Skip to main content
Log in

Invariance analysis, optimal system and conservation laws of \((2+1)\)-dimensional non-linear Vakhnenko equation

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

In this article, we study a \((2+1)\)-dimensional non-linear Vakhnenko equation which appears in a relaxing high rate active barothropic medium. The Lie symmetry analysis is applied to study the solution of \((2+1)\)-dimensional Vakhnenko equation and also used to find the optimal system of one-dimensional Lie subalgebra. The two-dimensional non-linear Vakhnenko equation is reduced to various ordinary differential equations by applying similarity transformation method, and solving these reduced ordinary differential equations, we find invariant solutions of \((2+1)\)-dimensional Vakhnenko equation. Moreover, the adjoint equation of the Vakhnenko equation and its conservation laws are also obtained, which can be used for mathematical analysis.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. X-B Hu, J. Phys. A 27, 201 (1994)

  2. I Aslan, Comput. Math. Appl. 61, 1700 (2011)

    Article  MathSciNet  Google Scholar 

  3. M L Wang, Phys. Lett. A 199, 169 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  4. P A Clarkson and M D Kruskal, J. Math. Phys. 30, 2201 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  5. A R Adem and C M Khalique, Comput. Fluids 81, 10 (2013)

    Article  MathSciNet  Google Scholar 

  6. G W Bluman and J D Cole, Similarity methods for differential equations (Springer Verlag, New York, 1974)

    Book  MATH  Google Scholar 

  7. S Sahoo, G Garai and S S Ray, Non-linear Dyn. 87, 1995 (2017)

    Article  Google Scholar 

  8. S Kumar and D Kumar, Int. J. Dynam. Control. 7, 496 (2018)

    Article  Google Scholar 

  9. D Baleanu, M Inc, A I Aliyu and A Yusuf, Superlatt. Microstruct. 111, 546 (2017)

    Article  ADS  Google Scholar 

  10. G Wang and K Fakhar, Comput. Fluids 119, 143 (2015)

    Article  MathSciNet  Google Scholar 

  11. G M Wei, Y L Yu, Y Q Xie and W X Zheng, Comput. Math. Appl. 75, 3420 (2018)

    Article  MathSciNet  Google Scholar 

  12. R Arora and A Chauhan, Int. J. Appl. Comput. Math. 5, 1 (2019)

    Google Scholar 

  13. D Kumar and S Kumar, Comput. Math. Appl. 78, 857 (2018)

    Article  Google Scholar 

  14. X Hu, Y Li and Y Chen, J. Math. Phys. 56, 053504 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  15. S V Coggeshall and J Meyerter , J. Math. Phys. 33, 3585 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  16. D Tanwar and A M Wazwaz, Phys. Scr. 95, 1 (2020)

    Article  Google Scholar 

  17. J Manafian, O A Ilhan and A Alizadeh, Phys. Scr. 95, 1 (2020)

    Article  Google Scholar 

  18. V A Vakhnenko, J. Phys. A 25, 4181 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  19. A J Morrison and E J Parkes, Glasg. Math. J. 43 65 (2001)

    Article  Google Scholar 

  20. A J Morrison and E J Parkes, Chaos Solitons Fractals 16, 13 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  21. K K Victor, B B Thomas and T C Kofane, Chin. Phys. Lett. 25, 425 (2008)

    Article  Google Scholar 

  22. E J Parkes, J. Phys. A 26, 6469 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  23. V O Vakhnenko and E J Parkes, Nonlinearity 11, 1457 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  24. V O Vakhnenko, E J Parkes and A J Morrison, Chaos Solitons Fractals 17, 683 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  25. A M Wazwaz, Phys Scr. 82, 065006 (2010)

    Article  ADS  Google Scholar 

  26. Y Wang and Y Chen, J. Math. Phys. 53, 123504 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  27. J C Brunelli and S Sakovich, Commun. Nonlinear Sci. Numer. Simulat. 18, 56 (2013)

    Article  ADS  Google Scholar 

  28. M S Hashemi, M C Nucci and S Abbasbandy, Commun. Nonlinear Sci. Numer. Simulat. 18, 867 (2013)

    Article  ADS  Google Scholar 

  29. J J Xiao, D H Feng, X Meng and Y Q Cheng, Pramana – J. Phys. 88: 1 (2017)

    Article  Google Scholar 

  30. C Xiang and H Wang, J. Appl. Math. Phys. 8, 793 (2020)

    Article  Google Scholar 

  31. Q Meng and He Bin, Complexity 2020, 1 (2020)

  32. S Kumar, A Kumar and H Kharbanda, Phys. Scr. 95, 1 (2020)

    Google Scholar 

  33. S Kumar and D Kumar, Comput. Math. Appl. 77, 2096 (2018)

    Article  Google Scholar 

  34. P J Olver, Applications of Lie groups to differential equations (Springer, New York, 1993) Vol. 107

  35. A Chauhan, K Sharma and R Arora, Math. Meth. Appl. Sci. 43, 8823 (2020)

    Article  Google Scholar 

  36. N H Ibragimov, J. Math. Anal. Appl. 333, 311 (2007)

    Article  MathSciNet  Google Scholar 

  37. N H Ibragimov, J. Phys. A 44, 432002 (2011)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rajan Arora.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yadav, S., Chauhan, A. & Arora, R. Invariance analysis, optimal system and conservation laws of \((2+1)\)-dimensional non-linear Vakhnenko equation. Pramana - J Phys 95, 8 (2021). https://doi.org/10.1007/s12043-020-02059-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-020-02059-9

Keywords

PACS Nos

Navigation