Influence of the $\beta$-time fractional derivative on soliton structures and stability in nonlinear polarization-preserving optical fibers

This paper investigates the influence of the $\beta$-time fractional derivative on Kudryashov's formulation of the nonlinear refractive index in polarization-preserving optical fiber systems. The $\beta$-fractional derivative framework offers a more general and memory-inclusive model of wave propagation compared to classical integer-order approaches. Using the improved modified extended tanh method, we derive several exact analytical solutions, including dark solitons, singular periodic solutions, and Jacobi elliptic function solutions. These solutions reveal the rich nonlinear dynamics introduced by the fractional temporal operator and provide insights into the modulation and stability of optical solitons in birefringent media. We demonstrate that the $\beta$-fractional derivative significantly modifies soliton behavior, especially affecting amplitude, width, and propagation speed. The originality of this work lies in introducing the $\beta$-time fractional derivative into the polarization-preserving optical fiber equation with Kudryashov-type nonlinearity-an approach not previously reported-and in obtaining new exact analytical solutions via the improved modified extended tanh method. These solutions, together with a detailed stability analysis, extend the current understanding of fractional-order soliton dynamics in nonlinear optical media.