Supratransmission of rogue wave-like breathing solitons

Supratransmission is a fascinating and counterintuitive nonlinear wave phenomenon that enables energy transmission through frequency band gaps. Recent studies have suggested that supratransmission in a damped-driven Klein-Gordon equation can lead to the formation of rogue waves. In this work, we reduce the same model to a damped and driven discrete nonlinear Schr\"odinger equation and demonstrate that the claimed rogue waves are unstable solitons exhibiting breathing dynamics. Through extensive numerical simulations, we thoroughly investigate and analyze the behavior of solitons and supratransmission in these lattices. Additionally, we perform asymptotic analysis to explain the underlying mechanism responsible for supratransmission. Our findings provide new insights into the dynamics of nonlinear waves and their stability in such systems.