Optimal system, dynamical behaviors and exact solution of a nonlinear transmission line model by applying the Lie symmetry method

Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunk lining routing calls between telephone switching centers, computer network connections and high speed computer data buses. In this paper, we seek the solutions of a nonlinear transmission line model by applying the Lie symmetry method. Corresponding to the optimal system of Lie subalgebras, similarity reductions and a variety of new exact solutions in the form of trigonometric functions and hyperbolic functions are obtained. Further, power series solution is obtained, and the convergence of the power series solution is also shown. Corresponding to one similarity reduction, by bifurcation of dynamical system, the stable and unstable regions are determined, which shows the existence of soliton solutions from the nonlinear dynamics view point.