Frenkel–Kontorova model

The Frenkel–Kontorova model, also known as the FK model, is a fundamental model of low-dimensional nonlinear physics.

The generalized FK model describes a chain of classical particles with nearest neighbor interactions and subjected to a periodic on-site substrate potential. In its original and simplest form the interactions are taken to be harmonic and the potential to be sinusoidal with a periodicity commensurate with the equilibrium distance of the particles. Different choices for the interaction and substrate potentials and inclusion of a driving force may describe a wide range of different physical situations.

Originally introduced by Yakov Frenkel and Tatiana Kontorova in 1938 to describe the structure and dynamics of a crystal lattice near a dislocation core, the FK model has become one of the standard models in condensed matter physics due to its applicability to describe many physical phenomena. Physical phenomena that can be modeled by FK model include dislocations, the dynamics of adsorbate layers on surfaces, crowdions, domain walls in magnetically ordered structures, long Josephson junctions, hydrogen-bonded chains, and DNA type chains. A modification of the FK model, the Tomlinson model, plays an important role in the field of tribology.

The equations for stationary configurations of the FK model reduce to those of the standard map or Chirikov–Taylor map of stochastic theory.

In the continuum-limit approximation the FK model reduces to the exactly integrable sine-Gordon (SG) equation, which allows for soliton solutions. For this reason the FK model is also known as the "discrete sine-Gordon" or "periodic Klein–Gordon equation".