Vector solitons and localized waves of two coupled nonlinear Schrödinger equations in the nonlinear electrical transmission line lattice

The study examines modulation instability and localized wave structures in a nonlinear electrical transmission line with next-neighbor couplings. By employing an expansion method, coupled nonlinear Schrödinger equations are derived to analyze the system. The influence of next-neighbor coupling on the perturbed plane wave is highlighted, demonstrating unstable modes arising from modulation instability. Notably, a stronger next-neighbor coupling significantly enhances the amplitude of modulation instability, confirming that the nonlinear electrical lattice supports localized nonlinear waves. Analytical analysis, considering the self-phase modulation parameter, reveals the existence of three types of coupled soliton modes: bright-bright solitons, dark-bright solitons, and bright-dark solitons, influenced by the nearest neighbor coupling. Numerical simulations further illustrate the development of modulation instability through modulated wave patterns. Additionally, at a specific propagation time, another structure is identified, confirming the formation of rogue waves with crests and troughs in the network. These wave phenomena are characteristic of nonlinear systems where dispersion and nonlinearity interact.