Analytical soliton solutions and stability analysis of the (3 + 1)-dimensional fractional nonlinear Schrödinger equation using advanced expansion methods

The fractional nonlinear Schrödinger equation (FNLSE) describes the wave propagation in nonlinear optical fibers, ion-acoustic waves in plasmas, propagation of ultrashort laser pulses, and signal processing. In this study, we investigate the (3 + 1)-dimensional FNLSE with beta fractional derivative to explore soliton dynamics in diverse physical and engineering domains. A couple of analytical approaches, namely the extended sinh-Gordon expansion method and the two-variable (G′/G, 1/G)-expansion method, are used to determine the assorted soliton solutions expressed in trigonometric, hyperbolic, and rational forms. The obtained solutions include bright, bell-shaped, anti-peakon-shaped, anti-bell-shaped, periodic, singular, and singular periodic solitons. We examine the impact of the beta fractional parameter on soliton dynamics through graphical simulations and test the stability analysis using linear stability theory. We also conduct bifurcation analysis to investigate the qualitative behavior of the derived solutions, highlighting the emergence of static solitons via saddle–center bifurcation. The results indicate that the introduced methods generate a wide range of soliton structures and provide insights into their stability and dynamic properties. The results contribute to the theoretical understanding of soliton propagation in nonlinear optical fibers and other dispersive media.