First and second order rogue waves dynamics in a nonlinear electrical transmission line with the next nearest neighbor couplings

In this paper, we study the influence of second neighbors interactions on a one-dimensional nonlinear electrical transmission line. Using the reductive perturbation method in the semi-discrete limit, we show how wave dynamics in the network can be governed by a cubic-quintic nonlinear Schrödinger equation (CQ-NLSE). The introduction of second neighbors couplings increases the group velocity of the waves as well as the number of regions of modulational stability/instability. By means of a transformation, the CQ-NLSE is converted into the Kundu–Eckhaus equation. Using the modified Darboux transformation, we find first and second order rational solutions of the Kundu–Eckhaus equation. Our results reveal that only the β0 parameter responsible for quintic nonlinear effects and nonlinear dispersion affects the rogue waves velocity when second neighbors are neglected while taking into account their contribution, we notice that in addition to β0, the wave number k also influences the waves velocity. Moreover, we also show that the rogue waves energy increases with the dispersive capacity CS between first neighbors, while it decreases in the presence of second neighbors.