Computational study of soliton behavior in the simplified modified form of the Camassa-Holm equation

Hamida Parvin; Md. Nur Alam; Md. Farhad Hossain; Mohammad Hassan; Md. Jakir Hossen; Hamida Parvin; Md. Nur Alam; Md. Farhad Hossain; Mohammad Hassan; Md. Jakir Hossen

doi:10.3934/math.2025957

AIMS Mathematics

2025, Volume 10, Issue 9: 21533-21548. doi: 10.3934/math.2025957

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Computational study of soliton behavior in the simplified modified form of the Camassa-Holm equation

1.
Department of Mathematics, Pabna University of Science and Technology, Pabna-6600, Bangladesh
2.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamil Nadu, India
3.
Department of Mathematics and Scientific Computing, Madan Mohan Malaviya University of Technology, Gorakhpur, U. P. 273 016, India
4.
Center for Advanced Analytics (CAA), COE for Artificial Intelligence, Faculty of Engineering & Technology (FET), Multimedia University, 75450 Melaka, Malaysia

Received: 02 June 2025 Revised: 24 July 2025 Accepted: 04 August 2025 Published: 17 September 2025
MSC : 33F05, 35C08, 35E05, 35Q51, 37J25, 37L50

This study offers an in-depth exploration of traveling wave solutions to the simplified modified Camassa-Holm (SMCH) equation through the application of the modified S-expansion method. Utilizing a traveling wave transformation, the SMCH equation is converted into a nonlinear ordinary differential equation, from which a wide range of exact solutions is systematically obtained. The modified S-expansion method, implemented using the Maple software, proves to be a robust and efficient analytical tool, yielding a variety of soliton solutions such as kink, bright, and dark solitons. To capture the intricate behavior of these solutions, MATLAB is employed to produce detailed 2D, 3D, and contour visualizations that reveal their structural features and propagation dynamics. A comparative assessment with the modified simple equation method and the exp(-ϕ(η))-expansion method highlights the modified S-expansion method's superior accuracy, simplicity, and adaptability to solve nonlinear partial differential equations. Significantly, the method also extends to fractional-order equations, showcasing its broad applicability in nonlinear system analysis. Key solutions are graphically represented under constrained parameter values to emphasize the core propagation features. Overall, this work enhances the current analytical methods to solve both classical and fractional-order nonlinear partial differential equations (PDEs) and offers a valuable foundation for future research into closed-form traveling wave solutions across various disciplines.
- the simplified modified Camassa-Holm equation,
- modified S-expansion method,
- soliton solutions,
- nonlinear differential equations
Citation: Hamida Parvin, Md. Nur Alam, Md. Farhad Hossain, Mohammad Hassan, Md. Jakir Hossen. Computational study of soliton behavior in the simplified modified form of the Camassa-Holm equation[J]. AIMS Mathematics, 2025, 10(9): 21533-21548. doi: 10.3934/math.2025957

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Abstract

This study offers an in-depth exploration of traveling wave solutions to the simplified modified Camassa-Holm (SMCH) equation through the application of the modified S-expansion method. Utilizing a traveling wave transformation, the SMCH equation is converted into a nonlinear ordinary differential equation, from which a wide range of exact solutions is systematically obtained. The modified S-expansion method, implemented using the Maple software, proves to be a robust and efficient analytical tool, yielding a variety of soliton solutions such as kink, bright, and dark solitons. To capture the intricate behavior of these solutions, MATLAB is employed to produce detailed 2D, 3D, and contour visualizations that reveal their structural features and propagation dynamics. A comparative assessment with the modified simple equation method and the exp(-ϕ(η))-expansion method highlights the modified S-expansion method's superior accuracy, simplicity, and adaptability to solve nonlinear partial differential equations. Significantly, the method also extends to fractional-order equations, showcasing its broad applicability in nonlinear system analysis. Key solutions are graphically represented under constrained parameter values to emphasize the core propagation features. Overall, this work enhances the current analytical methods to solve both classical and fractional-order nonlinear partial differential equations (PDEs) and offers a valuable foundation for future research into closed-form traveling wave solutions across various disciplines.

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