Similarity reduction and novel Jacobi wave solutions for the variable (4+1)-dimensional Fokas equation

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

doi:10.3934/math.20251061

AIMS Mathematics

2025, Volume 10, Issue 10: 23869-23879. doi: 10.3934/math.20251061

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Research article

Similarity reduction and novel Jacobi wave solutions for the variable (4+1)-dimensional Fokas equation

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

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

Received: 12 July 2025 Revised: 19 September 2025 Accepted: 09 October 2025 Published: 21 October 2025
MSC : 35-XX, 35C08

In this study, the (4+1)-dimensional Fokas equation with variable coefficients, which describes water waves in deep and wider channels, was reduced to a sixth-order nonlinear ordinary differential equation using the direct similarity reduction method. Then, the Jacobi expansion method was used to obtain multiple novel types of traveling wave solutions, including solitons, periodic waves, and singular waves. The obtained Jacobi wave solutions were considered new and have never been obtained before. Last, the dynamic behavior of the periodic and soliton wave solutions was explained according to different variable coefficient values and visualized by 3D plots.
- (4+1)-dimensional variable-coefficient Fokas,
- similarity reduction,
- novel Jacobi elliptic wave solutions
Citation: Rehab M. El-Shiekh, Mahmoud Gaballah. Similarity reduction and novel Jacobi wave solutions for the variable (4+1)-dimensional Fokas equation[J]. AIMS Mathematics, 2025, 10(10): 23869-23879. doi: 10.3934/math.20251061

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Abstract

In this study, the (4+1)-dimensional Fokas equation with variable coefficients, which describes water waves in deep and wider channels, was reduced to a sixth-order nonlinear ordinary differential equation using the direct similarity reduction method. Then, the Jacobi expansion method was used to obtain multiple novel types of traveling wave solutions, including solitons, periodic waves, and singular waves. The obtained Jacobi wave solutions were considered new and have never been obtained before. Last, the dynamic behavior of the periodic and soliton wave solutions was explained according to different variable coefficient values and visualized by 3D plots.

References

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