Exploring the truncated M-fractional exact solitons, modulation instability, and stability analysis of the Kudryashov–Sinelshchikov model

Haitham Qawaqneh; Mostafa Abotaleb; Mohammed Ahmed Alomair; Luai Abdulla Aldoghan; Haitham Qawaqneh; Mostafa Abotaleb; Mohammed Ahmed Alomair; Luai Abdulla Aldoghan

doi:10.3934/math.2026392

AIMS Mathematics

2026, Volume 11, Issue 4: 9470-9491. doi: 10.3934/math.2026392

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Exploring the truncated M-fractional exact solitons, modulation instability, and stability analysis of the Kudryashov–Sinelshchikov model

1.
Department of Basic Science, Al-Zaytoonah University of Jordan, Amman 11733, Jordan
2.
Engineering School of Digital Technologies, Yugra State University, Khanty Mansiysk 628012, Russia
3.
Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 18 January 2026 Revised: 12 March 2026 Accepted: 18 March 2026 Published: 09 April 2026
MSC : 35Q51, 35Q92, 35R11, 35R60, 60H15

In this paper, various types of exact soliton solutions of the truncated M-fractional Kudryashov–Sinelshchikov equation, a significant fluid surge model, were obtained. This model accounted for density and heat transfer effects while describing the propagation of pressure waves in mixtures of liquid–gas bubbles. By applying the modified $ (G'/G^2) $-expansion method and the extended Sinh–Gordon equation expansion method, we derived new solutions in the forms of trigonometric, hyperbolic, and rational functions. The obtained solutions were illustrated dynamically using 2D, 3D, and contour plots. These results were novel due to the use of a new definition of fractional derivatives. The effect of the fractional derivative on the solutions was demonstrated through 2D plots. To examine the stability of the obtained solutions, stability analysis was performed. Steady-state solutions were derived using modulation instability analysis. The obtained solutions may be useful in various fields of science and engineering.
- Kudryashov–Sinelshchikov model,
- truncated M-fractional derivative,
- qualitative analysis,
- analytical techniques,
- exact solitons solutions
Citation: Haitham Qawaqneh, Mostafa Abotaleb, Mohammed Ahmed Alomair, Luai Abdulla Aldoghan. Exploring the truncated M-fractional exact solitons, modulation instability, and stability analysis of the Kudryashov–Sinelshchikov model[J]. AIMS Mathematics, 2026, 11(4): 9470-9491. doi: 10.3934/math.2026392

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Abstract

In this paper, various types of exact soliton solutions of the truncated M-fractional Kudryashov–Sinelshchikov equation, a significant fluid surge model, were obtained. This model accounted for density and heat transfer effects while describing the propagation of pressure waves in mixtures of liquid–gas bubbles. By applying the modified $ (G'/G^2) $-expansion method and the extended Sinh–Gordon equation expansion method, we derived new solutions in the forms of trigonometric, hyperbolic, and rational functions. The obtained solutions were illustrated dynamically using 2D, 3D, and contour plots. These results were novel due to the use of a new definition of fractional derivatives. The effect of the fractional derivative on the solutions was demonstrated through 2D plots. To examine the stability of the obtained solutions, stability analysis was performed. Steady-state solutions were derived using modulation instability analysis. The obtained solutions may be useful in various fields of science and engineering.

References

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