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Investigation of the dynamical structures for nonlinear Vakhnenko-Parkes equation using two integration schemes

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Abstract

The dynamic behavior of the Vakhnenko-Parkes equation is examined in this manuscript. This is an important subject because of its implications for comprehending intricate mathematical models describing traveling wave phenomena and solitons. The construction of traveling wave solutions for the Vakhnenko-Parkes equation in closed form is the main issue addressed in the study. The modified auxiliary equation approach and the extended \((\frac{G'}{G^{2}})\)-expansion method are used to address this because they are effective in producing precise solutions of a large class of nonlinear partial differential equations. A visual component to comprehending the behavior of the equation is added by employing 3D-surface graphs, 2D-line graphs, and contour plots to explore these solutions graphically. A variety of traveling wave behavior is observed from the obtained solutions. These results imply that the Vakhnenko-Parkes equation and its solutions are complex, offering important insights into the underlying dynamics. The proposed techniques are applied for the first time to study the considered model in this work. A comparison of the obtained results with the previous works is presented to confirm the significance and novelty of the reported results.

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Acknowledgements

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TUDSPP-2024-257). The authors are also grateful to anonymous referees for their valuable suggestions, which significantly improved this manuscript.

Funding

This research was funded by Taif University, Saudi Arabia, Project No. (TU-DSPP-2024-257).

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Authors and Affiliations

  1. Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, 54590, Pakistan

    Saima Arshed, Ghazala Akram, Maasoomah Sadaf & Ejaz Hussain

  2. Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan

    Muhammad Abbas

  3. Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, 21944, Taif, Saudi Arabia

    Ahmed S. M. Alzaidi

  4. IT4Innovations, VSB–Technical University of Ostrava, 708 00, Ostrava-Poruba, Czech Republic

    Muhammad Bilal Riaz

Authors
  1. Saima Arshed
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  4. Ejaz Hussain
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  6. Ahmed S. M. Alzaidi
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Contributions

SA: Methodology, Writing original draft. GA: Supervision, Methodology, Writing review & editing. MS: Methodology, Writing original draft, Formal analysis. EH: Software, Investigation, Visualization, Writing original draft. MA: Methodology, Writing review & editing. ASMA: Formal analysis, Writing original draft. MBR: Visualization, Software, Investigation, Validation, Writing review & editing. All of the authors read and approved the final manuscript.

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Correspondence to Muhammad Abbas or Ahmed S. M. Alzaidi.

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Arshed, S., Akram, G., Sadaf, M. et al. Investigation of the dynamical structures for nonlinear Vakhnenko-Parkes equation using two integration schemes. Opt Quant Electron 56, 1072 (2024). https://doi.org/10.1007/s11082-024-06953-z

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