Nonlinear dynamics of the Frenkel-Kontorova model with impurities

The nonlinear dynamics of the Frenkel-Kontorova (FK) model with local impurities is considered analytically. Impurity modes, i.e., nonlinear oscillations localized near the impurity, are studied. We show that the low-frequency impurity mode can be regarded as a breather trapped by the impurity, and a similar approach is possible for the high-frequency mode. The stability of the nonlinear modes is investigated, and laws of their decay caused by nonintegrability of the system are determined. Considering a single kink in the FK chain, we derive the effective equation of motion for its collective coordinate, which takes into account inhomogeneities and the discreteness of the model. The adiabatic interaction of the kink with an impurity is analyzed. We show that the chain's discreteness plays an important role in the kink scattering. In particular, the reflection of the kink by a heavy-mass impurity is stipulated by the Peierls-Nabarro (PN) potential arising from the lattice discreteness, so that the reflection does not occur in the continuum model. The threshold velocity (or the threshold impurity mass) for kink reflection is determined by the amplitude of the PN potential. We also demonstrate that local impurities modify transport properties of one-dimensional systems, and the change of the kink diffusion coefficient depends on the character of the kink interaction with a separate impurity (repulsion or attraction).