin and Feir (1967): a uniform Stokes wave train is unstable to long-wave perturbations. For steepness ε=k₀a₀ and sideband Δk/k₀ < √2·ε, perturbations grow exponentially at rate σ. The instability leads to: sideband growth → nonlinear focusing → Peregrine soliton (rogue wave prototype: amplitude 3× carrier, localized in both space and time). Governing equation: focusing NLS. Time-reversal experiment: Peregrine soliton observed in water waves (Chabchoub 2011) and optics.