Computational study of soliton behavior in the simplified modified form of the Camassa-Holm equation

This study offers an in-depth exploration of traveling wave solutions to the simplified modified Camassa-Holm (SMCH) equation through the application of the modified S-expansion method. Utilizing a traveling wave transformation, the SMCH equation is converted into a nonlinear ordinary differential equation, from which a wide range of exact solutions is systematically obtained. The modified S-expansion method, implemented using the Maple software, proves to be a robust and efficient analytical tool, yielding a variety of soliton solutions such as kink, bright, and dark solitons. To capture the intricate behavior of these solutions, MATLAB is employed to produce detailed 2D, 3D, and contour visualizations that reveal their structural features and propagation dynamics. A comparative assessment with the modified simple equation method and the exp(-ϕ(η))-expansion method highlights the modified S-expansion method's superior accuracy, simplicity, and adaptability to solve nonlinear partial differential equations. Significantly, the method also extends to fractional-order equations, showcasing its broad applicability in nonlinear system analysis. Key solutions are graphically represented under constrained parameter values to emphasize the core propagation features. Overall, this work enhances the current analytical methods to solve both classical and fractional-order nonlinear partial differential equations (PDEs) and offers a valuable foundation for future research into closed-form traveling wave solutions across various disciplines.