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Exact traveling wave solutions of the van der Waals normal form for fluidized granular matter

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  • Abourabia, A.M.
  • Morad, A.M.

Abstract

Analytical solutions of the van der Waals normal form for fluidized granular media have been done to study the phase separation phenomenon by using two different exact methods. The Painlevé analysis is discussed to illustrate the integrability of the model equation. An auto-Bäcklund transformation is presented via the truncated expansion and symbolic computation. The results show that the exact solutions of the model introduce solitary waves of different types. The solutions of the hydrodynamic model and the van der Waals equation exhibit a behavior similar to the one observed in molecular dynamic simulations such that two pairs of shock and rarefaction waves appear and move away, giving rise to the bubbles. The dispersion properties and the relation between group and phase velocities of the model equation are studied using the plane wave assumption. The diagrams are drawn to illustrate the physical properties of the exact solutions, and indicate their stability and bifurcation.

Suggested Citation

  • Abourabia, A.M. & Morad, A.M., 2015. "Exact traveling wave solutions of the van der Waals normal form for fluidized granular matter," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 437(C), pages 333-350.
    • Handle: RePEc:eee:phsmap:v:437:y:2015:i:c:p:333-350
    DOI: 10.1016/j.physa.2015.06.005
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    References listed on IDEAS

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    1. He, Ji-Huan & Wu, Xu-Hong, 2006. "Exp-function method for nonlinear wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 700-708.
    2. Abourabia, A.M. & El-Danaf, T.S. & Morad, A.M., 2009. "Exact solutions of the hierarchical Korteweg–de Vries equation of microstructured granular materials," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 716-726.
    3. Garzó, Vicente & Santos, Andrés & Montanero, José María, 2007. "Modified Sonine approximation for the Navier–Stokes transport coefficients of a granular gas," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 94-107.
    4. Clerc, M.G. & Escaff, D., 2006. "Solitary waves in van der Waals-like transition in fluidized granular matter," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 371(1), pages 33-36.
    5. Abourabia, Aly M. & Hassan, Kawsar M. & Morad, Adel M., 2009. "Analytical solutions of the magma equations for molten rocks in a granular matrix," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1170-1180.
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