Solitary waves and modulation instability with the influence of fractional derivative order in nonlinear left-handed transmission line

The resolution of the reduced fractional nonlinear Schrodinger equation obtained from the model describing the wave propagation in the left-handed nonlinear transmission line presented by Djidere et al recently, allowed us in this work through the Adomian decomposition method (ADM) to highlight the behavior and to study the propagation process of the dark and bright soliton solutions with the e ect of the fractional derivative order as well as the Modulation Instability gain spectrum (MI) in the LHNLTL. By inserting fractional derivatives in the sense of Caputo, we used ADM to structure the approximate solitons solutions of the fractional nonlinear Schrodinger equation reduced with fractional derivatives. The pipe is obtained from the bright and dark soliton by the fractional derivatives order (see Figures 2 and 5). By the bias of MI gain spectrum the instability zones occur when the value of the fractional derivative order tends to 1. Furthermore, when the fractional derivative order takes small values, stability zones appear. These results could bring new perspectives in the study of solitary waves in left-handed metamaterials, as the memory e ect could have a better future for the propagation of modulated waves because we also show in this article that the stabilization of zones of the dark and bright solitons could be described by a fractional nonlinear Schrodinger equation with small values of fractional derivatives order. In addition, the obtained signi cant results are new and could nd applications in many research areas such as in the eld of information and communication technologies.