Nonlinear dynamics of the Frenkel-Kontorova model

An overview of the dynamics of one of the fundamental models of low-dimensional nonlinear physics, the Frenkel-Kontorova (FK) model, is presented. In its simplest form, the FK model describes the motion of a chain of interacting particles ("atoms") subjected to an external on-site periodic potential. Physically important generalizations of the FK model are discussed including nonsinusoidal on-site potentials and anharmonic (e.g., nonconvex, Kac-Baker, power-law) interactions between the particles. The results are summarized for the one-dimensional dynamics of kinks - topological excitations, including the kink diffusion and effects of disorder, and also for nonlinear localized modes, discrete breathers. A special attention is paid to the numerous applications of the FK model in the problems of low-dimensional solid state physics.