Abstract
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.
Highlights
In physics, nonlinearity is observed in many areas of physics [2, 10, 11, 13, 18,19,20,21, 25, 28, 31, 32]
To understand the nonlinear phenomena, which are described by nonlinear parial differential equations (NLPDEs), we need to obtain their exact solutions
Among the methods in literature, the Lie symmetry method [16, 22, 26, 27, 29] is one of the most effective methods for finding the exact solutions of NLPDEs by which it is possible to reduce the number of independent variables and further to reduce the order of ordinary differential equations, making them easier to solve
Summary
Nonlinearity is observed in many areas of physics [2, 10, 11, 13, 18,19,20,21, 25, 28, 31, 32]. Work has been done to study the exact solutions of these NLPDEs. Some of them include the Lie symmetry method [22, 26], Direct method for symmetries [8], Non-classical symmetry method [3], Backlund transformation method [9], solitary wave ansatz method [4], Hirota’s bilinear method [6, 15, 30], the modified simple equation method [17], the. Among the methods in literature, the Lie symmetry method [16, 22, 26, 27, 29] is one of the most effective methods for finding the exact solutions of NLPDEs by which it is possible to reduce the number of independent variables and further to reduce the order of ordinary differential equations, making them easier to solve. We consider a non-linear transmission line [12] of the following form: L
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