Computational study of the nonlinear bistability in a relativistic wave equation with anomalous diffusion

Abstract

In this work, we investigate computationally the dynamics of a nonlinear partial differential equation with anomalous diffusion that extends the well-known double sine–Gordon equation from relativistic quantum mechanics. The problem under consideration includes the presence of constant damping along with anomalous spatial derivatives. The model is defined on a close and bounded interval of the real line, and it is at rest at the initial time. One end of the interval is subject to sinusoidal driving, and the other considers the presence of an absorbing boundary in order to simulate a semi-infinite medium. The simulation of this system is carried out using a numerical method that resembles the energy properties of the continuous medium. The computational results shown in this work establish the presence of the nonlinear phenomenon of bistability in the system considered. We obtain hysteresis cycles for some particular scenarios, and employ the bistability of the system to simulate the transmission of binary signals from the driving boundary to the opposite end.