Sign Up Submit Enquiry

Investigating nonlinear wave structures via auto-Bäcklund transformation and Hirota bilinear method in the coupled Boussinesq system

Home Investigating nonlinear wave structures via auto-Bäcklund transformation and Hirota bilinear method in the coupled Boussinesq system

Related Articles

Contact us

Article Content

Abstract

We investigate various nonlinear wave structures of the coupled Boussinesq system (CBS) using two efficient techniques: the auto-Bäcklund transformation and the Hirota bilinear method (HBM). Various travelling wave solutions, including kink waves, solitary waves with lump excitation and multi-shock waves, are obtained for the CBS using the auto-Bäcklund transformation. We also obtain the interaction between two-soliton solutions utilising this technique. Initially, we have shown that the CBS passes the Painlevé test for integrability. Then, we derive the auto-Bäcklund transformation for CBS through the truncated Painlevé expansion. Next, we observe the interaction and propagation behaviours of multi-soliton solutions for CBS using the HBM. The existence of multi-soliton solutions establishes that the CBS is also integrable in the Hirota sense. Interestingly, we observe that the HBM offers a more comprehensive framework for exploring diverse interactions between bright solitons and dark solitons, compared to the auto-Bäcklund approach. The multi-solitons and solitary waves with lump excitation obtained in this study can be used to describe surface water waves, while the kink wave and multi-shock waves represent topological changes in the water surface modelled by the CBS. Our findings on nonlinear structures for the CBS, provide valuable insights into the behaviour of nonlinear waves in water dynamics and are important for analysing various phenomena, such as ocean waves and waves in coastal areas.

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

  • Nonlinear Dynamics and Chaos Theory
  • Nonlinear Optics
  • Partial Differential Equations
  • Phase Transitions and Multiphase Systems
  • Solitons
  • Waves, instabilities and nonlinear plasma dynamics

Data availability statement

The author confirms that there is no associated data available for the above research work. Data sharing does not apply to this article as no new data were created or analysed in this study.

References

  1. M Saqib, D Ahmad, A N Al-Kenani and T Allahviranloo, Alex. Eng. J. 74, 751 (2023)

    Google Scholar

  2. M Inc, S Hussain, A H Ali, M S Iqbal, R Ashraf, M A Tarar and M Adnan, Sci. Rep. 14, 10588 (2024)

    ADS Google Scholar

  3. P Chatterjee, K Roy and U N Ghosh, Waves and wave interaction in plasmas (World Scientific, 2023)

  4. K Shah, T Abdeljawad and B A Abdalla, AIMS Math. 8, 9890 (2023)

    MathSciNet Google Scholar

  5. A R Hadjesfandiari, Symmetry 16, 1046 (2024)

    Google Scholar

  6. G Sharma, J K Allen and F Mistree, Eng. Optim. 55, 1556 (2022)

    Google Scholar

  7. S Mesloub, Symmetry 15, 1952 (2023)

    ADS Google Scholar

  8. S Shagolshem, B Bira and N Kallur, Nonlinear Dyn. 113, 11999 (2024), https://doi.org/10.1007/s11071-024-10697-6

  9. A K Farhood and O H Mohammed, Part. Differ. Equ. Appl. Math. 7, 100513 (2023)

    Google Scholar

  10. M H Bashar, H Z Mawa, A Biswas, M M Rahman, M M Roshid and J Islam, Heliyon 9, e15662 (2023)

    Google Scholar

  11. O Kirci, Y Pandir, A Latifa and H Bulut, Opt. Quant. Electron. 56, 1019 (2024)

    ADS Google Scholar

  12. R Ansar, M Abbas, H Emadifar, T Nazir and A S M Alzaidi, PLOS ONE 19(1), e0296640 (2024)

    Google Scholar

  13. X Zeng, X Wu, C Liang, C Yuan and J Cai, Symmetry 15, 1021 (2023)

    ADS Google Scholar

  14. M I Khan, D N K Marwat, J Sabi’s and M Inc, Opt. Quant. Electron. 56, 459 (2024)

    ADS Google Scholar

  15. M Kapoor and V Joshi, Int. J. Appl. Mech. Eng. 27(3), 79 (2022)

    Google Scholar

  16. L A Sakkaf and U A Khawaja, Math. Meth. Appl. Sci. 43(17), 10168 (2020)

    Google Scholar

  17. M Sahoo and S Chakraverty, Int. J. Mod. Phys. B 38, 2440028 (2024)

    Google Scholar

  18. D H Peregrine, J. Fluid Mech. 27(4), 815 (1967)

    ADS MathSciNet Google Scholar

  19. M N Rafiq, M H Rafiq and H Alsaud, Chaos Solitons Fractals 190, 115755 (2025)

    Google Scholar

  20. M V Flamarion, R M Vargas-Magaña, Part. Differ. Equ. Appl. Math. 13, 101080 (2025)

    Google Scholar

  21. X Geng, M Jia, B Xue and Y Zhai, Lett. Math. Phys. 114, 30 (2024)

    ADS Google Scholar

  22. V Komornik, A F Pazoto and M D S Vieira, Math. Nachr. 297(12), 4478 (2024)

    MathSciNet Google Scholar

  23. X-Y Gao, Y-J Guo and W-R Shan, Qual. Theory Dyn. Syst. 23, 57 (2024)

    Google Scholar

  24. S Yan, L Wang and Q Zhang, Open Math. 22(1), 20230182 (2024)

    Google Scholar

  25. Z Yang, W-P Zhong and M Belić, Chin. Phys. Lett. 41, 044201 (2024)

    ADS Google Scholar

  26. H Qawaqneh, H A Jari, A Altalbe and A Bekir, Phys. Scr. 99, 125235 (2024)

    Google Scholar

  27. X Cheng and Q Huang, Commun. Theor. Phys. 76, 065001 (2024)

    ADS Google Scholar

  28. A Ali, J Ahmad, S Javed and S U Rehman, Phys. Scr. 98, 075217 (2023)

    ADS Google Scholar

  29. Y L Ma and B Q Li, Appl. Math. Lett. 124, 107677 (2022)

    MATH Google Scholar

  30. Z Ayati and M Moradi, Comput. Sci. Eng. 2(2), 275 (2022)

    Google Scholar

  31. E Yaşar and E Yaşar, Disc. Contin. Dyn. Syst. S 18, 1075 (2025)

    Google Scholar

  32. H F Ismael, H Bulut, H M Baskonus and W Gao, Commun. Theor. Phys. 72, 115002 (2020)

    ADS Google Scholar

  33. M M Rashid and W Khatun, J. Adv. Math. Comput. Sci. 35, 72 (2020)

    Google Scholar

  34. T Han and Z Li, Adv. Math. Phys. 3668063 (2021)

  35. S Kumar and V Gupta, J. Comput. Nonlinear Dynam. 19, 121002 (2024)

    Google Scholar

  36. T A Sulaiman, H Bulut, A Yokus and H M Baskonus, Indian J. Phys. 93(5), 647 (2019)

    ADS Google Scholar

  37. A Alharbi and M B Almatrafi, Symmetry 12(9), 1473 (2020)

    ADS Google Scholar

  38. R Kumar, K S Pandey and Avneesh Kumar, Braz. J. Phys. 52, 201 (2022)

    ADS Google Scholar

  39. R Gong, Y Shi and D S Wang, Disc. Contin. Dyn. Syst. 42(7), 3355 (2022)

    Google Scholar

  40. D Shi, Y Zhang, W Liu and J Liu, Symmetry 11(1), 77 (2019)

    ADS Google Scholar

  41. E Atilgan, M Senol, A Kurt and O Tasbozan, China Ocean Eng. 33(4), 477 (2019)

    ADS Google Scholar

  42. S Chen, Y Li, Y Li, B Guan and Y Liu, Results Phys. 24, 104147 (2021)

    Google Scholar

  43. Y-L Ma and B-Q Li, Theor. Dyn. Syst. 23, 137 (2024)

    Google Scholar

  44. B-Q Li and Y-L Ma, Comput. Math. Appl. 76, 204 (2018)

    MathSciNet Google Scholar

  45. Y-L Ma and B-Q Li, Qual. Theor. Dyn. Syst. 22, 146 (2023)

    Google Scholar

  46. Y-L Ma, A-M Wazwaz and B-Q Li, Math. Comput. Simul. 187, 505 (2021)

    Google Scholar

  47. N K Pal, S Nasipuri, P Chatterjee and S Raut, Pramana – J. Phys. 98, 110 (2024)

    Google Scholar

  48. F R Meng, B Tian and T Y Zhou, Int. J. Mod. Phys. B 38, 2450232 (2024)

    ADS Google Scholar

  49. J Weiss, M Tabor and G Carnevale, J. Math. Phys. 24, 522 (1983)

    ADS MathSciNet Google Scholar

  50. M Jimbo, M D Kruskal and T Miwa, Phys. Lett. A 92, 59 (1982)

    ADS MathSciNet Google Scholar

  51. B Sivatharani, J S Ranjore, N Asokan and K Subramanian, Nonlinear Dynam. 111, 18215 (2023)

    Google Scholar

  52. R K Gupta and M Sharma, Qual. Theory Dyn. Syst. 23, 115 (2024)

    Google Scholar

  53. C U Maheswari, N Muthuchamy, V K Chandrasekar, R Sahadevan and M Lakshmanan, J. Math. Phys. 65, 032702 (2024)

    ADS Google Scholar

  54. R Hirota, The direct method in soliton theory (Cambridge University Press, Cambridge, 2004)

    MATH Google Scholar

  55. A C Scott, The nonlinear Universe: Chaos, emergence, life (Springer, Berlin, 2007)

  56. R Hirota, Phys. Rev. Lett. 27(18), 1192 (1971)

    ADS Google Scholar

  57. J Hietarinta, Introduction to the Hirota bilinear method (Berlin, Heidelberg: Springer Berlin Heidelberg, 1997) pp. 95–105, https://doi.org/10.48550/arXiv.solv-int/9708006

  58. Y Guo, Y Chen, J Manafian, S Malmir, K H Mahmoud and A SA Alsubaie, Sci. Rep. 14, 26171 (2024)

    Google Scholar

  59. Y-Q Chen, B Tian, Y Shen and T-Y Zhou, Phys. Fluids 35, 127107 (2023)

    ADS Google Scholar

  60. L Yang and B Gao, Chaos Solitons Fractals 184, 115009 (2024)

    Google Scholar

  61. L Zhang and H Han, Commun. Nonlinear Sci. Numer. Simul. 138, 108185 (2024)

    Google Scholar

  62. X Wang and B Gao, Nonlinear Dynam. 112, 17321 (2024)

    Google Scholar

Download references

Acknowledgements

Snehalata Nasipuri is thankful to the Council of Scientific and Industrial Research (CSIR), India, for providing financial support under the Senior Research Fellowship (SRF) program (File No: 09/202(0120)/2021-EMR-I).

Author information

Authors and Affiliations

  1. Department of Mathematics, Visva-Bharati, Santiniketan, 731 235, India

    Snehalata Nasipuri & Prasanta Chatterjee

Corresponding author

Correspondence to Prasanta Chatterjee.

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

Cite this article

Nasipuri, S., Chatterjee, P. Investigating nonlinear wave structures via auto-Bäcklund transformation and Hirota bilinear method in the coupled Boussinesq system. Pramana – J Phys 99, 103 (2025). https://doi.org/10.1007/s12043-025-02936-1

Download citation

  • Received
  • Revised 
  • Accepted 
  • Published 
  • DOI  https://doi.org/10.1007/s12043-025-02936-1

Keywords

  • Painlevé integrability
  • kink solution
  • lump wave
  • multi-shock
  • multi-soliton
  • soliton interaction and phase shift

PACS Nos.

  • 02.30.Jr
  • 47.40.Nm
  • 05.45.Yv
  • 47.35.Fg
  • 03.40.Kf

Copyright © All rights reserved Designed and Developed by Digital Flavers

WhatsApp