We consider a generalized Frenkel-Kontorova model, describing the dynamics of a chain of particles in a periodic substrate potential, and analyze the effect of discreteness on the existence and properties of internal (or shape) modes of kinks, topological excitations of the chain. In particular, we show that kink’s internal modes can appear not only below but also above the phonon spectrum band and, in the latter case, the localized mode describes out-of-phase oscillations of the kink’s shape. For the sinusoidal on-site potential, when the model is described by the discrete sine-Gordon equation, we reveal, in sharp contrast with the continuum limit, the existence of the kink’s internal mode in a narrow region of the discreteness parameter. We apply two different analytical techniques to describe the cases of weak and strong coupling between particles, explaining qualitatively and even quantitatively the main features of the kink oscillations observed in numerical simulations. We also discuss the effect of nonlinearity on the existence and properties of kink’s internal modes and show, in particular, that a nonlinearity-induced frequency shift of the lattice vibrations can lead to the creation of the nonlinear kink’s internal modes, which, however, slowly decay due to a generation of radiation through higher-order harmonics.

• Received 29 April 1997

DOI:https://doi.org/10.1103/PhysRevE.56.6050