Modulational Instability and Localized Waves in the Monoatomic Chain with Anharmonic Potential

In this work, we use a monoatomic chain subject to an anharmonic potential with nearest neighbor couplings to derive the discrete nonlinear evolution equation. Thereafter, the modulational instability growth expression is derived by using the linear stability analysis. Our work displays the influence of the interaction potential and nearest neighbor couplings on the modulational instability and bandwidths. As a result, we show that the number of modulational unstable modes increases, while the number of modulational stable modes decreases when certain specific values are taken by the interaction potential. The same observation has been highlighted when the nearest neighbor couplings get high values. A numerical simulation of the continuous wave has been realized to confirm the fact that the modulational instability is sensitive to the nonlinear parameters due to the development of the modulated wave structures and trains of waves. Another particular result of this work has been displayed by driving the left and right ends of the chain into the lower and upper forbidden gaps. Supratransmission thresholds have been derived to get the corresponding driving amplitudes. It emerges that for the driving amplitude above the supratransmission threshold, the modulated waves occur in the lower forbidden frequency gap. At the same time, it results in a chaos-like motion in the propagation range of time in the upper forbidden gap. Such observations confirm the fact that in nonlinear structures, when the driving amplitude is considered above the supratransmission threshold, localized waves (gap solitons) are generated.