AbstractPerturbed sine‐Gordon equations are investigated numerically to see if they have breather solutions. It is shown that breathers radiate, blow up, and split into kink‐antikink pairs under most perturbations. The two perturbations proven by Birnir, McKean, and Weinstein not to produce radiation, of the first order in the perturbation parameter, a sin(u) + b ucos(u) and 1+3cos(u) ‐ 4cos(u/2) + 4cos(u)log(cos(u/4)), stop radiating first‐order radiation after adjusting the initial breather by the emission of such radiation. The first perturbation is a scaling of the breather, the second is shown to give a quasi‐periodic orbit, which is a two‐breather, on a torus. © 1994 John Wil