Investigation of more solitary waves solutions of the stochastics Benjamin-Bona-Mahony equation under beta operator

This study explores the stochastic Benjamin-Bona-Mahony (BBM) equation with a beta derivative (BD), thereby incorporating multiplicative noise in the Itô sense. We derive various analytical soliton solutions for these equations utilizing two distinct expansion methods: the $\frac{{\mathcal{G}}^{\mathrm{\prime }}}{{\mathcal{G}}^{\mathrm{\prime }}+\mathcal{G}+\mathcal{A}}$-expansion and the modified $\frac{{\mathcal{G}}^{\mathrm{\prime }}}{{\mathcal{G}}^2}$-expansion techniques, both within the framework of beta derivatives. A fractional multistep transformation is employed to convert the equations into nonlinear forms with respect to an independent variable. After performing an algebraic manipulation, the solutions are trigonometric and hyperbolic trigonometric functions. Our analysis demonstrates that the wave behavior is influenced by the fractional-order derivative in the proposed equations, thus providing deeper insights into the wave composition as the fractional order either increases or decreases. Additionally, we explore the effect of white noise on the propagation of the waves solutions. This study underscores the computational robustness and adaptability of the proposed approach to investigate various phenomena in the physical sciences and engineering.