The β Fermi-Pasta-Ulam-Tsingou recurrence problem

We perform a thorough investigation of the first Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in the β-FPUT chain for both positive and negative β. We show numerically that the rescaled FPUT recurrence time T_r=t_r/(N+1)³ depends, for large N, only on the parameter S≡Eβ(N+1). Our numerics also reveal that for small |S|, T_r is linear in S with positive slope for both positive and negative β. For large |S|, T_r is proportional to |S|^-1/2 for both positive and negative β but with different multiplicative constants. We numerically study the continuum limit and find that the recurrence time closely follows the |S|^-1/2 scaling and can be interpreted in terms of solitons, as in the case of the KdV equation for the α chain. The difference in the multiplicative factors between positive and negative β arises from soliton-kink interactions that exist only in the negative β case. We complement our numerical results with analytical considerations in the nearly linear regime (small |S|) and in the highly nonlinear regime (large |S|). For the former, we extend previous results using a shifted-frequency perturbation theory and find a closed form for T_r that depends only on S. In the latter regime, we show that T_r∝|S|^-1/2 is predicted by the soliton theory in the continuum limit. We then investigate the existence of the FPUT recurrences and show that their disappearance surprisingly depends only on Eβ for large N, not S. Finally, we end by discussing the striking differences in the amount of energy mixing between positive and negative β and offer some remarks on the thermodynamic limit.