New Solutions of Breaking Soliton Equation Using Softmax Method

This study presents the application of a novel Softmax method to obtain exact analytical solutions of the breaking soliton equation, a nonlinear partial differential equation that models complex wave phenomena. By transforming the governing equation into an ordinary differential equation using a traveling wave transformation and constructing a solution framework based on the softmax function, the method yields exact solutions in hyperbolic, trigonometric, exponential, rational, and polynomial forms. The derived solutions include diverse wave structures such as kink-lump solitons, rogue waves, and affine polynomial profiles, showcasing the method’s flexibility and capacity to capture complex nonlinear dynamics. The softmax method integrates classical techniques with modern symbolic computation and exhibits strong compatibility with algebraic solvers like Maple. Graphical simulations of the resulting wave profiles validate the accuracy and diversity of the solutions and highlight the potential of the method for applications in nonlinear wave theory, soliton dynamics, and computational mathematics. Compared to existing methods, the softmax approach provides an elegant, scalable, and exact mechanism for solving high-dimensional nonlinear systems. This framework may be extended to broader classes of integrable and nonintegrable PDEs in future work.