Phase portrait, sensitivity and chaotic analysis, variational principle, Hamiltonian and abundant wave solutions of the pulse narrowing transmission line model

Purpose The purpose of this paper is to give a detailed and in-depth research on the pulse narrowing transmission lines model qualitatively and quantitatively. Design/methodology/approach Applying the traveling wave transformation and semi-inverse method, the variational principle is established, and the Hamiltonian is extracted. The planar dynamical system is derived, then the phase portraits are plotted, and the bifurcation analysis is presented to discuss the existence conditions of the wave solutions with the different shapes. Furthermore, the chaotic phenomenon is probed via introducing the perturbed term, and the sensitivity analysis is given. Finally, the Hamiltonian-based method that is based on the energy conservation, as well as the variational method that originates from the variational principle and Ritz method, are adopted to develop the different wave solutions. Findings According to the theory of planar systems, the discussion on the existence conditions of the wave solutions with different wave shapes reveals that the considered model has bell-shaped solitary wave and periodic wave solutions. Meanwhile, it is found that the small changes in initial conditions can have a significant impact on the behavior of the solution. Furthermore, diverse wave solutions such as bell-shaped solitary waves, anti-bell-shaped solitary waves and periodic wave solutions are obtained by the variational method and Hamiltonian-based method, which are consistent with the discussion on the conditions for the existence of wave solutions with the different wave shapes. Originality/value To the best of the authors’ knowledge, the variational principle and Hamiltonian are first reported. The variational method and Hamiltonian-based method are first used to probe the diverse wave solutions. The qualitative analysis is presented to explore the dynamics of the model. The findings of this research are hoped to open some new perspectives toward the dynamics of the model under consideration.