The Effect of On-Site Potentials on Supratransmission in One-Dimensional Hamiltonian Lattices

We investigated a class of one-dimensional (1D) Hamiltonian N-particle lattices whose binary interactions are quadratic and/or quartic in the potential. We also included on-site potential terms, frequently considered in connection with localization phenomena, in this class. Applying a sinusoidal perturbation at one end of the lattice and an absorbing boundary on the other, we studied the phenomenon of supratransmission and its dependence on two ranges of interactions, $0<\alpha <\infty$ and $0<\beta <\infty$, as the effect of the on-site potential terms of the Hamiltonian varied. In previous works, we studied the critical amplitude $A_s(\alpha ,\mathsf{\Omega })$ at which supratransmission occurs, for one range parameter $\alpha$, and showed that there was a sharp threshold above which energy was transmitted in the form of large-amplitude nonlinear modes, as long as the driving frequency $\mathsf{\Omega }$ lay in the forbidden band-gap of the system. In the absence of on-site potentials, it is known that $A_s(\alpha ,\mathsf{\Omega })$ increases monotonically the longer the range of interactions is (i.e., as $\alpha ⟶0$). However, when on-site potential terms are taken into account, $A_s(\alpha ,\mathsf{\Omega })$ reaches a maximum at a low value of $\alpha$ that depends on $\mathsf{\Omega }$, below which supratransmission thresholds decrease sharply to lower values. In this work, we studied this phenomenon further, as the contribution of the on-site potential terms varied, and we explored in detail their effect on the supratransmission thresholds.