Bistability of a two-dimensional Klein-Gordon system as a reliable means to transmit monochromatic waves: A numerical approach

Departing from a reliable computational method to approximate solutions of dissipative, nonlinear wave equations, we study the bistability of a $(2+1)$-dimensional sine-Gordon system, spatially defined on a bounded square of the first quadrant of the Cartesian plane, and subject to Dirichlet boundary data in the form of harmonic driving on the coordinate axes, oscillating at a frequency in the forbidden band gap of the medium. It is shown numerically that, as its spatially discrete counterpart, the continuous $(2+1)$-dimensional sine-Gordon equation presents the process of nonlinear supratransmission, and that this phenomenon is independent of the discretization procedure. Moreover, our simulations show that a bistable region, where a conducting state and an insulating state may coexist, is present in this system, even in the presence of external damping. As an application, it is shown that the bistable regime may be properly employed in order to transmit certain monochromatic waves through these media.