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

Authors: S. H. Alfalqi; Mostafa M. A. Khater; J. F. Alzaidi; Dianchen Lu
Journal: Journal of Mathematics
 (2020)

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## Abstract

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.


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## Body

## 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]. These 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 (analytical, semianalytical, and numerical techniques) for constructing exact and numerical solutions [9–15]. These schemes include the modified and generalized Kudryashov methods, the extended tanh-function method, the improved tanϕ/2 expansion method, the novel, improved, extended, and generalized G′/G expansion method, the extended and generalized e−ϕξ 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 mathematical models in the optical fiber via the extended simplest equation method. The first model is Atangana conformable fractional SKdV equation that was derived by Krichever and Novikov in the following form [25]:(1)DtqUUx+UxxUxx−12UxxUx2=0,0<q<1,where U=Ux,t satisfies Newton’s equation of motion in a cubic potential. Equation (1) is also given by [26](2)DtqG+14Gxxz−GxGxz2G−GxxGz4G+Gx2Gz2G2−Gx8∫Gx2G2zdx=0.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]:(3)ℒ1ℬ2Cx−ℒ1ℬℬxC+ℬ2Cxxz−ℬℬxxℬz−ℒ2ℬℬxℬxz+ℒ2ℬx2ℬz−ℬ4ℬz=0,Dtqℬ−DtqC=0,where ℬ=ℬx,t,C=Cx,z,t. Additionally, ℒ1,ℒ2 are arbitrary constants. Using the following wave transformation ℬx,t=ℬb,Cx,z,t=vf,b=x+z−c/qt+1/Γqq 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(4)ℬ2ℬ‴−1+ℒ2ℬℬℬ′ℬ″+ℒ2ℬ′3−ℬ4ℬ′=0.While the second model is the Atangana conformable fractional (2 + 1) AKNS equation which is so close to the first model. This model is given by [28–30](5)4DtqQx+Qxxxz+8QxzQx+4QzQxx=0,where Q=Qx,z,t. Applying the next wave transformation Q=Qb,b=x+z−c/qt+1/Γqq to equation (5) gives(6)−4cQ′+Q‴+6Q′2=0.Balancing the terms in equations (4) and (6) based on the next principle of homogenous rule,(7)DdεPbdbε=N+ε,DPεdεPbdPεs=εN+sN+ε,leads to N=1.The rest sections are order in the following order. Section2 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.


## 2. 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.


### 2.1. The SKdV Equation

According to the extended simplest equation method and value of homogenous balance value, we get(8)ℬb=∑i=−NNaifbi=a−1fb+a0+fba1,where a−1,a0,anda1 are arbitrary constants. Also, fb follows the next ODE:(9)f′b=α+λfb+μfb2,where α,λ,andμ 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 fib,i=0,1,2,… lead to a system of equations. Solving this system with any computer software yields family one:(10)a0⟶12−λ−λ2−4αμ,a1⟶0,a−1⟶−α,ℒ2⟶3.Subsequently, the explicit solutions of the fractional SKdV equation are given as follows.Whenλ=0,αμ<0,(11)ℬ1x,t=−−αμ+−αμCothℋ−αμ∓Logϑ2,ℬ2x,t=−−αμ+−αμTanhℋ−αμ∓Logϑ2.When4αμ>λ2,(12)ℬ3x,t=−λ2−12λ2−4αμ+2αμλ−−λ2+4αμTan1/2ℋ+ϑ−λ2+4αμ,ℬ4x,t=−λ2−12λ2−4αμ+2αμλ−−λ2+4αμCot1/2ℋ+ϑ−λ2+4αμ,where ℋ=x+z−c/qt+1/Γqq.Family two:(13)a0⟶12λ−λ2−4αμ,a1⟶μ,a−1⟶0,ℒ2⟶3.Subsequently, the explicit solutions of the fractional SKdV equation are given as follows.Whenλ=0,αμ<0,(14)ℬ5x,t−−αμ+−αμTanhℋ−αμ∓Logϑ2,ℬ6x,t=−−αμ+−αμCothℋ−αμ∓Logϑ2.Whenα=0,Forλ>0,(15)ℬ7x,t=−λ2−λ22+λ1−eℋ+ϑλμ.Forλ<0,(16)ℬ8x,t=λ2−λ22−μ+μ1+eℋ+ϑλμ.When4αμ>λ2,(17)ℬ9x,t=−12λ2−4αμ+12−λ2+4αμTan12ℋ+ϑ−λ2+4αμ,ℬ10x,t=−12λ2−4αμ+12−λ2+4αμCot12ℋ+ϑ−λ2+4αμ,where ℋ=x+z−c/qt+q/Γqq


### 2.2. The (2 + 1)-AKNS Equation

According to the extended simplest equation method and value of homogenous balance value, we get(18)Qb=∑i=−NNaifbi=a−1fb+a0+fba1,where a−1,a0,anda1 are the arbitrary constants. Also, fb satisfies equation (9). Substituting equation (18) and its derivative along (9) into equation (6) and collecting all terms with the same power of fib,i=0,1,2,… lead to a system of algebraic equations. Solving this system with any computer software yields(19)a−1⟶0,a1⟶−μ,c⟶14λ2−4αμ.Subsequently, the explicit solutions of the fractional (2 + 1)-AKNS equation are given as follows.Whenλ=0,Forαμ>0,(20)Q1x,t=a0−αμTanαμx+z+ϑ+αμt+1/Γqqq,Q2x,t=−αμCotαμx+z+ϑ+αμt+1/Γqqq+a0.Forαμ<0,(21)Q3x,t=a0−−αμTanh−αμx+z+αμt+1/Γqqq∓Logϑ2,(22)Q4x,t=−−αμCoth−αμx+z+αμt+1/Γqqq∓Logϑ2+a0.Whenα=0,Forλ>0,(23)Q5x,t=λ+λ−1+eλx+z+ϑ−λ2t+1/Γqq/4qμ+a0.Forλ<0,(24)Q6x,t=μ−μ1+eλx+z+ϑ−λ2t+1/Γqq/4qμ+a0.When4αμ>λ2,(25)Q7x,t=λ2+a0−12−λ2+4αμTan12−λ2+4αμx+z+ϑ−λ2−4αμt+1/Γqq4q,Q8x,t=λ2−12−λ2+4αμCot12−λ2+4αμx+z+ϑ−λ2−4αμt+1/Γqq4q+a0.


## 2.1. The SKdV Equation

According to the extended simplest equation method and value of homogenous balance value, we get(8)ℬb=∑i=−NNaifbi=a−1fb+a0+fba1,where a−1,a0,anda1 are arbitrary constants. Also, fb follows the next ODE:(9)f′b=α+λfb+μfb2,where α,λ,andμ 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 fib,i=0,1,2,… lead to a system of equations. Solving this system with any computer software yields family one:(10)a0⟶12−λ−λ2−4αμ,a1⟶0,a−1⟶−α,ℒ2⟶3.Subsequently, the explicit solutions of the fractional SKdV equation are given as follows.Whenλ=0,αμ<0,(11)ℬ1x,t=−−αμ+−αμCothℋ−αμ∓Logϑ2,ℬ2x,t=−−αμ+−αμTanhℋ−αμ∓Logϑ2.When4αμ>λ2,(12)ℬ3x,t=−λ2−12λ2−4αμ+2αμλ−−λ2+4αμTan1/2ℋ+ϑ−λ2+4αμ,ℬ4x,t=−λ2−12λ2−4αμ+2αμλ−−λ2+4αμCot1/2ℋ+ϑ−λ2+4αμ,where ℋ=x+z−c/qt+1/Γqq.Family two:(13)a0⟶12λ−λ2−4αμ,a1⟶μ,a−1⟶0,ℒ2⟶3.Subsequently, the explicit solutions of the fractional SKdV equation are given as follows.Whenλ=0,αμ<0,(14)ℬ5x,t−−αμ+−αμTanhℋ−αμ∓Logϑ2,ℬ6x,t=−−αμ+−αμCothℋ−αμ∓Logϑ2.Whenα=0,Forλ>0,(15)ℬ7x,t=−λ2−λ22+λ1−eℋ+ϑλμ.Forλ<0,(16)ℬ8x,t=λ2−λ22−μ+μ1+eℋ+ϑλμ.When4αμ>λ2,(17)ℬ9x,t=−12λ2−4αμ+12−λ2+4αμTan12ℋ+ϑ−λ2+4αμ,ℬ10x,t=−12λ2−4αμ+12−λ2+4αμCot12ℋ+ϑ−λ2+4αμ,where ℋ=x+z−c/qt+q/Γqq


## 2.2. The (2 + 1)-AKNS Equation

According to the extended simplest equation method and value of homogenous balance value, we get(18)Qb=∑i=−NNaifbi=a−1fb+a0+fba1,where a−1,a0,anda1 are the arbitrary constants. Also, fb satisfies equation (9). Substituting equation (18) and its derivative along (9) into equation (6) and collecting all terms with the same power of fib,i=0,1,2,… lead to a system of algebraic equations. Solving this system with any computer software yields(19)a−1⟶0,a1⟶−μ,c⟶14λ2−4αμ.Subsequently, the explicit solutions of the fractional (2 + 1)-AKNS equation are given as follows.Whenλ=0,Forαμ>0,(20)Q1x,t=a0−αμTanαμx+z+ϑ+αμt+1/Γqqq,Q2x,t=−αμCotαμx+z+ϑ+αμt+1/Γqqq+a0.Forαμ<0,(21)Q3x,t=a0−−αμTanh−αμx+z+αμt+1/Γqqq∓Logϑ2,(22)Q4x,t=−−αμCoth−αμx+z+αμt+1/Γqqq∓Logϑ2+a0.Whenα=0,Forλ>0,(23)Q5x,t=λ+λ−1+eλx+z+ϑ−λ2t+1/Γqq/4qμ+a0.Forλ<0,(24)Q6x,t=μ−μ1+eλx+z+ϑ−λ2t+1/Γqq/4qμ+a0.When4αμ>λ2,(25)Q7x,t=λ2+a0−12−λ2+4αμTan12−λ2+4αμx+z+ϑ−λ2−4αμt+1/Γqq4q,Q8x,t=λ2−12−λ2+4αμCot12−λ2+4αμx+z+ϑ−λ2−4αμt+1/Γqq4q+a0.


## 3. Conclusion

This paper has investigated the exact traveling and solitary wave solutions of the fractional SKdV equation and the fractional (2 + 1)-AKNS equation. The 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) (Figures1 and 2). The powerful effect of the used method is illustrated. Moreover, the ability of applying to different types of nonlinear evolution equations has been verified.Figure 1
Periodic kink solitary wave solutions of equation (11) in three-dimensional, two-dimensional, and contour plot 3D forλ=6,a1=3,μ=−9,c=−3,andϑ=2.
(a)(b)(c)Figure 2
Kink solitary wave solutions of equation (21) in three-dimensional, two-dimensional and contour plot 3D for a0=1,α=−1,μ=4,andϑ=3.
(a)(b)(c)



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*Source: 8862484-2020-10-21.xml*