The β Fermi-Pasta-Ulam-Tsingou Recurrence Problem

One of the most remarkable and longest-studied problems in nonlinear dynamics is Fermi-Pasta-Ulam-Tsingou (FPUT) recurrences. We perform a thorough investigation of the first FPUT recurrence in the β-FPUT chain for both β > 0 and β < 0. 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 large |S|, T_r is proportional to |S|^-1/2 for both β > 0 and β < 0 but with different multiplicative constants. We numerically study the continuum limit and find the recurrence time closely follows the |S|^-1/2 scaling and can be interpreted in terms of mKdV solitons, and the difference in the multiplicative factors between positive and negative β arises from soliton-kink interactions. We complement our numerical results with analytical considerations in the nearly linear regime (small |S|) and 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, which depends only on S. In the latter regime, we show that the soliton theory correctly predicts T_r ~ |S|^-1/2 in the continuum limit.