Fractional Wave Structures in a Higher-Order Nonlinear Schrödinger Equation with Cubic–Quintic Nonlinearity and β-Fractional Dispersion

This study employs the improved modified extended tanh method (IMETM) to derive exact analytical solutions of a higher-order nonlinear Schrödinger (HNLS) model, incorporating β-fractional derivatives in both time and space. Unlike classical methods such as the inverse scattering transform or Hirota’s bilinear technique, which are typically limited to integrable systems and integer-order operators, the IMETM offers enhanced flexibility for handling fractional models and higher-order nonlinearities. It enables the systematic construction of diverse solution types—including Weierstrass elliptic, exponential, Jacobi elliptic, and bright solitons—within a unified algebraic framework. The inclusion of fractional derivatives introduces richer dynamical behavior, capturing nonlocal dispersion and temporal memory effects. Visual simulations illustrate how fractional parameters α (space) and β (time) affect wave structures, revealing their impact on solution shape and stability. The proposed framework provides new insights into fractional NLS dynamics with potential applications in optical fiber communications, nonlinear optics, and related physical systems.