Bifurcation and solutions of the fractional order Fokas system using reliable approaches

Asim Alawfi; M. B. Almatrafi; Asim Alawfi; M. B. Almatrafi

doi:10.3934/math.20261179

AIMS Mathematics

2026, Volume 11, Issue 9: 29727-29753. doi: 10.3934/math.20261179

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Bifurcation and solutions of the fractional order Fokas system using reliable approaches

Asim Alawfi ¹,
M. B. Almatrafi ^{2
,
,}

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia
2.
Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

Received: 12 May 2026 Revised: 15 August 2026 Accepted: 28 August 2026 Published: 14 September 2026
MSC : 34D05, 34D08, 34D35, 37N25, 37N30, 39A30

This research analyzed the bifurcation structure, stability dynamics, and traveling wave solutions of the nonlinear Fokas system utilizing the D-truncated fractional derivative. Nonlinear pulse transmission in monomode optical fibers was described using the Fokas model. The proposed nonlinear partial differential problem was successfully converted to an ordinary differential equation using a suitable wave transformation. The equilibrium points of the considered model were obtained. The stability and propagation properties of the derived system were shown graphically. In addition, the bifurcation investigation revealed the presence of numerous dynamical domains and critical parameter constraints linked to modifications in the system's qualitative behavior. Some accurate traveling wave solutions were successfully produced by employing a Sardar-subequation method, resulting in a variety of solitary, periodic, and singular wave structures that represented the model's dynamical behavior. Moreover, the qualitative features of the generated solutions were investigated using phase-plane analysis and bifurcation theory. We showed that the reduced amplitude equation, and, hence, its bifurcation structure, does not depend on the fractional orders; the D-truncated operator acted only on the traveling coordinate, rescaling where and when a given profile was realized. Parameter regimes in which each profile was real and bounded were stated explicitly, and only the bounded profiles were interpreted as physically admissible optical pulses.
- Fokas system,
- equilibrium points,
- stability,
- traveling wave solutions,
- bifurcation,
- Sardar-subequation method
Citation: Asim Alawfi, M. B. Almatrafi. Bifurcation and solutions of the fractional order Fokas system using reliable approaches[J]. AIMS Mathematics, 2026, 11(9): 29727-29753. doi: 10.3934/math.20261179

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Abstract

This research analyzed the bifurcation structure, stability dynamics, and traveling wave solutions of the nonlinear Fokas system utilizing the D-truncated fractional derivative. Nonlinear pulse transmission in monomode optical fibers was described using the Fokas model. The proposed nonlinear partial differential problem was successfully converted to an ordinary differential equation using a suitable wave transformation. The equilibrium points of the considered model were obtained. The stability and propagation properties of the derived system were shown graphically. In addition, the bifurcation investigation revealed the presence of numerous dynamical domains and critical parameter constraints linked to modifications in the system's qualitative behavior. Some accurate traveling wave solutions were successfully produced by employing a Sardar-subequation method, resulting in a variety of solitary, periodic, and singular wave structures that represented the model's dynamical behavior. Moreover, the qualitative features of the generated solutions were investigated using phase-plane analysis and bifurcation theory. We showed that the reduced amplitude equation, and, hence, its bifurcation structure, does not depend on the fractional orders; the D-truncated operator acted only on the traveling coordinate, rescaling where and when a given profile was realized. Parameter regimes in which each profile was real and bounded were stated explicitly, and only the bounded profiles were interpreted as physically admissible optical pulses.

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