Resonant kink-impurity interactions in theφ4model

We study kink-impurity interactions in the ${\mathrm{\phi }}^4$ model, extending our previous results [Zhang Fei, Y. Kivshar, and L. Vázquez, Phys. Rev. A 45, 6019 (1992)] to the case where the kink has an internal degree of freedom which may be excited due to scattering. We demonstrate numerically that the kink may be reflected by an attractive impurity when its velocity lies in some ‘‘windows,’’ and this effect is due to a resonant energy exchange between the kink translational mode, its internal mode, and the impurity mode. We observe additional features in the resonant interactions, e.g., quasiresonances at some intermediate velocities and ‘‘three-bounce’’ resonances. We also develop an analytical approach taking into account three dynamical variables and show that such a collective-coordinate model may explain qualitatively the resonance structures observed in numerical simulations.