On the dynamics of solitary wave solutions supported by the model of mutually penetrating continua

TL;DR

This paper analyzes a mathematical model of mutually penetrating continua, deriving solitary wave solutions including infinite and finite support waves, and studies their interactions through numerical simulations.

Contribution

It introduces a Hamiltonian reduction of the model to find solitary wave solutions and develops a numerical scheme to study wave interactions.

Findings

Existence of solitary waves with infinite support

Existence of compacton solutions with finite support

Solitary wave collisions are non-elastic but preserve wave shape

Abstract

The model we deal with is the mathematical model for mutually penetrating continua one of which is the carrying medium obeying the wave equation whereas the other one is the oscillating inclusion described by the equation for oscillators. These equations of motion are closed by the cubic constitutive equation for the carrying medium. Studying the wave solutions we reduce this model to a plane dynamical system of Hamiltonian type. This allows us to derive the relation describing the homoclinic trajectory going through the origin and obtain the solitary wave with infinite support. Moreover, there exist a limiting solitary wave with finite support, i.e. compacton. To model the solitary waves dynamics, we construct the three level finite-difference numerical scheme and study its stability. We are interested in the interaction of the pair of solitary waves. It turns out that the collisions of solitary waves have non-elastic character but the shapes of waves after collisions are preserved.