Dynamics of rogue waves and modulational instability with the Manakov system in a nonlinear electric transmission line with second couplings

In this study, we investigate rogue wave dynamics and modulational instability using the Manakov system in a nonlinear electrical transmission line with second couplings. Using semi-discrete approximation, we demonstrate how the dynamics of rogue waves in this type of transmission line can be governed by the Manakov system. To study the dynamics of rogue waves in this structure via this approximation, we used the parameters of this transmission line and derived new forms of propagating rogue wave solutions. The solutions obtained are presented as new rogue waves of types I and II. In this work, we show that the dynamics of different types of rogue waves in different types of nonlinear electrical transmission lines can be studied using the Manakov system. Indeed, with the choice of small values of inductance (L3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(L_{3})$$\end{document} in the two types of rogue waves, the effects of the second coupling are clearly visible during the formation of these waves, namely at the level shapes, hollows, and amplitude. Additionally, it can be observed that the dispersion capacity (CS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C_{S})$$\end{document} also affects the shapes, troughs, peaks, and widths of these rogue waves as the troughs gradually disappear, and the peak widths decrease when the dispersion capacity (CS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C_{S})$$\end{document} increases. Finally, concerning the modulational instability in this structure, the essential information that we can retain is that these second couplings (L3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(L_{3})$$\end{document} would impact the zones of instability, which could gradually disappear along this line. To avoid overload, we limited ourselves to these major effects. The results obtained by this Manakov system show not only its efficiency and robustness, but also its potential applicability to other types of useful nonlinear electrical transmission lines, and that these new forms of rogue waves do indeed exist in nonlinear electrical transmission lines with second couplings. This feature has not been sufficiently addressed in this type of nonlinear electrical transmission line and will be useful in many branches of physics.