Modeling and analysis of new rogue wave solutions in a nonlinear left-handed electrical transmission line with the Josephson junction

In electrical transmission systems, rogue waves, which are nonlinear phenomena, manifest as sudden and unpredictable voltage and current spikes. In this study, we propose new solution models for rogue waves propagating in a nonlinear electrical transmission line with a Josephson junction. To model the transmission line behavior and analyze the conditions for rogue wave formation, we used an approach based on nonlinear discrete differential equations and the Manakov system, and to formulate the expression for the modulational instability gain spectrum, linear stability was used. Our results show that the Josephson junction and the left-handed nature of this structure play crucial roles in the formation and propagation of these new rogue wave patterns and that the parameters of the Josephson junction and the electrical transmission line can be adjusted to control the occurrence of these phenomena. The Benjamin-Feir instability flourished in the electrical transmission line, and for decreasing the effects of the Josephson junction, the unstable modes increased. We observe an exponential growth of the continuous wave that appears for large values of the perturbed wavenumber, confirming that this structure is open to new coherent localized wave models. We also discuss the implications of our results for the design and management of nonlinear electrical-transmission systems.