MI gain rate g_max
g = 0.000
Critical wavenumber
K_c = 0.000
MI: γA²>0 → instability
Benjamin & Feir (1967): ocean waves in deep water are modulationally unstable. NLS: i∂ₜψ + β₂/2·∂ₓₓψ + γ|ψ|²ψ = 0.
Linearize ψ = (A+a)e^{iγA²t}: perturbation a grows as e^{g·t} for |K|<K_c=2A√(γ/β₂).
Maximum gain: g_max = γA² (at K_opt). Fermi-Pasta-Ulam recurrence follows the initial energy to side bands and back.
Linearize ψ = (A+a)e^{iγA²t}: perturbation a grows as e^{g·t} for |K|<K_c=2A√(γ/β₂).
Maximum gain: g_max = γA² (at K_opt). Fermi-Pasta-Ulam recurrence follows the initial energy to side bands and back.