Nonlinear supratransmission and bistability in the Fermi-Pasta-Ulam model

Abstract

The recently discovered phenomenon of nonlinear supratransmission consists in a sudden increase of the amplitude of a transmitted wave triggered by the excitation of nonlinear localized modes of the medium. We examine this process for the Fermi-Pasta-Ulam chain, sinusoidally driven at one edge and damped at the other. The supratransmission regime occurs for driving frequencies above the upper band-edge and originates from direct moving discrete breather creation. We derive approximate analytical estimates of the supratransmission threshold, which are in excellent agreement with numerics. When analysing the long-time behavior, we discover that, below the supratransmission threshold, a conducting stationary state coexists with the insulating one. We explain the bistable nature of the energy flux in terms of the excitation of quasi-harmonic extended waves. This leads to the analytical calculation of a lower-transmission threshold which is also in reasonable agreement with numerical experiments.