Nonlinear response of the sine-Gordon breather to an a.c. driver

We investigate the nonlinear response of the continuum sine-Gordon (SG) breather to an a.c. driver. We use an ansatz by Matsuda which is an exact collective variable (CV) solution for the unperturbed SG breather and uses only a single CV, r(t), which is the separation between the center of masses of the kink and antikink that make up the breather. We show that in the presence of a driver with an amplitude below the breakup threshold of the breather into kink and antikink, the a.c.-driven SG is quite accurately described by the r(t), which is a solution of an ordinary differential equation for a one-dimensional point particle in a potential V(r) driven by an a.c. driver and with an r-dependent mass, M(r). That is below the threshold for breakup, the solution for a driven r(t) and the use of the Matsuda identity gives a solution for the a.c.-driven SG, which is very close to the exact simulation of the a.c.-driven SG. We use a wavelet transform to analyze the frequency dependence of the time-dependent nonlinear response of the SG breather to the a.c. driver. We find the wavelet transforms of the CV solution and of the simulation of the a.c.-driven SG are qualitatively very similar to each other and often agree quite well quantitatively. In cases of breakup of the breather into K and A, where there is no appreciable radiation of phonons, we find the CV solution is very close to the exact simulation result.