Nonlinear wave transmission in harmonically driven hamiltonian sine-Gordon regimes with memory effects

In this work, for the first time in the literature, we provide solid computational evidence that the phenomenon of nonlinear supratransmission is present in time-fractional sine-Gordon equations with damping. The model under study considers a Caputo-type partial derivative of order α∈(1,2]with respect to the temporal variable, and it possesses an associated continuous energy density function which is dissipated throughout time. In the present work, we determine the dispersion relation of the physical model and employ a reliable numerical technique to approximate the solutions. Associated with the finite-difference scheme, we also propose a discrete energy density function. This methodology was implemented computationally to approximate the solution of the model on a closed and bounded interval, considering sinusoidal perturbations on one end, and homogeneous Neumann conditions on the other. As the most relevant physical conclusions of our experiments, we have confirmed the presence of nonlinear supratransmission in nonlinear wave equations with time-fractional derivatives. Additionally, we have found out that nonlinear supratransmission gradually ceases to be present in the system as αtends from 2 to 1. Moreover, we provide some bifurcation diagrams depicting the relation between the driving amplitude at which supratransmission is triggered versus driving frequency. To that end, we follow a systematic approach which is relatively standard nowadays. For the sake of convenience, we provide a Matlab implementation of our code as an appendix at the end of this manuscript.