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.

The dynamics of an FK model with a modified law of spring dispersion Φ(x) is considered. Φ(x) has two local minima. A dynamic structural phase transition between them was observed.

In [1], the FK model (β=1) was written in the continuum approximation, after which an exact analytical solution was found for a soliton moving at speed w (Frenkel-Kontorova dislocation).

In [1] there is no answer to the question about the presence in the system of other non-dislocation solutions that are localized in space and do not decay in time.

Such solutions were found in exact numerical calculations.

Let us write the Hamiltonian as the sum of kinetic K and potential U energies, with the law of elastic dispersion of the general form:

                                                               (1)

Let us consider not only Hooke’s law, but also a function that does not allow the intersection of atoms in space (x≠o), for example, with two local minima x≈α and x=1+β:

                                                                                 (2)

Assuming discrete time - t = j h, j = 1.2 ..., where h is the time step - we have solutions to the system of Newton’s equations for the k-th CH atom of the form:

                   (3)

System of equations (3) is an algorithm for constructing dynamic solutions of the FK model.

In [2], the “Chain” program with algorithm (2)-(3) constructs dynamic solutions of the FK model.

An example of a dynamic solution with Hooke's law of elastic dispersion is considered and in it an energy excitation that does not decay in time, moving with a non-uniform speed and with an energy lower than the rest energy of the dislocation, is found.

When considering the dynamics of the FK model [2]: - α=0.5, β=0, γ=0.044, V₀=0.03, Δ=1, a phase transition from β- to α-phase was found, with a decrease in the size of the CH by almost two times.

Conclusions:

1 The dynamics of the FK model and the dynamics of its continuum approximation do not always and everywhere coincide.

2 For the original discrete model, the result of an exact solution of the string limit may turn out to be erroneous. For example, in [3] an exact expression was obtained for a statistical sum of the FK model in the continuum approximation. Based on the above, it can be argued that this solution is not applicable to the original FK model.

3 If we accept that in local field theories ∇φ - this is a gradient analogue of Hooke’s chain, then we assume that in the center of the black hole matter with a changed metric and with fields collapsing to the size ɑN is grouped.