Wave-shape profiles in a coupled inductor-capacitor network with nonlinear dispersion

In this work, the qualitative structures of traveling waves are investigated in a bidimensional inductor-capacitor network with quadratic nonlinear dispersion. Applying the continuum limit approximation, we show that the dynamics of small-amplitude signals in the network can be governed by a (2+1)-dimensional partial differential equation. Using a simple transformation, we reduce the given equation to a nonlinear ordinary differential equation. By means of the phase plane analysis and depending on the wave velocity of the signals that are to propagate in the lattice, we present all phase portraits of the dynamical system. Parametric representations for solitary-wave solutions corresponding to the various phase portrait trajectories under different parameter conditions are derived. The results of our study demonstrate that the nonlinear dispersion in the network leads to a number of interesting solitary-wave profiles, e.g., bright-dark solitons and gray-gray solitons, which have not been observed for the same model when the dispersion is assumed linear. The two-dimensional graphics of all the solutions obtained in this paper are given.