Skip to main content
SpringerLink
Log in

Waves in motion: unraveling nonlinear behavior through the Gilson–Pickering equation

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

This study addresses the challenge of solving the nonlinear Gilson–Pickering \( \mathbb{G}\mathbb{P}\) equation, employing the unified method for analytical investigation and verifying the results through the He’s variational iteration method. The primary aim is to offer a robust solution for this complex equation. The research methodology integrates the unified method for analytical purposes and validates the outcomes through numerical approaches. The study’s significance lies in successfully resolving the \(\mathbb{G}\mathbb{P}\) equation, showcasing its applicability in various scientific contexts. This research’s implications are substantial, as it introduces an effective solution method with broad interdisciplinary relevance. In conclusion, the study underscores the compatibility of the proposed unified method with numerical solutions, contributing a novel perspective to mathematical modeling. This work pertains to the field of mathematical sciences and is theoretical, involving no specific subjects or participants.

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

Similar content being viewed by others

Data Availability Statement

This manuscript has associated data in a data repository. [Authors’ comment: Data will be made available on request.]

References

  1. Y. Mo, R. Kishek, D. Feldman, I. Haber, B. Beaudoin, P. O’Shea, J. Thangaraj, Experimental observations of soliton wave trains in electron beams. Phys. Rev. Lett. 110(8), 084802 (2013)

    Article  ADS  Google Scholar 

  2. M.M.A. Khater, Computational simulations of propagation of a tsunami wave across the ocean. Chaos Solitons Fractals 174, 113806 (2023)

    Article  MathSciNet  Google Scholar 

  3. M.M.A. Khater, Soliton propagation under diffusive and nonlinear effects in physical systems; (1+1)-dimensional MNW integrable equation. Phys. Lett. A 480, 128945 (2023)

    Article  MathSciNet  Google Scholar 

  4. S. Sarwar, New soliton wave structures of nonlinear (4+ 1)-dimensional fokas dynamical model by using different methods. Alex. Eng. J. 60(1), 795–803 (2021)

    Article  Google Scholar 

  5. M.M.A. Khater, Horizontal stratification of fluids and the behavior of long waves. Eur. Phys. J. Plus 138(8), 715 (2023)

    Article  Google Scholar 

  6. M.M.A. Khater, Characterizing shallow water waves in channels with variable width and depth; computational and numerical simulations. Chaos Solitons Fractals 173, 113652 (2023)

    Article  MathSciNet  Google Scholar 

  7. M.M.A. Khater, Advancements in computational techniques for precise solitary wave solutions in the (1 + 1)-dimensional Mikhailov–Novikov–Wang equation. Int. J. Theor. Phys. 62(7), 152 (2023)

    Article  MathSciNet  Google Scholar 

  8. M.M.A. Khater, Numerous accurate and stable solitary wave solutions to the generalized modified equal-width equation. Int. J. Theor. Phys. 62(7), 151 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  9. H.F. Ismael, W.-X. Ma, H. Bulut, Dynamics of soliton and mixed lump-soliton waves to a generalized Bogoyavlensky–Konopelchenko equation. Phys. Scr. 96(3), 035225 (2021)

    Article  ADS  Google Scholar 

  10. M.M.A. Khater, Long waves with a small amplitude on the surface of the water behave dynamically in nonlinear lattices on a non-dimensional grid. Int. J. Mod. Phys. B 37(19), 2350188 (2023)

    Article  ADS  Google Scholar 

  11. M.M.A. Khater, Abundant and accurate computational wave structures of the nonlinear fractional biological population model. Int. J. Mod. Phys. B 37(18), 2350176 (2023)

    Article  ADS  Google Scholar 

  12. M.M.A. Khater, Novel computational simulation of the propagation of pulses in optical fibers regarding the dispersion effect. Int. J. Mod. Phys. B 37(9), 2350083 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  13. M.M.A. Khater, In surface tension; gravity-capillary, magneto-acoustic, and shallow water waves’ propagation. Eur. Phys. J. Plus 138(4), 320 (2023)

    Article  Google Scholar 

  14. M.M. Khater, R.A. Attia, S.S. Alodhaibi, D. Lu, Novel soliton waves of two fluid nonlinear evolutions models in the view of computational scheme. Int. J. Mod. Phys. B 34(10), 2050096 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  15. M.M.A. Khater, A hybrid analytical and numerical analysis of ultra-short pulse phase shifts. Chaos Solitons Fractals 169, 113232 (2023)

    Article  MathSciNet  Google Scholar 

  16. M.M.A. Khater, Prorogation of waves in shallow water through unidirectional Dullin–Gottwald–Holm model; computational simulations. Int. J. Mod. Phys. B 37(8), 2350071 (2023)

    Article  ADS  Google Scholar 

  17. M.M.A. Khater, In solid physics equations, accurate and novel soliton wave structures for heating a single crystal of sodium fluoride. Int. J. Mod. Phys. B 37(7), 2350068–139 (2023)

    Article  ADS  Google Scholar 

  18. M.M. Khater, D. Lu, R.A. Attia, Lump soliton wave solutions for the (2+ 1)-dimensional Konopelchenko–Dubrovsky equation and KdV equation. Mod. Phys. Lett. B 33(18), 1950199 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  19. M.M.A. Khater, Nonlinear elastic circular rod with lateral inertia and finite radius: dynamical attributive of longitudinal oscillation. Int. J. Mod. Phys. B 37(6), 2350052 (2023)

    Article  ADS  Google Scholar 

  20. M.M.A. Khater, Computational and numerical wave solutions of the Caudrey–Dodd–Gibbon equation. Heliyon 9, e13511 (2023)

    Article  Google Scholar 

  21. M.M.A. Khater, Physics of crystal lattices and plasma; analytical and numerical simulations of the Gilson–Pickering equation. Res. Phys. 44, 106193 (2023)

    Google Scholar 

  22. J.-Q. Sun, X.-Y. Gu, Z.-Q. Ma, Numerical study of the soliton waves of the coupled nonlinear schrödinger system. Physica D 196(3–4), 311–328 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  23. M.M.A. Khater, Hybrid accurate simulations for constructing some novel analytical and numerical solutions of three-order GNLS equation. Int. J. Geom. Methods Mod. Phys. 20(9), 2350159–12 (2023)

    Article  MathSciNet  Google Scholar 

  24. M.M. Khater, Nonlinear elastic circular rod with lateral inertia and finite radius: dynamical attributive of longitudinal oscillation. Int. J. Mod. Phys. B 37(06), 2350052 (2023)

    Article  ADS  Google Scholar 

  25. M.M. Khater, X. Zhang, R.A. Attia, Accurate computational simulations of perturbed Chen–Lee–Liu equation. Res. Phys. 45, 106227 (2023)

    Google Scholar 

  26. R.A. Attia, J. Tian, D. Lu, J.F.G. Aguilar, M.M. Khater, Unstable novel and accurate soliton wave solutions of the nonlinear biological population model. Arab J. Basic Appl. Sci. 29(1), 19–25 (2022)

    Article  Google Scholar 

  27. M.M. Khater, Recent electronic communications; optical quasi-monochromatic soliton waves in fiber medium of the perturbed Fokas–Lenells equation. Opt. Quant. Electron. 54(9), 586 (2022)

    Article  Google Scholar 

  28. H.M. Baskonus, M. Osman, H.U. Rehman, M. Ramzan, M. Tahir, S. Ashraf, On pulse propagation of soliton wave solutions related to the perturbed Chen–Lee–Liu equation in an optical fiber. Opt. Quant. Electron. 53, 1–17 (2021)

    Article  Google Scholar 

  29. J. Manafian, M.F. Aghdaei, M. Khalilian, R.S. Jeddi, Application of the generalized \((g^{\prime }/g)\)-expansion method for nonlinear PDES to obtaining soliton wave solution. Optik 135, 395–406 (2017)

    Article  ADS  Google Scholar 

  30. L.-L. Feng, S.-F. Tian, X.-B. Wang, T.-T. Zhang, Rogue waves, homoclinic breather waves and soliton waves for the (2+ 1)-dimensional b-type Kadomtsev–Petviashvili equation. Appl. Math. Lett. 65, 90–97 (2017)

    Article  MathSciNet  Google Scholar 

  31. N.K. Vitanov, Breather and soliton wave families for the sine–Gordon equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 454(1977), 2409–2423 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  32. M.M. Khater, D. Lu, Diverse soliton wave solutions of for the nonlinear potential Kadomtsev–Petviashvili and Calogero–Degasperis equations. Res. Phys. 33, 105116 (2022)

    Google Scholar 

  33. G. Shen, J. Manafian, D.T.N. Huy, K.S. Nisar, M. Abotaleb, N.D. Trung, Abundant soliton wave solutions and the linear superposition principle for generalized (3+ 1)-d nonlinear wave equation in liquid with gas bubbles by bilinear analysis. Res. Phys. 32, 105066 (2022)

    Google Scholar 

  34. M.M. Khater, L. Akinyemi, S.K. Elagan, M.A. El-Shorbagy, S.H. Alfalqi, J.F. Alzaidi, N.A. Alshehri, Bright-dark soliton waves’ dynamics in pseudo spherical surfaces through the nonlinear Kaup–Kupershmidt equation. Symmetry 13(6), 963 (2021)

    Article  ADS  Google Scholar 

  35. M.M. Khater, S. Muhammad, A. Al-Ghamdi, M. Higazy, Novel soliton wave solutions of the Vakhnenko–Parkes equation arising in the relaxation medium. J. Ocean Eng. Sci. (2022)

  36. O. Nikan, Z. Avazzadeh, M. Rasoulizadeh, Soliton wave solutions of nonlinear mathematical models in elastic rods and bistable surfaces. Eng. Anal. Bound. Elem. 143, 14–27 (2022)

    Article  MathSciNet  Google Scholar 

  37. K. Hosseini, E. Hincal, S. Salahshour, M. Mirzazadeh, K. Dehingia, B. Nath, On the dynamics of soliton waves in a generalized nonlinear Schrödinger equation. Optik 272, 170215 (2023)

    Article  ADS  Google Scholar 

  38. M.M. Khater, In solid physics equations, accurate and novel soliton wave structures for heating a single crystal of sodium fluoride. Int. J. Mod. Phys. B 37, 2350068 (2022)

    Article  ADS  Google Scholar 

  39. W.-Q. Peng, S.-F. Tian, T.-T. Zhang, Dynamics of the soliton waves, breather waves, and rogue waves to the cylindrical Kadomtsev–Petviashvili equation in pair-ion-electron plasma. Phys. Fluids 31(10), 102107 (2019)

    Article  ADS  Google Scholar 

  40. M.M. Khater, Physics of crystal lattices and plasma; analytical and numerical simulations of the Gilson–Pickering equation. Res. Phys. 44, 106193 (2023)

    Google Scholar 

  41. H.M. Baskonus, Complex soliton solutions to the Gilson–Pickering model. Axioms 8(1), 18 (2019)

    Article  MathSciNet  Google Scholar 

  42. A. Rani, A. Zulfiqar, J. Ahmad, Q.M.U. Hassan, New soliton wave structures of fractional Gilson–Pickering equation using tanh-coth method and their applications. Res. Phys. 29, 104724 (2021)

    Google Scholar 

Download references

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Author information

Authors and Affiliations

  1. School of Medical Informatics and Engineering, Xuzhou Medical University, 209 Tongshan Road, Xuzhou, 221004, Jiangsu Province, People’s Republic of China

    Mostafa M. A. Khater

  2. Department of Basic Science, Obour High Institute for Engineering and Technology, Cairo, 11828, Egypt

    Mostafa M. A. Khater

Authors
  1. Mostafa M. A. Khater
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

MMAK conceived and designed the experiments, performed the experiments, analyzed and interpreted the data, contributed reagents, materials, analysis tools or data, and wrote the paper.

Corresponding author

Correspondence to Mostafa M. A. Khater.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

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

Khater, M.M.A. Waves in motion: unraveling nonlinear behavior through the Gilson–Pickering equation. Eur. Phys. J. Plus 138, 1138 (2023). https://doi.org/10.1140/epjp/s13360-023-04774-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-023-04774-9

Search

Navigation