In this study, we investigate the Lonngren wave equation by means of the generalized the -expansion method to obtain exact traveling-wave solutions for a variety of parameter regimes. This technique produces fresh analytical expressions that reveal the equation’s dynamic behavior. We verify the correctness of these solutions by substituting them back into the original equation. In addition, three-dimensional surface plots and contour diagrams are presented to illustrate the physical characteristics of the resulting waveforms. Overall, our results demonstrate the effectiveness of the generalized expansion approach for solving nonlinear partial differential equations and deepen the theoretical understanding of wave propagation in applied settings.
Koç et al. (Mon,) studied this question.