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

Abstract

The resolution of the reduced fractional nonlinear Schrödinger 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 effect 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 and in order to structure the approximate soliton solutions of the fractional nonlinear Schrödinger equation reduced, ADM is used. The pipe is obtained from the bright and dark soliton by the fractional derivatives order. 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 effect could have a better future for the propagation of modulated waves because in this paper the stabilization of zones of the dark and bright solitons which could be described by a fractional nonlinear Schrödinger equation with small values of fractional derivatives order has been revealed. In addition, the obtained significant results are new and could find applications in many research areas such as in the field of information and communication technologies.