Advertisement

Pramana

, 91:48 | Cite as

Lie point symmetries, conservation laws and exact solutions of (\(1+ n\))-dimensional modified Zakharov–Kuznetsov equation describing the waves in plasma physics

  • Muhammad Nasir Ali
  • Aly R Seadawy
  • Syed Muhammad Husnine
Article

Abstract

In this study, we explore the modified form of (\(1+n\))-dimensional Zakharov–Kuznetsov equation, which is used to investigate the waves in dusty and magnetised plasma. It is proved that the equation follows the property of nonlinear self-adjointness. Lie point symmetries are calculated and conservation laws in the framework of the new general conservation theorem of Ibragimov are obtained. The \((1/G^{\prime })\), \((G^{\prime }/G)\)-expansion and modified Kudryshov methods are applied to extract exact analytical solutions. The so-called bright, dark and singular solutions are also found using the solitary wave ansatz method. The results obtained in this study are new and may be of significant importance where this model is used to study the waves in different plasmas.

Keywords

Modified Zakharov–Kuznetsov equation formal Lagrangian nonlinear self-adjointness conservation laws modified Kudryshov method solitary wave ansatz method \((G^{\prime }/G)\)-expansion 

PACS Nos

02.30.Jr 47.10.A− 52.25.Xz 52.35.Fp 

References

  1. 1.
    A R Seadawy and K El-Rashidy, Pramana – J. Phys. 87, 20 (2016)ADSCrossRefGoogle Scholar
  2. 2.
    A R Seadawy, Pramana – J. Phys. 89(3): 49 (2017)ADSCrossRefGoogle Scholar
  3. 3.
    M Khater, A R Seadawy and D Lu, Pramana – J. Phys. 90: 59 (2018)Google Scholar
  4. 4.
    Abdullah, Aly Seadawy and Jun Wang, Pramana – J. Phys. 91: 26 (2018)Google Scholar
  5. 5.
    V E Zakharov and E A Kuznetsov, Zh. Eksp. Teor. Fiz. 66, 594 (1974)Google Scholar
  6. 6.
    M Eslami, B Fathi Vajargah and M Mirzazadeh, Ain Shams Eng. J. 5, 221 (2014)CrossRefGoogle Scholar
  7. 7.
    S T Mohyud Din, A Ayyaz, M A Iqbal, A Esen and S Kutluay, Asia Pac. J. Comput. Eng. 2(2), 1 (2015)Google Scholar
  8. 8.
    A Seadawy, Eur. Phys. J. Plus 132, 518 (2017)CrossRefGoogle Scholar
  9. 9.
    M N Alam, M G Hafez, M A Akbar and H Roshid, Alex. Eng. J. 54(3), 635 (2015)CrossRefGoogle Scholar
  10. 10.
    C T Sindi and J Manafian, Eur. Phys. J. Plus 132, 67 (2017)Google Scholar
  11. 11.
    Y L Jiang, Y Lu and C Chen, J. Nonlinear Math. Phys. 23(2), 1 (2016)Google Scholar
  12. 12.
    D M Mothibi and C M Khalique, Symmetry 7(2), 949 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    M A Abdou, Chaos Solitons Fractals 31, 95 (2007)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    A R Seadawy and S Z Alamri, Results Phys. 8, 286 (2018)ADSCrossRefGoogle Scholar
  15. 15.
    M Al-Amr, Comput. Math. Appl. 69, 390 (2013)Google Scholar
  16. 16.
    Q M Al-Mdallal and M I Syam, Chaos Solitons Fractals 33(5), 1610 (2007)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    A R Seadawy, Int. J. Comput. Methods 15(1), 1 (2018)Google Scholar
  18. 18.
    J H He and X H Wu, Chaos Solitons Fractals 30, 700 (2006)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    C Chen and Y L Jiang, Commun. Nonlinear Sci. Numer. Simul. 26, 24 (2015)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    A R Seadawy, Comput. Math. Appl. 71, 201 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Z Y Zhang, J Zhong, S S Dou, J Liu, D Peng and T Gao, Romanian Rep. Phys. 65(4), 1155 (2013)Google Scholar
  22. 22.
    H Jafari, N Kadkhoda and A Biswas, J. King Saud Univ. Sci. 25(1), 57 (2013)CrossRefGoogle Scholar
  23. 23.
    M N Ali, A R Seadawy, S M Husnine and K U Tariq, Optik – Int. J. Light Electron Opt. 156, 356 (2018)CrossRefGoogle Scholar
  24. 24.
    S Dinarvand and M M Rashidi, Nonlinear Anal. Real World Appl. 11(3), 1502 (2010)CrossRefGoogle Scholar
  25. 25.
    M Kaplan, A Bekir and A Akbulut, Nonlinear Dyn. 85, 2843 (2016)CrossRefGoogle Scholar
  26. 26.
    U T Kalim and A R Seadawy, Results Phys. 7, 1143 (2017)ADSCrossRefGoogle Scholar
  27. 27.
    A R Seadawy, O H El-Kalaawy and R B Aldenari, Appl. Math. Comput. 280, 57 (2016)MathSciNetGoogle Scholar
  28. 28.
    M M Rashidi and E Erfani, Int. J. Numer. Methods Heat Fluid Flow 21(7), 1 (2011)Google Scholar
  29. 29.
    Z Ali, S M Husnine and I Naeem, J. Appl. Math. 2013, 1 (2013)Google Scholar
  30. 30.
    A R Seadawy and K El-Rashidy, Math. Comput. Modell. 57, 1371 (2013)CrossRefGoogle Scholar
  31. 31.
    A R Seadawy, Physica A 455, 44 (2016)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    D Lu, A R Seadawy and M Arshad, Optik 140, 136 (2017)ADSCrossRefGoogle Scholar
  33. 33.
    A R Seadawy and D Lu, Results Phys. 7, 43 (2017)ADSCrossRefGoogle Scholar
  34. 34.
    A R Seadawy, Comput. Math. Appl. 67(1), 172 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    G W Bluman and S Kumei, Symmetries and differential equations (Springer Science & Business Media, New York, 2013) Vol. 81Google Scholar
  36. 36.
    S C Anco and G W Bluman, Eur. J. Appl. Math. 13(05), 545 (2002)Google Scholar
  37. 37.
    A R Seadawy, Phys. Plasmas 21(5), Article ID 052107 (2014)Google Scholar
  38. 38.
    R Naz, D P Mason and F M Mahomed, Nonlinear Anal. Real World Appl. 10(5), 2641 (2009)MathSciNetCrossRefGoogle Scholar
  39. 39.
    E Noether, Transp. Theory Stat. Phys. 1(3), 186 (1971)ADSCrossRefGoogle Scholar
  40. 40.
    A H Kara and F M Mahomed, Nonlinear Dyn. 45(3), 367 (2006)CrossRefGoogle Scholar
  41. 41.
    N H Ibragimov, J. Math. Anal. Appl. 333(1), 311 (2007)MathSciNetCrossRefGoogle Scholar
  42. 42.
    P J Olver, Applications of Lie groups to differential equations (graduate texts in mathematics, 2nd edn (Springer-Verlag, Berlin, Germany, 1993)CrossRefGoogle Scholar
  43. 43.
    N H Ibragimov, J. Math. Anal. Appl. 318(2), 742 (2006)MathSciNetCrossRefGoogle Scholar
  44. 44.
    M A Helal, A R Seadawy and R S Ibrahim, Appl. Math. Comput. 219, 5635 (2013)MathSciNetGoogle Scholar
  45. 45.
    D Daghan and O Adonmez, Braz. J. Phys. 46, 321 (2016)ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  • Muhammad Nasir Ali
    • 1
  • Aly R Seadawy
    • 2
    • 3
  • Syed Muhammad Husnine
    • 1
  1. 1.Department of Sciences and HumanitiesNational University of Computer and Emerging SciencesLahore Pakistan
  2. 2.Mathematics Department, Faculty of ScienceTaibah UniversityAl-Madinah Al-MunawarahSaudi Arabia
  3. 3.Mathematics Department, Faculty of ScienceBeni-Suef UniversityBeni SuefEgypt

Personalised recommendations