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

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 beenverified 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.