Dynamical Behaviour of the Light Pulses through the Optical Fiber: Two Nonlinear Atangana Conformable Fractional Evolution Equations

This 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. These models show the movement of the waves in optical ﬁber mathematically. The 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.

In this research, we investigate two primary mathematical 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]: where U � U(x, t) satisfies Newton's equation of motion in a cubic potential. Equation (1) is also given by [26] 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]: where B � B(x, t), C � C(x, z, t). Additionally, L 1 , L 2 are arbitrary constants. Using the following wave transformation ) q and then integrating the second equation of the transformed system once with zero constant of integration and substituting the result into the first equation of the same system lead to While the second model is the Atangana conformable fractional (2 + 1) AKNS equation which is so close to the first model. is model is given by [28][29][30] where Q � Q(x, z, t). Applying the next wave transforma- Balancing the terms in equations (4) and (6) based on the next principle of homogenous rule, leads to N � 1. e rest sections are order in the following order. Section 2 gives the implementation of the extended simplest equation method for the Atangana conformable fractional SKdV equation (8) and the Atangana conformable fractional (2 + 1)-AKNS equation (9). Also, some solutions are sketched to illustrate the physical behaviour of the wave solutions. Section 3 produces a conclusion of our paper.

Application
In this section, we apply the extended simplest equation method to the SKdV equation and the (2 + 1)-AKNS equation for constructing the exact traveling and solitary wave solutions.

e SKdV Equation.
According to the extended simplest equation method and value of homogenous balance value, we get where a − 1 , a 0 , and a 1 are arbitrary constants. Also, f(b) follows the next ODE: where α, λ, and μ are the arbitrary constants. Substituting equation (8) and its derivative along equation (9) into equation (4) and collecting all terms with the same power of f i (b), i � (0, 1, 2, . . .) lead to a system of equations. Solving this system with any computer software yields family one: Subsequently, the explicit solutions of the fractional SKdV equation are given as follows.

e (2 + 1)-AKNS Equation.
According to the extended simplest equation method and value of homogenous balance value, we get where a − 1 , a 0 , and a 1 are the arbitrary constants. Also, f(b) satisfies equation (9). Substituting equation (18) and its derivative along (9) into equation (6) and collecting all terms with the same power of f i (b), i � (0, 1, 2, . . .) lead to a system of algebraic equations. Solving this system with any computer software yields Subsequently, the explicit solutions of the fractional (2 + 1)-AKNS equation are given as follows.

Conclusion
is paper has investigated the exact traveling and solitary wave solutions of the fractional SKdV equation and the fractional (2 + 1)-AKNS equation. e extended simplest equation method has successfully been implemented and some new distinct optical solitary wave solutions are obtained for both models. Some solutions have been sketched in three types (three-dimensional, two-dimensional, and contour plots) (Figures 1 and 2). e powerful effect of the used method is illustrated. Moreover, the ability of applying to different types of nonlinear evolution equations has been verified.
Data Availability e data that support the findings of this study are available from the corresponding author (Mostafa M. A. Khater) upon reasonable request.

Conflicts of Interest
e authors declare that there are no conflicts of interest.

Authors' Contributions
All the authors conceived the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.