Dynamics of Low-Pass Electrical Transmission Lines: Lie Symmetries, Solitons, Bifurcations, and Modulation Instability

Lie symmetry analysis is an effective method for solving differential equations and simplifying them by reducing their nonlinearity and order. Symmetries have a crucial role not only in differential equations but also in different scientific fields. In this study, we apply Lie symmetry analysis to investigate the symmetries of the low-pass electrical transmission lines model. Additionally, we derive the transformed equivalent forms of the equation and obtain some invariant solutions. The article discusses the discovery of novel soliton wave solutions and their propagation characteristics within nonlinear low-pass electrical transmission lines. By employing a new mapping method, the study establishes various types of soliton solutions for the nonlinear evolution equations governing the behavior of nonlinear low-pass electrical transmission lines. The governed equation in this study can be used to improve signal transmission in optical fiber communication, electrical networks, power nonlinear signal filtering and grid stability. The study also incorporates a bifurcation analysis to explore different solution branches of the governing model, shedding light on the system’s complex dynamics. By the Hamiltonian system, the study investigates phase portraits that illustrate the flow of solutions. The linear stability analysis is also examined to check the modulation instability of the solitons. This analysis is significant for understanding the small disturbances in the system that affect the wave propagation. The results of the study are visually presented via 3D plots, 2D and contour plots generated with Maples software. These visual aids validate the analytical findings and provide a clear representation of the wave behavior in nonlinear low-pass electrical transmission lines. The soliton solutions derived are new and contribute to a deeper understanding of nonlinear wave phenomena in electrical transmission lines.