Length-scale competition in the damped sine-Gordon chain

TL;DR

This paper investigates the dynamics of the damped sine-Gordon chain under spatio-temporal forcing, revealing two regimes with distinct length-scale controls and analyzing their stability, velocity, and energy balance.

Contribution

It identifies two different regimes in the driven sine-Gordon chain and elucidates the role of multiple length-scales in controlling system dynamics, supported by stability analysis and numerical spectral methods.

Findings

Two regimes with distinct length-scale controls identified.

Breather frequency is governed by driving frequency  and velocity by group velocity.

Energy balance estimates match numerical critical forcing values.

Abstract

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