Exploring the truncated M-fractional exact soliton solutions and modulation instability of the Heimburg model

Haitham Qawaqneh; Abdulaziz Khalid Alsharidi; Haitham Qawaqneh; Abdulaziz Khalid Alsharidi

doi:10.3934/math.2026333

AIMS Mathematics

2026, Volume 11, Issue 3: 8104-8133. doi: 10.3934/math.2026333

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Exploring the truncated M-fractional exact soliton solutions and modulation instability of the Heimburg model

Haitham Qawaqneh ¹,
Abdulaziz Khalid Alsharidi ^{2
,
,}

1.
Department of Basic Science, Al-Zaytoonah University of Jordan, Amman 11733, Jordan
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia

Received: 01 January 2026 Revised: 16 February 2026 Accepted: 24 February 2026 Published: 26 March 2026
MSC : 35Q51, 35Q92, 35R11, 35R60, 60H15

This research investigated the different types of exact soliton solutions of the biomathematics model known as the truncated M-fractional Heimburg model. This concerned equation describes the transmission of electromechanical pulses in nerves as well as to describe the flow of blood through blood vessels. For our purpose, we used the modified extended tanh function scheme and the modified $ (G'/G^2) $-expansion scheme. The obtained exact soliton solutions were the periodic, kink, singular, dark, bright, and other soliton solutions. The obtained solutions were dynamically explained by using 2D, 3D, and contour graphs. The obtained results were nonexistent due to the use of a novel definition of fractional derivatives. Both schemes were not used for the concerned model in the literature. The effect of the fractional derivative on the solutions was explained by using 2D graphs. Next we gained the steady-state solutions with the help of modulation instability analysis. The obtained solutions are useful for various purposes like blood flow simulation, vascular disease modeling, hemodynamic analysis, medical device design, physiological research, etc.
- Heimburg model,
- novel analytical techniques,
- dynamical analysis,
- modulation instability,
- exact solitons
Citation: Haitham Qawaqneh, Abdulaziz Khalid Alsharidi. Exploring the truncated M-fractional exact soliton solutions and modulation instability of the Heimburg model[J]. AIMS Mathematics, 2026, 11(3): 8104-8133. doi: 10.3934/math.2026333

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Abstract

This research investigated the different types of exact soliton solutions of the biomathematics model known as the truncated M-fractional Heimburg model. This concerned equation describes the transmission of electromechanical pulses in nerves as well as to describe the flow of blood through blood vessels. For our purpose, we used the modified extended tanh function scheme and the modified $ (G'/G^2) $-expansion scheme. The obtained exact soliton solutions were the periodic, kink, singular, dark, bright, and other soliton solutions. The obtained solutions were dynamically explained by using 2D, 3D, and contour graphs. The obtained results were nonexistent due to the use of a novel definition of fractional derivatives. Both schemes were not used for the concerned model in the literature. The effect of the fractional derivative on the solutions was explained by using 2D graphs. Next we gained the steady-state solutions with the help of modulation instability analysis. The obtained solutions are useful for various purposes like blood flow simulation, vascular disease modeling, hemodynamic analysis, medical device design, physiological research, etc.

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