The Frenkel-Kontorova model is one of the most simple and rich models of classical mechanics. The nonlinearity of this model leads either to the exactly integrable sine-Gordon equation which admits topological and dynamical solitons, or to the important equation of stochastic theory, the Taylor-Chirikov map, which involves such issues as fractal structures, commensurate-incommensurate transitions, glass-like behavior, etc. The FK model describes a number of physical objects such as dislocations and crowdions in solids, domain walls, Josephson junctions, biological molecules, and crystal surfaces.

A rather detailed description of the FK model may be found in the monograph by O.M. Braun and Yu.S. Kivshar "The Frenkel-Kontorova Model: Concepts, Methods, and Applications" (Springer, 2004) or in the survey by  O.M. Braun and Yu.S. Kivshar, Physics Reports 306 (1998) 1 "Nonlinear dynamics of the Frenkel-Kontorova model"

Below I present a brief introduction into the FK model and some results of our studies on this topic.