Multistructure Novel Stochastic Soliton Solutions for the (2+1)-Dimensional Breaking Wave Equation Within the Fractional Derivative Influence

In this paper, we investigate the exact stochastic solutions of the (2 + 1)-dimensional stochastic fractional-space breaking soliton equation (SFSBSE) involving the truncated M-fractional derivative in space. This equation models a range of physical phenomena, including fluid wave propagation, shallow water dynamics, and plasma physics, under the influence of stochastic fluctuations. To derive analytical solutions, we apply the Kumar–Malik approach, which transforms the SFSBSE into an ordinary differential equation via a stochastic wave transformation. A wide class of exact solutions is obtained, including Jacobi elliptic, hyperbolic, trigonometric, and exponential function solutions. The effects of multiplicative noise and the fractional-order operator on the solution behavior are analyzed both analytically and graphically. Several 3D, 2D, and contour plots are provided to illustrate the qualitative dynamics of the stochastic soliton profiles. In particular, we highlight the transition from deterministic to stochastic behavior as the noise intensity increases and emphasize the role of the fractional order in modulating the soliton structure. To the best of our knowledge, this is the first time the Kumar–Malik method has been implemented to obtain exact stochastic solutions of the SFSBSE with a truncated M-fractional derivative. The correctness of all obtained solutions is verified symbolically using Maple. The presented framework is expected to provide new insights into the dynamics of stochastic fractional soliton models.