Modulation instability gain and localized waves in the modified Frenkel–Kontorova model with high-order nonlinearities

Abstract

In this paper, modulation instability and nonlinear supratransmission are investigated in a one-dimensional chain of atoms using cubic–quartic nonlinearity coefficients. As a result, we establish the discrete nonlinear evolution equation by using the multi-scale scheme. To compute the modulation instability gain, the linearizing technique is employed. The effect of the higher nonlinear component on modulation instability is particularly examined. Following that, full numerical integration was performed to identify modulated wave patterns as well as the appearance of a rogue wave. Through the nonlinear supratransmission phenomenon, one end of the discrete model is driven into the forbidden bandgap. As a consequence, for driving amplitudes above the supratransmission threshold, the bright soliton and modulated wave patterns are satisfied. An important behavior is observed in the transient range of time of propagation when the bright soliton wave turns into a chaotic soliton wave. These results corroborate our analytical investigations on the modulation instability and show that the one-dimensional chain of atoms is a fruitful medium to generate long-lived modulated waves.