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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Dynamics of Waves in the Cubically Nonlinear Model for Mutually Penetrating Continua

Discontinuity, Nonlinearity, and Complexity 6(4) (2017) 425--433 | DOI:10.5890/DNC.2017.12.002

Vjacheslav Danylenko; Sergii Skurativskyi

Subbotin Institute of Geophysics, NAS of Ukraine, Bohdan Khmelnytskyi str. 63-G, Kyiv, Ukraine

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Abstract

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.

Acknowledgments

One of the authors (Skurativskyi S.I.) is grateful to Prof. Jan Awrejcewicz and other Organizers of the 13th International Conference “Dynamical Systems - Theory and Applications”, December 7-10, 2015, Lodz, Poland, where the main part of this report had been presented. Also he appreciates the helpful discussions with and the valuable remarks of Prof. Vladimirov V.A. (AGH, Krakow, Poland).

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