Frenkel–Kontorova model

The Frenkel–Kontorova model, also known as the FK model, is a fundamental model of low-dimensional nonlinear physics. H = m a 2 ∑ n ( d x n d t ) 2 + U {displaystyle {cal {H}}={frac {m_{a}}{2}}sum _{n}{igg (}{frac {dx_{n}}{dt}}{igg )}^{2}+U}     (1) 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 which 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 equation or 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. A simple model of a harmonic chain in a periodic substrate potential was proposed by Ulrich Dehlinger in 1928. Dehlinger derived an approximate analytical expression for the stable solutions of this model which he termed Verhakungen which correspond to what is today called kink pairs. An essentially similar model was developed by Ludwig Prandtl in 1912/13 but did not see publication until 1928. The model was independently proposed by Yakov Frenkel and Tatiana Kontorova in their 1938 paper On the theory of plastic deformation and twinning to describe the dynamics of a crystal lattice near a dislocation and to describe crystal twinning. In the standard linear harmonic chain any displacement of the atoms will result in waves and the only stable configuration will be the trivial one.For the nonlinear chain of Frenkel and Kontorova there exist stable configurations beside the trivial one. For small atomic displacements the situation resembles the linear chain, however for large enough displacements it is possible to create a moving single dislocation for which an analytical solution was derived by Frenkel and Kontorova. The shape of these dislocations is defined only by the parameters of the system such as the mass and the elastic constant of the springs. Dislocations, also called solitons, are distributed non-local defects and mathematically they are a type of topological defect. The defining characteristic of solitons/dislocations is that they behave much like stable particles, they can move while maintaining their overall shape. Two solitons of equal and opposite orientation may cancel upon collision but a single soliton can not annihilate spontaneously. The generalized FK model treats a one-dimensional chain of atoms with nearest neighbor interaction in periodic on-site potential, the Hamiltonian for this system is