Kink dynamics in the spatially modulated ${\mathrm{\phi }}^4$ model is studied analytically with the Born approximation and a collective-coordinate approach, and numerically. Four types of kink behavior can be distinguished: (i) radiation at high velocity, (ii) strong resonant beating in the kink velocity, (iii) propagation of the kink with almost periodic velocity oscillations, and (iv) trapping at low velocity. Although some of these regimes exist also in the sine-Gordon (SG) case, the ${\mathrm{\phi }}^4$ model exhibits a much richer dynamics due to the existence of a localized internal mode of the kink, which can exchange energy with the translational mode. The propagation in a modulated substrate is very sensitive to this effect, and in this respect the ${\mathrm{\phi }}^4$ model could be more representative of real systems which may not have the peculiar properties of the integrable SG model.

• Received 11 November 1992

DOI:https://doi.org/10.1103/PhysRevE.48.548