In this paper, the exact explicit traveling wave solutions to the simplified modified Camassa–Holm (SMCH) equation. The SMCH equation is a significant nonlinear evolution equation because it can be used to describe certain physical processes in oceanography and fluid dynamics. The SMCH equation has important applications in mathematics, physics, and engineering. Using the modified auxiliary equation method and the extended \(\left( \frac{G'}{G^{2}}\right)\) -expansion method, we achieve precise solutions with traveling wave behavior. The proposed methods are applied for the first time in this work to examine the considered mathematical model. The solutions are extracted in terms of trigonometric, hyperbolic, and rational functions. The obtained results not only confirm the previously reported solutions of SMCH equation but also offer some new results. To clearly illustrate the physical implications of the examined equation, we provide graphical representations using 3D, 2D and contour plots for certain values of the parameters. The illustrations help to clarify the diverse characteristics and behaviors, associated with the solutions, providing useful insights for researchers as well as practitioners across a variety of scientific and technical disciplines.
