A new solitary wave solution of the fractional phenomena Bogoyavlenskii equation via Bäcklund transformation

In this paper, we use the Riccati–Bernoulli sub-ODE method in conjunction with the Bäcklund transformation to find out the exact solutions of the nonlinear time–space fractional Bogoyavlenskii equation. The obtained solutions encompass multiple kink solitary wave solutions that are quite unique and important in addition to solutions presented in hyperbolic, trigonometric, and rational function forms. This equation describes central factors influencing its behavior including fluid dynamics in shallow water waves and plasma, which demonstrates our conclusions have broad applications for such systems. We also study the effect of the fractional order parameter ( $\alpha$) on solutions and plot their behavior using MATLAB in two dimensions. This work also contributes to the knowledge of the physical structures of the fractional Bogoyavlenskyi equation apart from showcasing the potential of the Riccati–Bernoulli sub-ODE method when applied to nonlinear fractional differential equations.