Energy transfer into a semi-infinite $β$-Fermi-Pasta-Ulam-Tsingou chain under periodic force loading

Sergei D. Liazhkov

Published 2022-12-19Version 1

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 at large times. In the harmonic case, the total energy, transmitted into the chain, grows in time linearly at non-zero group velocities, corresponding to the excitation frequency, and grows in time as the square root of time at zero group velocity. Based on the concept of the renormalized dispersion relation, large-time asymptotic approximation for the total energy is obtained in the weakly anharmonic case. Using the approximation, we analyze energy transfer at the frequencies both in the pass-band (dispersion relation) and in the stop-band of the harmonic chain. Consistency of the asymptotic estimates with the results of numerical simulations is discussed.