Critical curves for the soliton-antisoliton annihilation on the infinitely long Josephson junction under the perturbation of bias current, shunt loss, and surface-impedance loss are 'obtained by means of the perturbation theory presented by McLaughlin and Scott. It is demonstrated that the presence of the surface-impedance loss severely reduces the stability of antipolar Auxons, especially when the shunt loss and bias current are small. The results are compared with experiment, and good agreement is found in most cases.