Prof.
Nikolay Kudryashov
(National Research Nuclear University MEPhI)

We consider the following dynamical system:
\begin{equation}\begin{gathered}
\label{I_1}
m\,\frac{d^2y_{i}}{d t}=F_{i+1,i}-F_{i,i-1}-f_0\,\sin{\left(\frac{2\,\pi\,y_i}{a}\right)}, \qquad
(i=1,\ldots,N),
\end{gathered}\end{equation}
where $y_i$ measures the displacement of the i-th mass from equilibrium in time $t$, the force $F_{i+1,i}$ describes the nonlinear interaction between atpms dislocations in the crystal lattice in case of dislocations
\begin{equation}\begin{gathered}
\label{I_2}
F_{i+1,i}=\gamma\,(y_{i+1}-y_i)+\alpha\,(y_{i+1}-y_i)^2+\beta\,(y_{i+1}-y_i)^3,
\end{gathered}\end{equation}
and $f_0$, $a$, $\gamma$, $\alpha$, $\beta$ are constant parameters of system \eqref{I_1}.
The system of equations \eqref{I_1} is the generalization of some well-known dynamical systems. At $\alpha=0$ and $\beta=0$ the system of equations \eqref{I_1} is the mathematical model introduced by Frenkel and Kontorova for the description of dislocations in the rigid body \cite{Frenkel}. In this model it was suggested that the influence of atoms in the crystal is taken into account by term $f_0\,\sin {\frac{2\,\pi\,y_i}{a}}$ but the atoms in case of dislocations interact by means of linear low. Assuming that $N\rightarrow \infty$ and $h\rightarrow 0$ where $h$ is the distance between atoms, we can get the Sine-Gordon equation.
In case of $f=0$ and $\beta=0$ system of equations \eqref{I_1} is the well-known Fermi-Pasta-Ulam model \cite{Fermi} which was studied many times. It is known that the Fermi-Pasta-Ulam model is transformed at $N\rightarrow \infty$ and $h\rightarrow 0$ to the Korteweg-de Vries equation \cite{Kruskal}.
The main result of work \cite{Kruskal} was the introduction of solitons as solutions of the Koryeweg-de Vries equation.
It was shown in 1967 that the Cauchy problem for this equation can be solved by the Inverse Scattering transform \cite{Gardner}.
Assuming $f_0=0$, $\alpha \neq 0$ and $\beta \neq 0$ at $N \rightarrow \infty$ and $h \rightarrow 0$ one can find the modified Korteweg-de Vries equation for the description of nonlinear waves.
In papers \cite{ Kudr15, Kudr17} the author took into account high order terms in the Taylor series for the description of nonlinear waves in the Fermi-Pasta-Ulam and the Kontorova-Frenkel models assuming that $\alpha \neq 0$ and $\beta \neq 0$ and did not obtain nonlinear integrable differential equations in mass chain. Here we assume that the interaction between dislocations in crystal is described by means of nonlinear low at $\alpha \neq 0 $ and $\beta \neq 0$ and consider the other equations. The aim of this talk is to present the nonlinear partial differential equations corresponding to dynamical system \eqref{I_1} and to discuss the properties of these equations.

### Short biography note

\bibitem[1]{Frenkel} T.A. Kontorova, Ya. I. Frenkel, On theory of plastic deformation, JETP {\bf 89} (1938) 1340, 1349 (in Russian)

\bibitem[2]{Fermi} E. Fermi, J.R. Pasta, S.Ulam, Studies of nonlinear problems, Report LA-1940, 1955. Los Alamos: Los Alamos Scientific Laboratory

\bibitem [3]{Kruskal} N.J. Zabusky, M.D. Kruskal, Interactions of solitons in a collisionless plasma and the recurrence of initial states, Physical Review Letters, {\bf 15} (1965) 240.

\bibitem [4]{Gardner} C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Method for solving the Korteweg de Vries equation. J. Math. Phys. {\bf 19} (1967) 1095

\bibitem[5]{Kudr15} N.A. Kudryashov, Refinment of the Korteweg-de Vries equation from the Fermi-Pasta-Ulam. Phys. Lett. A {\bf 379} (2015) 2610

\bibitem[6]{Kudr17} N.A. Kudryashov, Analytical properties of nonlinear dislocaton equation, Appl. Math. Lett. {\bf 69} (2017) 29