Analytical insights and physical behavior of solitons in the fractional stochastic Allen-Cahn equations using a novel method.

This study investigates the space-time fractional stochastic Allen-Cahn (STFSAC) equation, a novel extension of the classical Allen-Cahn equation incorporating fractional derivatives and stochastic noise. The model is designed to capture soliton dynamics in complex systems where non-local interactions and randomness are critical, such as plasma physics and materials science. For the first time, we propose the fractional extended sinh-Gordon method (FESGM) and employ the modified [Formula: see text]-expansion method (MGM) to derive exact analytical soliton solutions. Our results demonstrated that noise intensity and fractional parameters significantly influence soliton amplitude, stability, and pattern formation, with increasing stochasticity leading to more complex behavior. The FESGM offered a robust framework for handling fractional stochastic systems, while the MGM provided complementary insights into nonlinear dynamics. The findings were validated through 2D and 3D visualizations, highlighting the interplay between fractional effects and noise. This work advances the understanding of soliton behavior in stochastic fractional systems and provides a foundation for applications in nonlinear optics, disordered media, and phase transitions.