EXPLORING SOLITON FAMILIES IN $ \beta $ FRACTIONAL COUPLED NLSES WITH VARIABLE COEFFICIENTS USING IMPROVED MODIFIED EXTENDED TANH-FUNCTION METHOD

This research presents a groundbreaking analytical investigation of optical soliton dynamics within the framework of variable-coefficient coupled higher-order nonlinear Schrödinger equation (NLSE) incorporating $ \beta $-fractional derivatives. The study introduces a novel application of the improved modified extended tanh-function (IMETF) method to derive exact analytical solutions that capture the complex interplay between fractional derivatives, variable coefficients, and higher-order nonlinear effects. Our methodology enables the systematic transformation of the intricate fractional NLSE system into mathematically tractable forms, yielding an unprecedented spectrum of soliton solutions including bright, dark, singular, periodic, Jacobi elliptic, rational, and hyperbolic wave structures. The incorporation of $ \beta $-fractional derivatives introduces fundamentally new propagation characteristics and non-local effects that significantly enhance soliton controllability and stability. Through comprehensive numerical simulations and graphical analysis, we demonstrate how fractional-order parameters and variable coefficients collectively govern soliton dynamics, enabling precise manipulation of wave properties in inhomogeneous media. This work makes substantial contributions to both theoretical and applied nonlinear optics by establishing a robust analytical framework for solving complex fractional NLSE systems, revealing novel soliton behaviors with enhanced stability properties, and providing practical insights for advanced applications in optical communications, photonic device engineering, and nonlinear wave control technologies. The results demonstrate the IMETF method's exceptional capability in addressing challenging nonlinear wave phenomena while opening new avenues for soliton manipulation in fractional-order systems with variable parameters.