Exact Solutions and Stability Thresholds for the Fractional Gardner Equation with High-Order Dispersion

Authors

  • Wafaa B. Rabie Department of Basic Sciences, Higher Institute of Engineering and Technology, Menoufia, Egypt,
  • Hadeel Bin Amer Department of Computer Science, College of Computer and Information Sciences, Majmaah University, Al-Majmaah 11952, Saudi Arabia
  • Hasib Khan Prince Sultan University, Riyadh https://orcid.org/0000-0002-7186-8435
  • Jehad Alzabut Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
  • Dalia l. Elimy Department of Basic Science, Higher Institute of Engineering and Technology, Tanta, Egypt.

DOI:

https://doi.org/10.29020/nybg.ejpam.v19i1.6805

Keywords:

Fractional Gardner equation; Higher-order dispersion effects; mEDA method; Multi-waveform solutions; Fractional parameter analysis

Abstract

This study investigates the fractional Gardner equation with high-order dispersion, a fundamental model for nonlinear wave propagation in plasmas, optical fibers, and fluid systems. Using the modified extended direct algebraic (mEDA) method, we derive exact analytical solutions including bright/dark solitons, singular waves, and periodic patterns. All solutions have been
verified numerically using Mathematica to ensure their validity, as process innovation. The analysis reveals that the fractional order β significantly influences wave decay rates and memory effects, while specific parameter constraints govern solution existence and stability. A comprehensive linear stability analysis examines the modulation instability of the obtained solutions, revealing distinct regimes of marginal stability, instability, and stability based on dispersion relation characteristics.
Physically, these solutions model wave phenomena in nonlinear optics, Bose–Einstein condensates, and oceanic systems, with the fractional order β providing crucial insights into non-local and memory-dependent processes. The stability analysis provides essential insights for practical wave manipulation applications. The combined analytical and stability approaches offer significant value for understanding nonlinear wave dynamics across various physical contexts.

Downloads

Published

2026-02-16

Issue

Vol. 19 No. 1: (January 2026)

Section

Mathematical Analysis

License

Copyright (c) 2026 Wafaa B. Rabie, Hadeel Bin Amer, Hasib Khan, Jehad Alzabut, Dalia l. Elimy

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Upon acceptance of an article by the European Journal of Pure and Applied Mathematics, the author(s) retain the copyright to the article. However, by submitting your work, you agree that the article will be published under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). This license allows others to copy, distribute, and adapt your work, provided proper attribution is given to the original author(s) and source. However, the work cannot be used for commercial purposes.

By agreeing to this statement, you acknowledge that:

  • You retain full copyright over your work.
  • The European Journal of Pure and Applied Mathematics will publish your work under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).
  • This license allows others to use and share your work for non-commercial purposes, provided they give appropriate credit to the original author(s) and source.

How to Cite

Exact Solutions and Stability Thresholds for the Fractional Gardner Equation with High-Order Dispersion. (2026). European Journal of Pure and Applied Mathematics, 19(1), 6805. https://doi.org/10.29020/nybg.ejpam.v19i1.6805