Skip to main content
SpringerLink
Log in

Exact solutions to the \((3 + 1)\)-dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients

  • Published:
Pramana Aims and scope Submit manuscript

Abstract

The \((3 + 1)\)-dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients, which have very crucial applications in various areas, such as fluid dynamics, plasma physics and so on, are studied via unified and generalised unified method in this work. Solitary, soliton, elliptic, singular (periodic type) and non-singular (soliton-type) solutions of these two equations are extracted using the unified method. Otherwise, polynomial solutions in double-wave form and rational solutions in double-soliton form of the modified KdV equation with variable coefficients are acquired by exploiting the generalised unified method. The dynamical demeanours of these solutions help to comprehend the physical phenomena reflected by the equations, are depicted and analysed graphically for specific values of randomly undetermined parameters, which are diverse in each solution. The outcomes show that these two methods are quite trustworthy and effective to explore numerous solutions of nonlinear partial differential equations. We recognise that two approaches have never been utilised to study these two equations and work carried out in this paper is fresh and handy. Compared to previous methods, more comprehensive solutions can be obtained using them in this paper.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. K H Kim, K Lee and J Seo, J. Diff. Eqs. 340(15), 463 (2022)

    Article  ADS  Google Scholar 

  2. M S Osman, Comput. Math. Appl. 75(1), 1 (2018)

    Article  MathSciNet  Google Scholar 

  3. M S Osman, H Rezazadeh and M Eslami, Nonlinear Eng. 8(1), 559 (2019)

    Article  ADS  Google Scholar 

  4. M S Osman, Pramana – J. Phys. 93(2), 26 (2019)

    Google Scholar 

  5. L D Zhang, S F Tian, T T Zhang and X J Yan, EPL 135(2), 20003 (2021)

    Article  ADS  Google Scholar 

  6. A M Wazwaz and M S Osman, Comput. Math. Appl. 76(2), 276 (2018)

    Article  MathSciNet  Google Scholar 

  7. H M Jaradat, S Al-Shara, F Awawdeh and M Alquran, Phys. Scr. 85(3), 035001 (2012)

    Article  ADS  Google Scholar 

  8. G W Wang, X Wang, F Guan and H F Song, Optik 279, 170768 (2023)

    Article  ADS  Google Scholar 

  9. M A Akbar, A M Wazwaz, F Mahmud, D Baleanu, R Roy, H K Barman, W Mahmoud, M A Al Sharif and M S Osman, Results Phys. 43, 106079 (2022)

    Google Scholar 

  10. H F Ismaela, H Bulutb, C Parkc and M S Osman, Results Phys. 19, 103329 (2020)

    Article  Google Scholar 

  11. N Das and S S Ray, J. Nonlinear Opt. Phys. Mater.https://doi.org/10.1142/S0218863523500753

  12. R Mia, M M Miah and M S Osman, Heliyon 9, e15690 (2023)

    Article  Google Scholar 

  13. S Quresh, M A Akanbi, A A Shaikh, A S Wusu, O M Ogunlaran, W Mahmoud and M S Osman, Alex. Eng. J. 74, 585 (2023)

    Article  Google Scholar 

  14. H I Abdel-Gawad and M Osman, J. Adv. Res. 6(4), 593 (2015)

    Article  Google Scholar 

  15. M S Osman, Nonlinear Dyn. 87(2), 1209 (2017)

    Article  Google Scholar 

  16. N Das and S S Ray, Optik 287, 171060 (2023)

    Article  ADS  Google Scholar 

  17. Y Asghari, M Eslami1 and H Rezazadeh, Opt. Quantum Electron. 55, 318 (2023)

    Article  Google Scholar 

  18. E Az-Zo’bi, A F Al-Maaitah, M A Tashtoush and M S Osman, Pramana – J. Phys. 96, 184 (2022)

    Google Scholar 

  19. F Tasnim, M A Akbar and M S Osman, Fractal Fractional 7, 426 (2023)

    Article  Google Scholar 

  20. S Djennadi, N Shawagfeh, M Inc, M S Osman, J F Gómez-Aguilar and O A Arqub, Phys. Scr. 96, 094006 (2021)

    Article  ADS  Google Scholar 

  21. X K Liu and X Y Wen, Commun. Theor. Phys. 74(6), 065001 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  22. S Naowarat, S Saifullah, S Ahmad and M De la Sen, Symmetry-Basel 15(1), 135 (2023)

    Article  ADS  Google Scholar 

  23. M L Wang, J L Zhang, Z G Liu and E Q Li, Appl. Math. Lett. 129, 107929 (2022)

    Article  Google Scholar 

  24. S A El-Tantawy and W M Moslem, Phys. Plasmas 21(5), 052112 (2014)

    Article  ADS  Google Scholar 

  25. B A van Tiggelen and S E Skipetrov, Phys. Rev. B 103(17), 174204 (2021)

    Article  ADS  Google Scholar 

  26. Y L Shen and R X Yao, Math. Probl. Eng. 2018, 4014382 (2018)

    Google Scholar 

  27. W X Zheng and G M Wei, AIP Conf. Proc. 1820(1), 080016 (2017)

    Google Scholar 

  28. X F Cao, Math. Probl. Eng. 2017, 5157123 (2017)

    Google Scholar 

  29. S Malik and S Kumar, A Das and Nonlinear Dyn. 107(3), 2689 (2022)

    Article  Google Scholar 

  30. S Saha Ray and S Sahoo, Commun. Theor. Phys. 63(1), 25 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  31. S Sahoo and S Saha Ray, Comput. Math. Appl. 70(2), 158 (2015)

    Article  MathSciNet  Google Scholar 

  32. S Kaewta, S Sirisubtawee, S Koonprasert and S Sungnul, Fractal Fractional 5(3), 88 (2021)

    Article  Google Scholar 

  33. P Halder, A Bandyopadhyay, S Dalui and S Sardar, Z. Naturforschung. A 77(7), 659 (2022)

    Article  ADS  Google Scholar 

  34. O Guner, E Aksoy, A Bekir and A C Cevikel, AIP Conf. Proc. 1738(1), 290013 (2016)

    Google Scholar 

  35. M H Islam, K Khan, M A Akbar and M A Salam, SpringerPlus 3(1), 105 (2014)

    Article  Google Scholar 

  36. Y M Zhao, Y H He and H Z Yang, AIMS Math. 5(5), 4121 (2020)

    Article  MathSciNet  Google Scholar 

  37. S Bibi, S T Mohyud-Din, R Ullah, N Ahmed and U Khan, Results Phys. 7, 4434 (2017)

    Article  ADS  Google Scholar 

  38. S Javeed, M Inc, M A Abbasi, K H Mahmoud, Z U Abadin Zafar and S Razzaq, Results Phys. 38, 105546 (2022)

    Article  Google Scholar 

  39. C M Khalique and O D Adeyemo, Results Phys. 18, 103197 (2020)

    Article  Google Scholar 

  40. X X Zheng, Y D Shang and Y Huang, Abst. Appl. Anal. 2013, 109690 (2013)

    Google Scholar 

  41. H Kumar, A Malik and F Chand, Pramana – J. Phys. 80(2), 361 (2013)

    Google Scholar 

  42. Y Zhang, J Li and Y N Lv, Ann. Phys. 323(12), 3059 (2008)

    Article  ADS  Google Scholar 

  43. Y Zhang, Z L Cheng and X H Hao, Chin. Phys. B 21(12), 120203 (2012)

    Article  ADS  Google Scholar 

  44. X R Hu and Y Chen, J. Nonlinear Math. Phys. 19(1), 16 (2012)

    Article  Google Scholar 

  45. X L Gai, Y T Gao, L Wang, D X Meng, X Lü, Z Y Sun and X Yu, Commun. Nonlinear Sci. Numer. Simulat. 16(4), 1776 (2011)

    Article  ADS  Google Scholar 

  46. M A Kawser, M A Akbar and M A Khan, Results Phys. 50, 106587 (2023)

    Article  Google Scholar 

  47. S Hristova, S Tersian and R Terzieva, Fractal Fractional 5(2), 37 (2021)

    Article  Google Scholar 

  48. T Y Han, Z Li and K Zhang, Results Phys. 44, 106174 (2023)

    Article  Google Scholar 

  49. M S Osman, Pramana – J. Phys. 88(4), 67 (2017)

    MathSciNet  Google Scholar 

  50. L K Bazaglia Kuroda, A V Gomes, R Tavoni, P F de Arruda Mancera, N Varalta and R D Figueiredo Camargo, Comput. Appl. Math. 36(3), 1173 (2017)

    MathSciNet  Google Scholar 

  51. R Khalil, M Al-Horani, A Yousef and M Sababheh, J. Comput. Appl. Math. 264, 65 (2014)

    Article  MathSciNet  Google Scholar 

  52. H I Abdel-Gawad, J. Stat. Phys. 147(3), 506 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  53. L H Zhang, Appl. Math. Comput. 208(1), 144 (2009)

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of Shanxi (No. 202103021224068).

Author information

Authors and Affiliations

  1. College of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, People’s Republic of China

    Yating Hao & Ben Gao

Authors
  1. Yating Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ben Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ben Gao.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hao, Y., Gao, B. Exact solutions to the \((3 + 1)\)-dimensional time-fractional KdV–Zakharov–Kuznetsov equation and modified KdV equation with variable coefficients. Pramana - J Phys 98, 8 (2024). https://doi.org/10.1007/s12043-023-02693-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12043-023-02693-z

Keywords

PACS

Search

Navigation