Dynamics of Waves in the Cubically Nonlinear Model for Mutually Penetrating Continua - Danylenko, Vjacheslav; Skurativskyi, Sergii (Discontinuity, Nonlinearity, and Complexity, 2026)
In this report we study the mathematical model for mutually penetrating continua. This model consists of the wave equation describing the carrying medium and the equation for oscillators forming the oscillating inclusion. Prescribing the constitutive equation of the carrying medium and kinetics of oscillator’s dynamics for model in question, the cubic nonlinearity is accounted. We are interested in the structure of wave solutions obeying the dynamical system of Hamiltonian type. This allows us to determine the peculiarities of the phase space of dynamical system, namely, the relation describing the homoclinic trajectories, the division of phase plane into the parts with equivalent orbits’ behavior, the conditions of bifurcations. To simulate the wave dynamics, we construct the three level finite-difference numerical scheme and study the evolution of solitary waves, their pair interactions and stability. Propagation of periodic waves is modeled as well.