Long time stability of low--frequency packets of modes at finite total energy

Abstract

We consider the FPUT $α$-model with $N$ particles and initial data with small total energy $E$, independent of $N$. In particular, we prove that if the energy is initially given to Fourier modes of wave number $πk/N\leq θ_*$, then it does not flow to modes of wave number larger than $θ_*$ up to long times of order $E^{-(1-ε/2)}$, $θ_*$ and $ε$ being small numbers. The proof relies on the closeness of the Toda lattice to the FPUT system. Unlike previous approaches, we do not make use of the action-angle variables of the Toda system, but we exploit the Flaschka integrals of the latter. Actually, we find a suitable recombination of the Flaschka integrals whose quadratic part is used to control the packet, the higher order part fulfilling very good estimates.