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The model we deal with is the mathematical model for mutually penetratingcontinua one of which is the carrying medium obeying the wave equation whereasthe other one is the oscillating inclusion described by the equation foroscillators. These equations of motion are closed by the cubic constitutiveequation for the carrying medium. Studying the wave solutions we reduce thismodel to a plane dynamical system of Hamiltonian type. This allows us to derivethe relation describing the homoclinic trajectory going through the origin andobtain the solitary wave with infinite support. Moreover, there exist alimiting solitary wave with finite support, i.e. compacton. To model thesolitary waves dynamics, we construct the three level finite-differencenumerical scheme and study its stability. We are interested in the interactionof the pair of solitary waves. It turns out that the collisions of solitarywaves have non-elastic character but the shapes of waves after collisions arepreserved.

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