Research Article Dynamical Behaviour of the Light Pulses through the Optical Fiber: Two Nonlinear Atangana Conformable Fractional Evolution Equations S. H. Alfalqi, 1 Mostafa M. A. Khater , 2,3 J. F. Alzaidi, 1 and Dianchen Lu 2 1 Department of Mathematics, Faculty of Science and Arts, Mahayil Asir King Khalid University, Abha, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Jiangsu University, Jiangsu 212013, China 3 Department of Mathematics, El Obour Institutes, Cairo 11828, Egypt Correspondence should be addressed to Mostafa M. A. Khater; mostafa.khater2024@yahoo.com Received 9 September 2020; Accepted 4 October 2020; Published 21 October 2020 Academic Editor: Hijaz Ahmad Copyright © 2020 S. H. Alfalqi et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is study, using the extended simplest method of equation, examines the explicit movement solutions of both the Schwarzian Korteweg-de Vries (SKdV) and (2 + 1)-Ablowitz-Kaup-Newell-Segur (AKNS.) equation. ese models show the movement of the waves in optical fiber mathematically. e SKdV equation explains the movement of the isolated waves in diverse fields and on the site in a small space microsection. Some solutions obtained have been developed to show the physical and dynamic behaviors of these solutions in the obtained wave. 1. Introduction Partial differential equations (PDEs) have been playing an essential role in describing and studying some complex phenomena in distinct branches of science [1–5]. ese phenomena have been formulated in nonlinear PDEs with an integer order or fractional order [6–8]. Studying these mathematical models have been forcing many research groups in physics, chemistry, mathematics, and so on to derive practical and powerful computational schemes (an- alytical, semianalytical, and numerical techniques) for constructing exact and numerical solutions [9–15]. ese schemes include the modified and generalized Kudryashov methods, the extended tanh-function method, the improved tan(ϕ/2) expansion method, the novel, improved, extended, and generalized (G ′ /G) expansion method, the extended and generalized e − ϕ(ξ) expansion method, the Khater method, the modified Khater method, the Adomian decomposing method, the B-spline schemes, and so on [16–24]. In this research, we investigate two primary mathe- matical models in the optical fiber via the extended simplest equation method. e first model is Atangana conformable fractional SKdV equation that was derived by Krichever and Novikov in the following form [25]: D q t U U x + U xx U x  x − 1 2 U xx U x  2 � 0, (0 < q < 1), (1) where U � U(x, t) satisfies Newton’s equation of motion in a cubic potential. Equation (1) is also given by [26] D q t G + 1 4 G xxz − G x G xz 2G − G xx G z 4G + G 2 x G z 2G 2 − G x 8  G 2 x G 2  z dx � 0. (2) Equation (2) has an essential role in a right-moving soliton and the nonlocal form. However, we study a new form of equation (2) that is given in the following system [27]: Hindawi Journal of Mathematics Volume 2020, Article ID 8862484, 6 pages https://doi.org/10.1155/2020/8862484