Recursive behavior in a diatomic FPUT lattice

Abstract

We study the diatomic FPUT lattice with cubic anharmonic potential, and analyze the recurrent behaviour of its solutions. We find that two distinct types of recurrence occur. One type is the classic FPUT recurrence; for such recurrence, we find that the relation between recurrence period and nonlinear strength is similar to that in the monatomic case. The other type, which cannot exist in the monatomic lattice, is the recurrence due to the interactions between modes in the two branches of the dispersion relation. Indeed, we prove the existence of the optical-acoustical-acoustical resonant interaction between three Fourier modes for which a recurrent behavior in the distribution of the energy is observed. In addition, we develop a reduced Fourier-space dynamical model that reproduces the same recurrent behavior. We assess the robustness of our results through numerical simulations of the diatomic Toda lattice and the diatomic granular chain; in both cases, the same recursive behavior is observed. Finally, in the continuous limit, we derive from the diatomic model a system of three coupled PDEs which are known to be integrable.