Brazilian Journal of Physics

, Volume 49, Issue 1, pp 67–78 | Cite as

Three-Dimensional Nonlinear Extended Zakharov-Kuznetsov Dynamical Equation in a Magnetized Dusty Plasma via Acoustic Solitary Wave Solutions

  • Abdullah
  • Aly R. SeadawyEmail author
  • Jun Wang
General and Applied Physics


The propagation of nonlinear three-dimensional dust-ion-acoustic solitary waves in a magnetized two-ion-temperature dusty plasma is analyzed. Modified extended mapping method is further modified to discover dust-ion-acoustic solitary wave solutions of the nonlinear three-dimensional extended Zakharov-Kuznetsov dynamical equation. Consequently, different kinds of solitary wave solutions representing electric potential, electric and magnetic fields, and electron fluid pressure, are obtained with the help of Mathematica. The new dispersive solitary wave solutions are found in various shapes such as bright and dark solitons, periodic solitary wave solutions, and dark and bright solitary waves, that are expressed in different forms such as hyperbolic, rational, exponential, and trigonometric functions. These results demonstrate the efficiency and accuracy of the proposed method that can be applied to other nonlinear models. The results are shown graphically.


Magnetized two-ion-temperature dusty plasma Extended Zakharov-Kuznetsov equation Solitary wave solutions Electric potential Electric and magnetic field Electron fluid pressure Graphical representation 


  1. 1.
    P.K. Shukla, A.A. Mamun. Introduction to dusty plasma physics (Institute of Physics Publishing, Bristol, 2002)CrossRefGoogle Scholar
  2. 2.
    A.P. Misra, Y. Wang, Dust-acoustic solitary waves in a magnetized dusty plasma with nonthermal electrons and trapped ions. Commun. Nonlinear Sci. Numer. Simul. 22, 1360–9 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abdullah, A.R. Seadawy, J. Wang, Modified KdV-Zakharov-Kuznetsov dynamical equation in a homogeneous magnetised electron-positron-ion plasma and its dispersive solitary wave solutions. Pramana J. Phys. 91, 26 (2018)ADSCrossRefGoogle Scholar
  4. 4.
    A. Seadawy, D. Lu, Ion acoustic solitary wave solutions of three-dimensional nonlinear extended Zakharov-Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma. Results Phys. 6, 590 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    A.R. Seadawy, Exact solutions of a two-dimensional nonlinear Schrödinger equation. Appl. Math. Lett. 25, 687 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Coely et al. (eds.), Backlund and Darboux Transformations. Providence, RI: Amer. Math. Soc (2001)Google Scholar
  7. 7.
    Abdullah, A.R. Seadawy, J. Wang, Mathematical methods and solitary wave solutions of three-dimensional Zakharov-Kuznetsov-Burgers equation in dusty plasma and its applications. Result Phys. 7, 4269 (2017)ADSCrossRefGoogle Scholar
  8. 8.
    A.R. Seadawy, K. El-Rashidy, Traveling wave solutions for some coupled nonlinear evolution equations by using the direct algebraic method. Math. Comput. Model. 57, 1371 (2013)CrossRefGoogle Scholar
  9. 9.
    I. Kourakis, W.M. Moslem, U.M. Abdelsalam, R. Sabry, P.K. Shukla, Nonlinear dynamics of rotating multi-component pair plasmas and epi plasmas. Plasma Fusion Res. 4, 111 (2009)CrossRefGoogle Scholar
  10. 10.
    Seadawy A.R., Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions. J. Electromagn. Waves Appl. 31(14), 1353–1362 (2017)CrossRefGoogle Scholar
  11. 11.
    Y. Chen, Z. Yan, H. Zhang, Exact solutions for a family of variable-coefficient reaction-Duffing equations via the Backlund transformation. Theor. Math. Phys. 132, 9705 (2002)CrossRefGoogle Scholar
  12. 12.
    A.M. Wazwaz, A sine-cosine method for handling nonlinear wave equations. Math. Comput. Model. 40, 499 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M.A. Helal, A. Seadawy, M.H. Zekry, Stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation. Appl. Math. Comput. 232, 1094 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    N.M. Aslam, M.-D.S. Tauseef, W. Asif, A. Al-Said Eisa, Exp-function method for traveling wave solutions of nonlinear evolution equations. Appl. Math. Comput. 216, 47783 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    H.A. Ghany, Exact solutions for stochastic fractional Zakharov-Kuznetsov equations. Chin. J. Phys. 51, 875 (2013)MathSciNetGoogle Scholar
  16. 16.
    A.H. Khater, D.K. Callebaut, M.A. Helal, A.R. Seadawy, Variational method for the nonlinear dynamics of an elliptic magnetic stagnation line. Eur. Phys. J. D. 39, 237–245 (2006)ADSCrossRefGoogle Scholar
  17. 17.
    A. Seadawy, The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions. Optik – Int. J. Light Electron. Opt. 139, 31–43 (2017)CrossRefGoogle Scholar
  18. 18.
    A.R. Seadawy, Traveling wave solutions of the Boussinesq and generalized fifth-order KdV equations by using the direct algebraic method. Appl. Math. Sci. 6(82), 4081–4090 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    A. Seadawy, Approximation solutions of derivative nonlinear Schrödinger equation with computational applications by variational method. Eur. Phys. J. Plus. 130(9), 182 (2015)CrossRefGoogle Scholar
  20. 20.
    A. Seadawy, Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its a solitary wave solutions via mathematical methods. Eur. Phys. J. Plus. 132(12), 518 (2017)CrossRefGoogle Scholar
  21. 21.
    A.R. Seadawy, Ion acoustic solitary wave solutions of two-dimensional nonlinear KadomtsevPetviashviliBurgers equation in quantum plasma. Math. Methods Appl. Sci. 40(5), 1598–1607 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    A. Asghar, A.R. Seadawy, L. Dianchen, New solitary wave solutions of some nonlinear models and their applications. Adv. Diff. Equa. 1, 232 (2018)Google Scholar
  23. 23.
    M.A. Helal, A.R. Seadawy, Variational method for the derivative nonlinear Schrödinger equation with computational applications. Phys. Scr. 80, 350–360 (2009)CrossRefzbMATHGoogle Scholar
  24. 24.
    A.H. Khater, D.K. Callebaut, A.R. Seadawy, General soliton solutions of an n-dimensional complex Ginzburg-Landau equation. Phys. Scr. 62, 353–357 (2000)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    M. Arshad, A. Seadawy, L. Dianchen, Exact bright–dark solitary wave solutions of the higher-order cubic–quintic nonlinear Schrödinger equation and its stability. Optik. 138, 409 (2017)CrossRefGoogle Scholar
  26. 26.
    Abdullah, A. Seadawy, J. Wang, New mathematical model of vertical transmission and cure of vector-borne diseases and its numerical simulation. Adv. Diff. Equa. 66, 1–15 (2018)MathSciNetGoogle Scholar
  27. 27.
    A. Asghar, A.R. Seadawy, L. Dianchen, Soliton solutions of the nonlinear Schrödinger equation with the dual power law nonlinearity and resonant nonlinear Schrödinger equation and their modulation instability. Optik. 7988, 79–88 (2017)Google Scholar
  28. 28.
    L. Dianchen, A.R. Seadawy, M. Arshad, W. Jun, New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified KdV-Zakharov-Kuznetsov equations and their applications. Results Phys. 7, 899–909 (2017)ADSCrossRefGoogle Scholar
  29. 29.
    M. Arshad, A. Seadawy, L. Dianchen, W. Jun, Travelling wave solutions of generalized coupled Zakharov-Kuznetsov and dispersive long wave equations. Results Phys. 6, 1136–1145 (2016)ADSCrossRefGoogle Scholar
  30. 30.
    N. Ali, G. Zaman, A.A.M Abdullah, Alshomrani A.S., The effects of time lag and cure rate on the global dynamics of HIV-1 model. BioMed Res. Int. Article ID 8094947 (2017)Google Scholar
  31. 31.
    Z.M. Liu, W.S. Duan, G.J. He, Effects of dust size distribution on dust acoustic waves in magnetized two-ion-temperature dusty plasmas. Phys. Plasmas. 15, 083702 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    A.R. Seadawy, Stability analysis for Zakharov–Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl. 67, 17280 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    A.R. Seadawy, Stability analysis for two-dimensional ion-acoustic waves in quantum plasmas. Phys. Plasmas. 21, 052107 (2014)ADSCrossRefGoogle Scholar
  34. 34.
    Z. Hui-Ling, T. Bo, W. Yu-Feng, S. Wen-Rong, L. Li-Cai, Soliton solutions and chaotic motion of the extended Zakharov–Kuznetsov equations in a magnetized two-ion-temperature dusty plasma. Phys. Plasmas. 21, 073709 (2014)ADSCrossRefGoogle Scholar
  35. 35.
    A.R. Seadawy, Three-dimensional weakly nonlinear shallow water waves regime and its travelling wave solutions. Int. J. Comput. Methods 15(1) (2018)Google Scholar
  36. 36.
    E. Yomba, A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equation. Phys. Lett. A. 372, 1048–106 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    H.C. Ma, Y.D. Yu, D.-J. Ge, New exact traveling wave solutions for the modified form of Degasperis–Procesi equation. Appl. Math. Comp. 203, 792–798 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Física 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceJiangsu UniversityZhenjiangPeople’s Republic of China
  2. 2.Mathematics Department, Faculty of ScienceTaibah UniversityAl-MadinahSaudi Arabia
  3. 3.Mathematics Department, Faculty of ScienceBeni-Suef UniversityBeni-SuefEgypt

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