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.
Highlights
The evolution of digital electronics and communication has made enormous advances, in the transmission of information
We focus on the study of the propagation of dark and bright solitons in the left-handed nonlinear transmission line (LHNLTL) by the Adomian decomposition method with the effect of the fractional derivative order (α)
The approximate solution for the fractional nonlinear Schrodinger equation from equation (5) is obtained by adding all solutions obtained above equation (23), equation (24), and adding the expression of the initial condition U0 = tanh(kz)ejz of the dark soliton, we obtain
Summary
The evolution of digital electronics and communication has made enormous advances, in the transmission of information. The expression ”metamaterial” was first mentioned in the field of optics by the Russian physicist [2] when he theoretically introduced 10 the concept of left-handed materials and its realization by [3] In this current work, we focus on the study of the propagation of dark and bright solitons in the left-handed nonlinear transmission line (LHNLTL) by the Adomian decomposition method with the effect of the fractional derivative order (α). The main purpose of this work is to investigate the behavior of the dark and bright solitons by a nonlinear evolution equation with fractional order describing the waves propagation in nonlinear left-handed electrical transmission line giving by [39, 40],. The technique consists of decomposition the linear and nonlinear parts of equation (6) to apply linear operator of the highest-ordered derivation of terms. Provide successive terms of series solution by recurrence using Adomian’s polynomials as
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