Invariant solutions and bifurcation analysis of the nonlinear transmission line model

In this paper, the nonlinear transmission line model with the power law nonlinearity and the constant capacitance and voltage relationship is studied using Lie symmetry analysis. Corresponding to the infinitesimals obtained, using commutation relations, abelian and non-abelian Lie subalgebras are obtained. Also, using the adjoint table, a one-dimensional optimal system of subalgebra is presented. Based on this optimal system, the corresponding Lie symmetry reductions are obtained. Moreover, variety of new similarity solutions in the form of trigonometric functions, hyperbolic functions, are obtained. Corresponding to one similarity reduction, by bifurcation analysis of dynamical system, the stable and unstable regions are determined, which show the existence of soliton solutions from the nonlinear dynamics point of view. Some of the obtained solutions represented graphically and observations are also discussed.