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Exact solutions of the hierarchical Korteweg–de Vries equation of microstructured granular materials

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  • Abourabia, A.M.
  • El-Danaf, T.S.
  • Morad, A.M.

Abstract

The problems under consideration are related to wave propagation in microstructured materials, characterized by higher-order nonlinear and higher-order dispersive effects; particularly, the wave propagation in dilatant granular materials. In the present paper the model equation is solved analytically by exact methods. The types of solutions are defined and discussed over a wide range of material parameters (two dispersion parameters and one microstructure parameter). The dispersion properties and the relation between group and phase velocities of the model equation are studied. The diagrams are drawn to illustrate the physical properties of the exact solutions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:2:p:716-726
    DOI: 10.1016/j.chaos.2008.03.015
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    References listed on IDEAS

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    1. El-Danaf, Talaat S. & Ramadan, Mohamed A. & Abd Alaal, Faysal E.I., 2005. "The use of adomian decomposition method for solving the regularized long-wave equation," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 747-757.
    2. Zayed, E.M.E. & Abourabia, A.M. & Gepreel, Khaled A. & El Horbaty, M.M., 2007. "Travelling solitary wave solutions for the nonlinear coupled Korteweg–de Vries system," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 292-306.
    3. Ilison, O. & Salupere, A., 2006. "On the propagation of solitary pulses in microstructured materials," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 202-214.
    4. Soliman, A.A., 2006. "The modified extended tanh-function method for solving Burgers-type equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(2), pages 394-404.
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    Cited by:

    1. 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.
    2. 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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