Breather resonant phase locking by an external perturbation

We study the effect of the resonant phase locking in the problem of the sine-Gordon equation breather under the action of a small oscillating external force with slowly varying frequency. We obtain equations determining the time evolution of the parameters of the perturbed breather. We describe the regular asymptotic procedure of averaging such equations and show that the averaged equations in the leading order already well describe the phenomenon of resonant phase locking in which the breather oscillations are strongly excited. We obtain necessary and sufficient conditions for the phase locking relating the rate of the perturbation frequency variation and its amplitude to the initial data of the breather.