  Citation: Elmandouh, A.; Fadhal, E. Bifurcation of Exact Solutions for the Space-Fractional Stochastic Modified Benjamin–Bona–Mahony Equation. Fractal Fract. 2022, 6, 718. https:// doi.org/10.3390/fractalfract6120718 Academic Editors: Asifa Tassaddiq, Muhammad Yaseen and Sania Qureshi Received: 7 November 2022 Accepted: 30 November 2022 Published: 2 December 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). fractal and fractional Article Bifurcation of Exact Solutions for the Space-Fractional Stochastic Modified Benjamin–Bona–Mahony Equation Adel Elmandouh 1,2, * and Emad Fadhal 1 1 Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt * Correspondence: aelmandouh@kfu.edu.sa Abstract: This paper studies the influence of space-fractional and multiplicative noise on the exact solutions of the space-fractional stochastic dispersive modified Benjamin–Bona–Mahony equation, driven in Ito’s sense by a multiplicative Wiener process. The bifurcation of the exact solutions is investigated, and novel fractional stochastic solutions are presented. The dependence of the solutions on the initial conditions is discussed. Due to the significance of the fractional stochastic modified Benjamin–Bona–Mahony equation in describing the propagation of surface long waves in nonlinear dispersive media, the derived solutions are significantly more helpful for and influential in comprehending diverse, crucial, and challenging physical phenomena. The effect of the Wiener process and the fractional order on the exact solutions are studied. Keywords: bifurcation; phase portrait; stochastic equations; conformable fractional derivative 1. Introduction Many problems in the world can be identified through physical and mathematical models. It has been shown that models are practically related to nonlinear partial dif- ferential equations (NPDEs), which can be used to describe many real-life phenomena. Several mathematical models have been determined for physical processes in the field of scientific research, and this has led to the comprehensive study of the behaviors of nonlinear waves receiving attention. The category of shallow water equations is one of the most important tools for modeling the behavior of waves, which has been applied and integrated with the mathematical process of the study of atmospheric and oceanic models. Several phenomena of shallow water waves have been obtained from NPDEs [1–4]. For more detail related to the equations closely connected to the developing topics, we direct the reader to [5–9]. Closed-form solutions for nonlinear partial differential equations (PDEs) are crucial for understanding intricate phenomena. In this regard, it is necessary to find wave solutions, in particular. Researchers have presented new methods and refined exist- ing approaches. Various significant methods have been introduced, such as the Darboux transformation [10], Weierstrass elliptic functions methods [11,12], Bäcklund transforma- tion [13], Lie group [14–17], Hirota’s method [18,19], and the bifurcation method [20–26]. The analytical and numerical solutions for various types of NPDEs have been investigated using traditional Lie symmetry approaches; see, for instance [27]. Numerous branches of science, including physics and engineering, have emphasized the benefits of taking random effects into account when analyzing, simulating, and mod- eling complicated processes. The reason for this is that the noise may provide statistical characteristics and significant phenomena that cannot be ignored [28–31]. Further, when stochastic terms are introduced to PDEs, exact solutions are more difficult to find than deterministic PDEs. Models based on fractional derivatives have been successful in de- scribing nonlinear physical phenomena. A continuing interest in fractional calculus resides in its applicability. In fields such as physics, mechanics, chemistry, and biology, fractional calculus is extremely useful [32–35]. Fractal Fract. 2022, 6, 718. https://doi.org/10.3390/fractalfract6120718 https://www.mdpi.com/journal/fractalfract