From the fact that the stretching of the elastically periodic Hooke chain (CH) is symmetrical relative to its center (odd symmetry), it follows that the coordinate of the center of gravity CH remains unchanged. This makes it possible to accurately solve the system of nonlinear equations for the stationary states of the FK model and select the ground state from them.

In 1938, the authors: Yakov Ilyich Frenkel and Tatyana Abramovna Kontorova (LFTI) put forward the Frenkel-Kontorova model (FK model) [1], which served as the basis for the creation of many theories of highly nonlinear processes [2].

The FK model is an elastically periodic chain of atoms (CH) in a periodic potential.

CH - is a one-dimensional sequence of N point atoms of masses m with coordinates {x_i} and period β, interconnected by elastic springs with the law of elastic dispersion Φ(x). Φ(x) - most often this is Hooke's law, .

The periodic potential V(x) has even symmetry and period a=1.

In most previous works [1,2], it was assumed N = ∞ that solutions would be simplified by eliminating boundary effects from consideration. As it turned out [3,4], this assumption is wrong - in the discrete FK model there is no small parameter 1/N, so it is necessary to find exact solutions taking into account the position of its boundary atoms.

The potential energy of the FK system has the form:

                                         (1)

where N is the number of atoms in the chain, N=2K+1; x_i is the distance of the i-th atom to the center of the chain.

In [1]

If β=0, then in the ground state of the FK model the point CH is at the minimum of the potential V(x=0). When β≠0 the position of the central atom does not change x₀=0. The positions of the remaining atoms relative to the center are not even and are determined from a system of nonlinear equilibrium equations.

  - system of N equilibrium equations. Taking into account x_-i = - x_i, we have:

                                                                                  (2)

At x₀=0, all coordinates x_i and β are functions of x₁, x₁ ∈ [0, 1], β ∈ [0, 1], therefore, solutions for the ground state of the FK model can be obtained by minimizing U(x₁) with respect to x₁ [3,4]. Comparing the exact solutions [3,4] with their continuum approximations [5], we found that the properties of the ground state of the discrete FK model coincide with the properties of its continuum approximation only in the homogeneity region.