Skip to main content
SpringerLink
Log in

Solutions of Zakharov-Kuznetsov equation with power law nonlinearity in (1+3) dimensions

  • Solitons and Chaos
  • Published:
Physics of Wave Phenomena Aims and scope Submit manuscript

Abstract

This paper studies the Zakharov-Kuznetsov equation in (1+3) dimensions with an arbitrary power law nonlinearity. The method of Lie symmetry analysis is used to carry out the integration of the Zakharov-Kuznetsov equation. The solutions obtained are cnoidal waves, periodic solutions, singular periodic solutions, and solitary wave solutions. Subsequently, the extended tanh-function method and the G′/G method are used to integrate the Zakharov-Kuznetsov equation. Finally, the nontopological soliton solution is obtained by the aid of ansatz method. There are numerical simulations throughout the paper to support the analytical development.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Biswas, “1-Soliton Solution of the Generalized Zakharov-Kuznetsov Equation with Nonlinear Dispersion and Time-Dependent Coefficients,” Phys. Lett. A. 373(33), 2931 (2009).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. A. Biswas and E. Zerrad, “1-Soliton Solution of the Zakharov-Kuznetsov Equation with Dual-Power Law Nonlinearity,” Commun. Nonlin. Sci. Num. Simul. 14(9–10), 3574 (2009).

    Article  MATH  Google Scholar 

  3. A. Biswas and E. Zerrad, “Solitary Wave Solution of the Zakharov-Kuznetsov Equation in Plasmas with Power Law Nonlinearity,” Nonlin. Analys. RealWorld Appl. 11(4), 3272 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Deng, “New Exact Solutions to the Zakharov-Kuznetsov Equation and Its Generalized Form,” Commun. Nonlin. Sci. Num. Simul. 15(4), 857 (2010).

    Article  MATH  Google Scholar 

  5. D. D. Ganji, E. M. Sadeghi, and M. G. Rahmat, “Modified Camassa-Holm and Degasperis-Proscesi Equations Solved by Adomian Decomposition Method and Comparison with HPM and Exact Solutions,” Acta Appl.Math. 104(3), 303 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  6. S. S. Ganji, Z. Z. Ganji, D. D. Ganji, and S. Karimpour, “Periodic Solution for Strongly Nonlinear Vibration System by He’s Energy Balance Method,” Acta Appl.Math. 106(1), 79 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Hirota, The Direct Method in Soliton Theory (Cambridge Univ. Press, Cambridge, 2004).

    Book  MATH  Google Scholar 

  8. C. M. Khalique and A. Biswas, “A Lie Symmetry Approach to Nonlinear Schrödinger’s Equation with Non-Kerr Law Nonlinearity,” Commun. Nonlin. Sci. Num. Simul. 14(12), 4033 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Kruskal and M. Schwarzchild, “Some Instabilities of a Completely Ionized Plasma,” Proc. R. Soc. London. Ser. A. 223, 348 (1954).

    Article  ADS  MATH  Google Scholar 

  10. N. Kudryashov, “Seven Common Errors in Finding Exact Solutions of Nonlinear Differential Equations,” Commun. Nonlin. Sci. Dif. Equat. 14(9–10), 3507 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  11. P. D. Lax and C. D. Levermore, “The Small Dispersion Limit of Korteweg-deVries Equation III,” in Selected Papers (Springer Verlag, N.Y., 2005). Vol. I. Pt. IV, pp. 524–544.

    Chapter  Google Scholar 

  12. C. Ling and Xiu-lian Zhang, “The Formally Variable Separation Approach for the Modified Zakharov-Kuznetsov Equation,” Commun. Nonlin. Sci. Num. Simul. 12(5), 636 (2007).

    Article  MathSciNet  Google Scholar 

  13. W. M. Moslem, S. Ali, P. K. Shukla, X. Y. Tang, and G. Rowlands, “Solitary, Explosive and Periodic Solutions of the Quantum Zakharov-Kuznetsov Equation and Its Transverse Instability,” Phys. Plasma. 14, 082308 (2007).

    Article  ADS  Google Scholar 

  14. K. Murawski and P. M. Edwin, “The Zakharov-Kuznetsov Equation for Nonlinear Ion-Acoustic Waves,” J. Plasma Phys. 47(1), 75 (1992).

    Article  ADS  Google Scholar 

  15. S. L. Musher, A. M. Rubenchik, and V. E. Zakharov, “Weak Langmuir Turbulence,” Phys. Rep. 252, 177 (1995).

    Article  ADS  Google Scholar 

  16. M. Toda, Nonlinear Waves and Solitons (Springer Verlag, N.Y., 1989).

    MATH  Google Scholar 

  17. A. M. Wazwaz, “Nonlinear Dispersive Special Type of the Zakharov-Kuznetsov Equation ZK(n, n) with Compact and Noncompact Structures,” Appl. Math. Comp. 161(2), 577 (2005).

    Article  MATH  Google Scholar 

  18. A. M. Wazwaz, “The Extended Tanh Method for Zakharov-Kuznetsov (ZK) Equation, the Modified ZK Equation and Its Generalized Forms,” Commun. Nonlin. Sci. Num. Simul. 13(6), 1039 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. M. Wazwaz, “Exact Solutions with Solitons and Periodic Structures for the Zakharov-Kuznetsov (ZK) Equation and Its Modified Form,” Commun. Nonlin. Sci. Num. Simul. 10(6), 597 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  20. V. E. Zakharov and E. A. Kuznetsov, “On Three-Dimensional Solitons,” Sov. Phys.-JETP. 39, 285 (1974).

    ADS  Google Scholar 

  21. Li-Ping Zhang and Ju-Kui Xue, “A Nonlinear Zakharov-Kuznetsov Equation in Dusty Plasmas with Dust Charge Fluctuation and Dust Size Distribution,” Physica Scripta. 76(3), 238 (2007).

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. International Institute for Symmetry Analysis and Mathematical Modeling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho, 2735, Republic of South Africa

    B. T. Matebese, A. R. Adem, C. M. Khalique & A. Biswas

  2. Department of Mathematical Sciences, Center for Research and Education in Optical Sciences and Applications, Delaware State University, Dover, DE, 19901-2277, USA

    A. Biswas

Authors
  1. B. T. Matebese
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. R. Adem
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. M. Khalique
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Biswas.

About this article

Cite this article

Matebese, B.T., Adem, A.R., Khalique, C.M. et al. Solutions of Zakharov-Kuznetsov equation with power law nonlinearity in (1+3) dimensions. Phys. Wave Phen. 19, 148–154 (2011). https://doi.org/10.3103/S1541308X11020117

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1541308X11020117

Keywords

Search

Navigation