Bifurcation of solitary and periodic waves of an extended cubic-quintic Schrödinger equation with nonlinear dispersion effects governing modulated waves in a bandpass inductor-capacitor network

The present work describes the behavior of solitary and periodic waves in a nonlinear electrical transmission line with linear dispersion. Based on the semidiscrete approximation, we show that the dynamics of modulated wave in the system can be described by an extended cubic-quintic nonlinear Schrödinger equation. Using a simple transformation, we reduce the given equation to a cubic-quintic Duffing oscillator equation. By means of the method of dynamical systems, we obtain bifurcations of the phase portraits of the traveling wave under different parameter conditions. Corresponding to the various phase portrait trajectories, we derive possible exact explicit parametric representations of solutions. The results of our study demonstrate that the additional imprint phase in the signal voltage leads to a number of interesting solitary-wave solutions, e.g., gray soliton and anti-gray soliton, which have not been observed for the same model without this parameter. These new obtained solutions are useful in better understanding of the dynamic of the considered network as well as of other systems that can be governed by a cubic-quintic nonlinear Schrödinger equation model.