https://doi.org/10.1140/epjd/s10053-025-01101-y
Research - Plasmas
The Tantawy technique for analyzing fractional Gardner equation and modeling fractional ion-acoustic solitary waves in electronegative plasmas
1
Department of Physics, Faculty of Science, Al-Baha University, P.O. Box 1988, Al-Baha, Saudi Arabia
2
Department of Physics, Faculty of Science, Port Said University, Port Said, 42521, Port Said, Egypt
3
Fizmako Research Group, Universidad Nacional de Colombia, 111321, Bogotá, Colombia
4
Department of Physics, Government Post Graduate College Nowshera, 24100, Nowshera, Pakistan
5
Department of Physics, Government Post Graduate College Mardan, 23200, Mardan, Pakistan
6
Department of Physics, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, 11671, Riyadh, Saudi Arabia
a tantawy@sci.psu.edu.eg, samireltantawy@yahoo.com
Received:
22
August
2025
Accepted:
20
November
2025
Published online:
17
December
2025
Building on our previous explorations of fractional KdV-solitary waves [Part (I)] ( El-Tantawy in Braz J Phys 55:123, 2025) and fractional modified KdV (mKdV)-solitary waves [Part (II)] (El-Tantawy in Braz J Phys 55:176, 2025), this research now explores the complex realm of fractional Gardner-solitary waves (FGSWs) in unmagnetized electronegative plasmas (ENPs) with nonthermal electrons. First, the reductive perturbation technique is applied to reduce the fluid model equations to the integer cubic-quadratic nonlinear Gardner/Extended KdV (EKdV) equation. This equation can describe the propagation of various nonlinear structures (e.g., solitary waves (SWs) and shock waves) in different plasma models and fluids, especially when the quadratic nonlinearity coefficient is close to zero. The second goal of the current study is to investigate the characteristics of fractional nonlinear structures that can arise and propagate in the current model. For this purpose, a novel technique, namely the Tantawy technique (TT), is applied to analyze the planar fractional EKdV (FEKdV) equation and model FEKdV-SWs. This approach produces accurate and stable analytical approximations. Additionally, the FEKdV equation is analyzed using the new iterative method (NIM) within the Laplace transform framework to compare its results with those of TT. To assess the accuracy of all generated approximations, the absolute error against the exact solution for the integer case is numerically estimated. Moreover, we numerically investigate the impact of different plasma parameters—negative ion concentration, ion mass ratio, and the fractional parameter—on the characteristic behavior of the FEKdV-SWs. This research provides important details about fractional nonlinear structures that can exist and propagate in laboratory, space, and astrophysical plasma systems.
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Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
