Abstract
In this study, solitary wave solutions of the Lonngren wave equation (a crucial nonlinear model describing signal propagation in electrical and telegraph lines) are investigated. The model is first transformed into a nonlinear ordinary differential equation using a traveling wave transformation. For the solutions, the sub-equation of an auxiliary equation method (SAEM26) and the Kudryashov auxiliary equation method (KAEM) are applied for the first time in the literature. Using a customized symbolic algorithm, an algebraic system involving both model and method parameters is solved. Based on these solutions, various novel soliton types are derived, including bright and dark solitary waves and singular solutions. Furthermore, trigonometric-type periodic solutions are obtained through a Galilean transformation. To deeply explore the model’s dynamics, the focus shifts to comprehensive qualitative analyses. Extensive studies, including phase portrait generation, chaotic behavior identification, sensitivity analysis, and modulation instability analysis, are performed. These analyses are supported by striking 2D and 3D graphical visualizations, offering a profound insight into the system’s dynamics and stability. Crucially, the study demonstrates that the proposed methods (SAEM26 and KAEM) are highly effective and flexible in generating diverse wave structures and uncovering complex nonlinear behaviors. It also provides a strong indication of the methods’ potential for future successful applications to other nonlinear evolution equations, including their stochastic and fractional forms.