Energy supply into a semi-infinite $\beta$-Fermi-Pasta-Ulam-Tsingou chain by periodic force loading

TL;DR

This paper investigates how energy is supplied and evolves in a semi-infinite $eta$-Fermi-Pasta-Ulam-Tsingou chain under periodic forcing, analyzing both harmonic and weakly anharmonic cases with theoretical and numerical methods.

Contribution

It provides a new asymptotic approximation for energy transfer in the chain, accounting for both pass-band and stop-band frequencies, and explains energy growth behavior over time.

Findings

Energy grows linearly at non-zero group velocities in the harmonic case.

Energy grows as $\sqrt{t}$ at zero group velocity.

Asymptotic approximation aligns with numerical simulations.

Abstract

We deal with dynamics of the~$\beta$-Fermi-Pasta-Ulam-Tsingou chain with one free end, subjected to the sinusoidal periodic force. We examine evolution of the total energy, supplied at large times. In the harmonic case~($\beta=0$), the energy grows in time linearly at non-zero group velocities, corresponding to the excitation frequency and grows in time as~$\sqrt{t}$ at zero group velocity. Explanation of behavior in time of the energy is proposed by analysis of obtained approximate closed-form expression for the field of particle velocities. In the weak anharmonic case, large-time asymptotic approximation for the total energy is obtained by using the renormalized dispersion relation. Operating of the approximation, we analyze energy transfer at the driving frequencies, lying both in the pass-band and in the stop-band of the harmonic chain. Consistency of the asymptotic assesses with the results of numerical simulations is discussed.