Full derivation of the wave kinetic equation

We provide the rigorous derivation of wave kinetic equation from cubic nonlinear Schrödinger (NLS) at timescale, under a particular scaling law that describes limiting process. This solves main conjecture in theory turbulence, i.e. systems. Our result is analog Lanford’s theorem on Boltzmann particle systems, where both cases one takes thermodynamic limit as size system diverges to infinity, and interaction strength waves/radius particles vanishes 0, according (Boltzmann-Grad case). More precisely, dimensions $d\geq 3$ , we consider large box $L$ with weak nonlinearity $\alpha $ . In $L\to \infty \to 0$ \sim L^{-1}$ show long-time behavior statistically described by equation, well justified approximation, up times are $O(1)$ (i.e. independent ) multiples timescale $T_{\text{kin}}\sim \alpha ^{-2}$ first its kind for any dispersive system.