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Abstract In this article, the modified Kudryashov and extended simple equation methods are employed to obtain analytical solutions for the Zhiber–Shabat problem. The outcomes of this study clearly indicate that the provided methodologies are appropriate techniques for generating some new exact solutions for nonlinear evolution equations. Furthermore, the nature of the solutions would be presented in three dimensions for various parameters applying the most advanced scientific instruments. The physical behavior of the solutions are graphically displayed, and it is established that the acquired solutions are newly constructed in the form of bright, dark, optical, singular, and bell-shaped periodic soliton wave structures. The properties of the nonlinear model have been illustrated using 3D, 2D, and contour plots by selecting an appropriate set of parameters, which is demonstrated to visualize the physical structures more productively. Finally, it is concluded that similar strategies can also be implemented to study many contemporary models. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the governing model appearing in diverse disciplines. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research.
Alhefthi et al. (Mon,) studied this question.