Abstract
In this study, we investigate the soliton solutions of the simplified modified Camassa–Holm equation, which plays a significant role in various applications within mathematical physics and engineering disciplines. By applying an appropriate wave transformation, the equation is reduced to an ordinary differential equation form. To obtain its analytical solutions, we employ the generalized Kudryashov method — a technique widely adopted by researchers for its strong ability to generate rational solutions and encompass a broad variety of functional forms. The application of this method produces diverse types of soliton solutions, including bright, dark, kink, and singular profiles. These solutions are verified through direct substitution and illustrated graphically, revealing distinct behaviors under varying parameter conditions. The results uncover new wave structures not previously reported for the simplified modified Camassa–Holm equation, offering deeper insight into its nonlinear dynamics. The novelty of this study lies in the first successful application of the generalized Kudryashov method to this model, highlighting its efficiency, practicality, and wide applicability to nonlinear differential equations.