Existence and dynamics of solitary waves in a two-dimensional Noguchi nonlinear electrical network

Abstract

In this work, we investigate solitary waves in a nonlinear two-dimensional discrete electrical lattice. It is made of several of the well-known Noguchi electrical transmission lines, that are transversely or longitudinally coupled to one another by an inductor ${L}_{2}$ or ${L}_{1}$ and a capacitor ${C}_{2}$ or ${C}_{1}$ mounted in parallel. The linear dispersion law of the network is given and the effects of the transverse coupling elements ${L}_{2}$ and ${C}_{2}$ on the allowed bandwidth frequencies are examined. Using the continuum limit approximation, we show that the dynamics of the small amplitude signals in the network can be governed by a (2+1)-dimensional generalized modified Zakharov-Kuznetsov equation. The fixed points of our model equation are examined and the bifurcations of its phase portrait are presented, as functions of the wave velocity of the signals that are to propagate in lattice. Likewise, we derive exact explicit solutions that are possible under different wave velocities and for physically realistic values of the network's parameters. These include pulse, kink, and anti-kink wave solutions and correspond to some special level curves of the first integral of the model equation. We find out that the transverse coupling parameters considerably affect the characteristics of the waves that are propagated throughout the system. Direct numerical simulations are also performed on the exact equations of the network and the results are in agreement with the analytical predictions.