An efficient approach is used in this article to solve and analyze the nonlinear autonomous equations. The Tanh method provides more general, precise traveling wave solutions with little additional work. The Tanh technique is demonstrated to be an effective mathematical tool for solving nonlinear autonomous equations and any model equation in mathematical physics. Additionally, the obtained solutions are physically represented and interpreted. Furthermore, the authors present the visual effects of the behavior of wave structures using various figures and contour plots, in the results to enhance our comprehensive understanding.