The β Fermi-Pasta-Ulam-Tsingou recurrence problem

Abstract

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 E β (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 E β 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.