Bifurcation analysis and novel traveling wave solutions for the (4+1)-dimensional Fokas equation with variable coefficients

Rehab M. El-Shiekh; Mahmoud Gaballah; Rehab M. El-Shiekh; Mahmoud Gaballah

doi:10.3934/math.2026346

AIMS Mathematics

2026, Volume 11, Issue 3: 8417-8427. doi: 10.3934/math.2026346

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Bifurcation analysis and novel traveling wave solutions for the (4+1)-dimensional Fokas equation with variable coefficients

Rehab M. El-Shiekh ^{1
,
,},
Mahmoud Gaballah ^2,3

1.
Department of Business Administration, College of Business Administration in Majmaah, Majmaah University, Al-Majmaah City, 11952, Saudi Arabia
2.
Department of Physics, College of Science at Al-Zulfi, Majmaah University, Al-Majmaah City, 11952, Saudi Arabia
3.
Geomagnetic and Geoelectric Department, National Research Institute of Astronomy and Geophysics (NRIAG), 11421 Helwan, Cairo, Egypt

Received: 04 January 2026 Revised: 07 March 2026 Accepted: 16 March 2026 Published: 30 March 2026
MSC : 35-XX, 35C08

In this paper, the variable-coefficient (4+1)-dimensional Fokas (4D-vc Fokas) equation, which describes the evolution of water waves with surface tension in ocean dynamics, is studied using bifurcation analysis. The dynamic system and phase portrait are presented and discussed graphically. Then, the whole discrimination system of the 4D-vc Fokas equation was discussed to classify the possible analytic traveling wave solutions; as a result, novel solitary waves and periodic waves were yielded by setting specific relationships between the coefficients. The soliton wave's motion was affected by the choices of the variable coefficients and took a parabolic and periodic shape; it also became a bright or dark soliton.
- (4+1)-dimensional variable-coefficient Fokas equation,
- bifurcation analysis,
- discrimination system,
- traveling waves
Citation: Rehab M. El-Shiekh, Mahmoud Gaballah. Bifurcation analysis and novel traveling wave solutions for the (4+1)-dimensional Fokas equation with variable coefficients[J]. AIMS Mathematics, 2026, 11(3): 8417-8427. doi: 10.3934/math.2026346

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Abstract

In this paper, the variable-coefficient (4+1)-dimensional Fokas (4D-vc Fokas) equation, which describes the evolution of water waves with surface tension in ocean dynamics, is studied using bifurcation analysis. The dynamic system and phase portrait are presented and discussed graphically. Then, the whole discrimination system of the 4D-vc Fokas equation was discussed to classify the possible analytic traveling wave solutions; as a result, novel solitary waves and periodic waves were yielded by setting specific relationships between the coefficients. The soliton wave's motion was affected by the choices of the variable coefficients and took a parabolic and periodic shape; it also became a bright or dark soliton.

References

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