Rigorous Derivation of the Wave Kinetic Equation for \beta-FPUT System

Abstract:

While the WKE has been rigorously derived for the cubic nonlinear Schrödinger equation in dimensions d\ge 2 and for the Majda–McLaughlin–Tabak model in d=1, there is still lack of rigorous justification for the \beta-FPUT model whose sinusoidal dispersion and unconserved frequency shift pose additional obstacles. In this thesis, we establish the WKE for a reduced evolution equation, removing the nonresonant terms, from the one‑dimensional \beta-FPUT chain. We work in the kinetic limit N \to \infty and \beta \to 0 under the scaling laws \beta=N^{-\gamma} with 0<\gamma<1. The result holds up to the sub‑kinetic time scale T=N^{-\epsilon}\min(N, N^{5/4\gamma})=N^{-\epsilon}T_{kin}^{5/8} for \epsilon\ll1, where T_{kin} represents the kinetic (thermalization) timescale. We also prove a sufficient upper bound for the nonlinearity parameter $\beta$ that allows one to perform the canonical transformation on the original evolution equation. This upper bound suggests a scaling between \beta and N, which governs the importance of the non-resonant terms in the original equation. By applying the symplectic integrator method, we further develop numerical studies on the \beta-FPUT model, comparing the magnitudes of resonant and nonresonant sums across various nonlinearity strengths and particle numbers to verify the predicted \beta-threshold.