Length-scale competition in the damped sine-Gordon chain with spatiotemporal periodic driving

It is shown that there are two different regimes for the damped sine-Gordon chain driven by the spatiotemporal periodic force sin(t-knx) with a flat initial condition. For >kn, the system first bifurcates at a critical c(n) to a translating two-breather excitation from a state locked to the driver. For <kn, the excitations of the system are the locked states with the phase velocity /kn in all the regions of studied. In the first regime, the frequency of the breathers is controlled by, and the velocity of the breathers, controlled by kn, is shown to be the group velocity determined from the linear dispersion relation for the sine-Gordon equation. A linear stability analysis reveals that, in addition to two competing length scales, namely, the width of the breathers and the spatial period of the driving, there is one more length scale which plays an important role in controlling the dynamics of the system at small driving. In the second regime the length scale kn controls the excitation. The above picture is further corroborated by numerical nonlinear spectral analysis. An energy-balance estimate is also presented and shown to predict the critical value of in good agreement with the numerical simulations.