The Benjamin–Feir instability (1967): a Stokes wave A₀e^{iγ|A₀|²t} is unstable to sidebands with |Ω| < 2γ^{1/2}|A₀|ω₀/√2. Growth rate: σ = Ω/2·√(4γ|A₀|²−Ω²).
Governed by the focusing NLS: i∂_t A + ½∂²_x A + γ|A|²A = 0. The exact spatially periodic breather is the Akhmediev breather, recurs with period T = 2π/σ — a mechanism for rogue waves.
Space-time plot shows the recurrence and amplitude focusing pattern.