Resonant kink-impurity interactions in the sine-Gordon model

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Abstract
We present results concerning kink-impurity interactions in the sine-Gordon (SG) model. In particular, we demonstrate that the SG kink may be totally reflected by an attractive impurity if its initial velocity is in some resonance windows. This effect can be predicted by a suitable collective-coordinate method, and the resonance structures can be explained with an energy exchange between the kink translational mode and the impurity mode. We also study the scattering of the kink by an excited impurity and show that such a scattering strongly depends on the amplitude and phase of the impurity mode. In particular, resonance phenomena are also observed in the scattering. In addition, we consider the interactions of the kink with an isotopic (heavy-mass) impurity. We demonstrate that if the impurity mass is not too large in comparison with the standard mass in the SG model, the kink can pass the impurity almost freely at any initial velocity. However, if the impurity mass is large enough, a higher-velocity kink will be reflected while a lower-velocity kink will pass. We explain this effect analytically and show that the impurity mode plays an important role in the scattering. In all the cases considered we find good agreements between the collective-coordinate analysis and the direct numerical simulations.