Nonlinear dynamics of a parametrically driven sine-Gordon system

We consider a sine-Gordon system, driven by an ac parametric force in the presence of loss. It is demonstrated that a breather can be maintained in a steady state at half of the external frequency. In the small-amplitude limit the effect is described by an effective nonlinear Schrödinger equation. For an arbitrary frequency of the applied field the threshold field for the breather stabilization is determined by a perturbation method and it is compared to direct numerical simulations. We also analyze the variation of the breather parameters in the stationary regime and calculate the frequency of such a modulation for a general type of driving force.