Vsevolod A. Vladimirov (AGH University of Science and Technology, Krakow, Poland)

On continual models connected with the chains of pre-stressed elastic bodies

Abstract:
We consider a one-dimensional chain of strongly pre-stressed bodies interacting wich each other by means of nonlinear forces. Passing to continual analog of such a system, we can obtain different nonlinear PDS, depending on a type of elastic force. Thus, in the case when the interaction has the form $F(z)=A z^n+B z$, $|A|=O|B|$, $n>1$, we get a Boussinesq equation, while in the case when $B=0$ Nesterenko's equation is obtained. However, if we assume that $$ F(z)=A \operatorname{sgn}[z] z^n+B z, $$ for $n=2 k,$ $|A|=O(1)$, while $|B|\ll 1$ then, using a formal multi-scaled decomposition, we get a nonlinear evolutionary PDE, describing compactons (both bright and dark ones). Next, we will show that the compacton solutions possess many interesting features. In particular, they evolve in a self-similar mode and restore their shapes after mutual collisions.