Soliton Solutions for a $\beta$-Fractional Highly Dispersive Equation Featuring Kudryashov’s Arbitrary Sextic Power Nonlinearity

Abstract

This study investigates a highly dispersive nonlinear model with an arbitrary refractive index, originally proposed by Kudryashov, which governs soliton propagation in polarization-preserving optical fibers. The model incorporates $\beta$-fractional derivatives, sextic-power nonlinearity, and generalized nonlinear laws, extending its applicability to complex dispersive media. By employing the generalized Kudryashov integration method, we derive exact solutions, including dark solitons and singularity-type solitons, for the governing equation. The obtained solutions are visualized through 2D and 3D plots under carefully selected parameter regimes, providing deeper physical insights into the wave dynamics and nonlinear interactions. The results highlight the impact of fractional-order dispersion and high-order nonlinearity on soliton evolution, offering potential applications in optical communications and nonlinear wave theory.