  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.
