Exploring soliton solutions and interesting wave-form patterns of the (1 + 1)-dimensional longitudinal wave equation in a magnetic-electro-elastic circular rod.

In this work, we use two newly powerful and efficient techniques, the unified method and the unified Riccati equation expansion (UREE) method, to derive solitary wave solutions for the (1 + 1)-dimensional longitudinal wave equation (LWE), presenting a systematic approach to reveal the underlying dynamics. By incorporating mathematical analysis and numerical simulations, we investigate the behavior and properties of these solitary wave solutions in various fields of science. The derived solutions are entirely new, and these findings show the essential complexities of the longitudinal wave equation. These obtained solutions shed light on the fundamental mechanisms governing solitary waves in the longitudinal wave equation, contributing to a deeper understanding of nonlinear wave phenomena. We present effective visualizations of the dynamical wave structures using various graphs, such as 3D, 2D, and contour plots in the obtained solutions to improve our comprehensive understanding. Consequently, numerous types of wave profiles, including periodic wave, interaction between periodic wave and kink wave, W-shape wave, kink wave, interaction between breather wave and kink wave, traveling wave, mixed periodic, singular soliton, and some new types of soliton profiles were found. Consequently, this work presents a comprehensive study of solitary wave solutions using the unified method and UREE method for the (1 + 1)-dimensional LWE equation. The outcomes of the current study manifest that the considered methods are significant and systematic in solving nonlinear evolution equations. [ABSTRACT FROM AUTHOR]