For a reduced beta-FPUT model, the wave kinetic equation is rigorously derived under the scaling beta=N^{-gamma} up to times of order N^{-epsilon} min(N, N^{5gamma/4}).
One-dimensional wave kinetic theory
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Rigorous Derivation of the Wave Kinetic Equation for $\beta$-FPUT System
For a reduced beta-FPUT model, the wave kinetic equation is rigorously derived under the scaling beta=N^{-gamma} up to times of order N^{-epsilon} min(N, N^{5gamma/4}).