Modulational instability and soliton propagation in an alternate right-handed and left-handed multi-coupled nonlinear dissipative transmission network

We investigate analytically the dynamics of modulated waves in an alternate right-handed and left-handed multi-coupled nonlinear dissipative discrete electrical lattice, made of several of the well-known modified Noguchi electrical transmission network that are transversely coupled to one another by a linear capacitor C2. The damped two-dimensional (2D) complex Ginzburg-Landau equation (CGLE) governing slowly modulated wave propagation in this network system is derived. Considering modulated Stokes waves propagating in the network, we establish their criteria of instability, derive the analytical expression of the modulational instability (MI) growth rate, and show that the presence of the dispersive element CS softens the instability of the network system. Through the MI growth rate, we establish that the instability features of our network system show the famous donut Benjamin–Feir instability region. With the help of the linear dispersion law, we show that the behavior of our network model depends on that of the propagating frequency fp. It can adopt purely right-handed (corresponding to monotone increasing fp), purely left-handed (associated to monotone decreasing fp) or composite right-/left-handed (for composite monotone increasing and monotone decreasing fp) behaviors without changing its structure. Also, the network system under consideration supports purely backward (when fp is positive), purely forward (when fp is negative) or composite backward/forward (when fp changes its sign) traveling waves. It appears that the network is right-handed (RH) and/or composite right-/left-handed (RH/LH) and supports backward traveling waves only for very low frequencies and becomes left-handed and supports only forward traveling waves for only high frequencies. Though the exact and approximative soliton-like solutions of the derived CGLE of the network, we investigate analytically RH, LH, as well as RH/LH behaviors of the system and show how to manipulate the transversal wave number to modify the behavior of the network, as well as the width and the motion of the bright solitary voltage signals in the network. We show that our network system supports RH and LH bright and kink soliton signals that simultaneously propagate at the same frequency; we also show that it supports backward and forward LH bright and kink soliton signals which simultaneously propagate at opposite frequency.