Length-scale competition in the damped sine-Gordon chain

Abstract

It is shown that there are two different regimes for the damped sine-Gordon chain driven by the spatio-temporal periodic force $Γsin(ωt - k_{n} x)$ with a flat initial condition. For $Γ_{c}(n)$ to a translating {\em 2-breather} excitation from a state locked to the driver. For $ω< k_{n}$, the excitations of the system are the locked states with the phase velocity $ω/k_{n}$ in all the region of $Γ$ studied. In the first regime, the frequency of the breathers is controlled by $ω$, and the velocity of the breathers, controlled by $k_{n}$, 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 $k_{n}$ 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 $Γ$ in good agreement with the numerics.

12 pages, REVTeX, 3 Figures (e-mail requests to A. Sanchez, anxo@ing.uc3m.es)