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Formation of advanced soliton dynamics through the M-fractional regularized long-wave equation
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  • Published: 09 February 2026

Formation of advanced soliton dynamics through the M-fractional regularized long-wave equation

  • Mohammad Mobarak Hossain1,
  • Harun-Or Roshid1,
  • Mohammad Safi Ullah2 &
  • …
  • Md. Abu Naim Sheikh3 

Scientific Reports , Article number:  (2026) Cite this article

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  • Engineering
  • Mathematics and computing
  • Physics

Abstract

This research addresses the modified F-expansion, the newly established extended modified F-expansion, and the unified method for computing the exact solution of the time-fractional regularized long-wave (Tf-RLW) model with Conformable fractional derivative. The Tf-RLW model is widely applicable to solitary wave propagation in shallow water waves, plasma waves, and ion-acoustic waves. The three analytical techniques are employed to simplify this nonlinear model and justify the obtained waves in relation to the existing waves of this model. Fractional derivatives are non-integer-order operators widely used to model complex phenomena in science and engineering, with notable applications in ocean and coastal engineering for tsunami-wave mitigation. This research presented the bright-dark bell waves, periodic rogue waves, periodic waves, singular kink waves, and kinky-periodic bell waves as solutions to the governing model. 3-D, density, and 2-D sketches are given to analyze the intense behavior of the attained waves. Stability investigation of the obtained solutions is also included. All the solutions obtained in this research are substantiated using Maple 18.

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Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

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Funding

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Author information

Authors and Affiliations

  1. Department of Mathematics, Sunamgonj Science and Technology University, Shantiganj, Sunamganj, 3000, Bangladesh

    Mohammad Mobarak Hossain & Harun-Or Roshid

  2. Department of Mathematics, Comilla University, Cumilla, 3506, Bangladesh

    Mohammad Safi Ullah

  3. Department of Mathematics, Dhaka University of Engineering & Technology, Gazipur, Bangladesh

    Md. Abu Naim Sheikh

Authors
  1. Mohammad Mobarak Hossain
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  2. Harun-Or Roshid
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  3. Mohammad Safi Ullah
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  4. Md. Abu Naim Sheikh
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Contributions

Mohammad Mobarak Hossain: conceptualization, visualization, software, methodology, data-curation, writing – original draft. Harun-Or-Roshid: formal analysis, validation, methodology, writing – review and editing. Mohammad Safi Ullah: formal analysis, review – original draft. Md. Abu Naim Sheikh: resources, investigation, validation.

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Correspondence to Mohammad Mobarak Hossain or Mohammad Safi Ullah.

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Hossain, M.M., Roshid, HO., Ullah, M.S. et al. Formation of advanced soliton dynamics through the M-fractional regularized long-wave equation. Sci Rep (2026). https://doi.org/10.1038/s41598-026-37284-6

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  • Received: 29 October 2025

  • Accepted: 21 January 2026

  • Published: 09 February 2026

  • DOI: https://doi.org/10.1038/s41598-026-37284-6

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Keywords

  • Tf-RLW equation
  • Conformable derivative
  • Soliton
  • Periodic wave
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