Kolmogorov–Zakharov spectrum E(k) ~ k^(−α) from resonant wave interactions
Wave turbulence theory (Zakharov, 1967) describes weakly nonlinear wave systems via the kinetic equation ∂n_k/∂t = St[n]. For four-wave resonances (ω₁+ω₂=ω₃+ω₄, k₁+k₂=k₃+k₄), the exact stationary solution is the Kolmogorov–Zakharov (KZ) spectrum — an analogue of Kolmogorov's k⁻⁵/³ for hydrodynamic turbulence but for dispersive waves (deep water: E(k) ~ k⁻⁵/², NLS: k⁻⁴/³).