Soliton structures, bifurcation patterns, and chaos in a nonlinear transmission line governed by the modified complex Ginzburg–Landau equation

This paper investigates the nonlinear dynamics of solitary electrical signals propagating along a nonlinear transmission line governed by the modified complex Ginzburg–Landau equation. By applying Lie symmetry analysis, the governing partial differential equation is reduced to a system of ordinary differential equations, enabling the derivation of exact traveling wave solutions. Two advanced analytical techniques, the new Kudryashov method and the generalized Arnous method, are employed to construct bright, singular, and rational soliton solutions. A comprehensive bifurcation analysis is conducted to classify equilibrium points and explore their stability through phase portraits. Additionally, modulational instability analysis is used to identify the conditions under which continuous wave backgrounds become unstable, offering theoretical support for soliton formation. To examine chaotic behavior, external perturbations of various forms including trigonometric, Gaussian, and hyperbolic types are introduced into the reduced system. The presence of chaos is confirmed through the computation of Lyapunov exponents, analysis of phase portraits, and application of the Melnikov method. The study provides new insights into the interplay between soliton dynamics, bifurcation structures, and perturbation-induced chaos in nonlinear electrical transmission networks.