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The scattering of kinks and low-frequency breathers of the nonlinearsine-Gordon (SG) equation on a spatially localized $\mathcal{PT}$-symmetricperturbation (defect) with a balanced gain and loss is investigatednumerically. It is demonstrated that if a kink passes the defect, it alwaysrestores its initial momentum and energy and the only effect of the interactionwith the defect is a phase shift of the kink. A kink approaching the defectfrom the gain side always passes, while in the opposite case it must havesufficiently large initial momentum to pass through the defect instead of beingtrapped in the loss region. The kink phase shift and critical velocity arecalculated with the use of the collective variable method. Kink-kink(kink-antikink) collisions at the defect are also briefly considered, showinghow their pairwise repulsive (respectively, attractive) interaction can modifythe collisional outcome of a single kink within the pair with the defect. Forthe breather, the result of its interaction with the defect strongly depends onthe breather parameters (velocity, frequency and initial phase) and on thedefect parameters. The breather can gain some energy from the defect and as aresult potentially even split into a kink-antikink pair or it can lose a partof its energy. Interestingly, the breather translational mode is very weaklyaffected by the dissipative perturbation, so that a breather penetrates moreeasily through the defect when it comes from the lossy side, than a kink. Inall studied soliton-defect interactions the energy loss to radiation ofsmall-amplitude extended waves is negligible.

Interaction of sine-Gordon kinks and breathers with a $\mathcal{PT}$-symmetric defect