Numerical study of the process of nonlinear supratransmission in Riesz space-fractional sine-Gordon equations

In this work, we consider a (1+1)-dimensional Riesz space-fractional damped sine-Gordon equation defined on a bounded spatial interval. Sinusoidal Dirichlet boundary data are imposed at one end of the interval and homogeneous Neumann conditions at the other. The system is initially at rest in the equilibrium position, and is discretized to simulate its complex dynamics. The method employed in this work is a finite-difference discretization of the mathematical model of interest. Our scheme is throughly validated against simulations on the dynamics of the classical and the space-fractional sine-Gordon equations, which are available in the literature. As the main result of this manuscript, we have found numerical evidence on the presence of the phenomenon of nonlinear supratransmission in Riesz space-fractional sine-Gordon systems. Simulations have been conducted in order to predict its occurrence for some values of the fractional order of the spatial derivative, and a wide range of values of the frequency of the sinusoidal perturbation at the boundary. As far as the author knows, this may be one of the first numerical reports on the existence of nonlinear supratransmission in sine-Gordon systems of Riesz space-fractional order.