Skip Navigation Links
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


Dumitru Baleanu (editor)

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


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

Download Full Text PDF



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.


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).


  1. [1]  Rodionov, V.N. (1996), An Essay of Geomechanics, NauchnyiMir, Moscow.
  2. [2]  Sadovskyi,M.A. (1986), Self-similarity of geodynamical processes, Herald of the RAS, 8, 3-12.
  3. [3]  Nazarov, V.E. and Radostin, A.V. (2015), Nonlinear Acoustic Waves in Micro-inhomogeneous Solids, John Wiley: United Kingdom.
  4. [4]  Guyer, R.A. and Johnson, P.A. (2009), Nonlinear Mesoscopic Elasticity: The Complex Behaviour of Granular Media including Rocks and Soil, John Wiley: Weinheim.
  5. [5]  Danylenko, V.A., Danevych, T.B., Makarenko, O.S., Skurativskyi, S.I., and Vladimirov, V.A. (2011), Self-organization in nonlocal non-equilibrium media, Subbotin in-t of geophysics NAS of Ukraine: Kyiv.
  6. [6]  Nikolaevskij, V.N. (1989), Dynamics of viscoelastic media with internal oscillators, Lecture Notes in Engineering 210, 39, Springer: Berlin.
  7. [7]  Slepjan, L.I. (1967),Wave of deformation in a rod with flexible mounted masses, Mechanics of Solids, 5, 34-40.
  8. [8]  Palmov, V.A. (1969), On a model of medium of complex structure, Journal of Applied Mathematics and Mechanics, 4, 768-773.
  9. [9]  Danylenko, V.A. and Skurativskyi, S.I (2008), Resonance regimes of the spreading of nonlinear wave fields in media with oscillating inclusions, Reports of NAS of Ukraine, 11, 108-112.
  10. [10]  Danylenko, V.A. and Skurativskyi, S.I. (2012), Travelling wave solutions of nonlocal models for media with oscillating inclusions, Nonlinear Dynamics and Systems Theory, 4(12), 365-374.
  11. [11]  Skurativskyi, S.I. (2014), Chaotic wave solutions in a nonlocal model for media with vibrating inclusions, Journal of Mathematical Sciences, 198(1), 54-61.
  12. [12]  Danylenko, V.A. and Skurativskyi, S.I. (2015), On the dynamics of solitary wave solutions supported by the model of mutually penetrating continua, Dynamical systems. Mechatronics and life sciences, 2, 453-460. (arXiv:1512.05226v1 [nlin.PS] 15 Dec 2015)
  13. [13]  Samsonov, A.M. (2001), Strain solitons in solids and how to construct them, Chapman and Hall CRC: New York.
  14. [14]  Nayfeh, A.H. and Mook, D.T. (1995), Nonlinear oscillations, JohnWiley: New York.
  15. [15]  Palmov, V. A. (1998), Vibrations of Elasto -Plastic Bodies, Springer-Verlag: Berlin.
  16. [16]  Nikolaevskiy, V.N. (1989), Mechanism and dominant frequencies of vibrational enhancement of yield of oil pools, USSR Acad. Sci., Earth Science Sections, 307, 570-575.
  17. [17]  Beresnev, I.A. and Nikolaevskiy, V.N. (1993), A model for nonlinear seismic waves in a medium with instability, Physica, 66D, 1-6.
  18. [18]  Nigmatulin, R.I. (1991), Dynamics of Multiphase Media, (Volume 2), Hemisphere Publishing Corporation: New York.
  19. [19]  Gamburtseva, N.G., Nikolaev, A.V., Khavroshkin, O.B. and Tsyplakov, V.V. (1986), Solitonic properties of teleseismic waves, Transactions of Acad. of Sc. of USSR, 291(4), 814-816.
  20. [20]  Danylenko,V.A., Skurativskyi, S.I. and Skurativska, I.A. (2014), Asymptotic wave solutions for the model of a medium with Van Der Pol oscillators, Ukrainian Journal of Physics, 59(9), 932-938.
  21. [21]  Skuratovskii, S.I. and Skuratovskaya, I.A. (2010), Localized autowave solutions of the nonlinear model of complex medium, Electronic Journal “Technical Acoustics”, 6, http:
  22. [22]  Hyman, J. and Rosenau, Ph. (1993), Compactons: solitons with finite wavelength, Phys. Rev. Lett., 70, 564.
  23. [23]  Vladimirov, V.A. and Skurativskyi, S.I. (2015), Solitary waves in one-dimensional pre-stressed lattice and its continual analog, Dynamical systems. Mechatronics and life sciences, 2, 531-542. (arXiv:1512.06125v1 [nlin.PS] 18 Dec 2015)
  24. [24]  Danylenko, V.A. and Skurativskyi, S.I. (2016), Peculiarities of wave dynamics in media with oscillating inclusions, International Journal of Non-Linear Mechanics, 84, 31-38.
  25. [25]  Porubov, A.V. and Maugin, G.A. (2009), Cubic non-linearity and longitudinal surface solitary waves, International Journal of Non-Linear Mechanics, 44, 552-559.
  26. [26]  Potapov, A.I. (2005), Strain waves in a medium with internal structure, Eds. Gaponov-Grekhov A.V., Nekorkin V.I., Nonlinear Waves’2004, Institute of Applied Physics RAS: Nizhny Novgorod.
  27. [27]  Payan, C., Garnier, V., andMoysan, J. (2009), Determination of third order elastic constants in a complex solid applying coda wave interferometry, Applied Phys. Lett., 94, 011904.
  28. [28]  Vladimirov, V., Maczka, C., Sergyeyev, A., and Skurativskyi, S.I (2014), Stability and dynamical features of solitary wave solutions for a hydrodynamic type system taking into account nonlocal effects, Commun. in Nonlinear Science and Num. Simul., 19(6), 1770-1782.