# Computational Simulations; Abundant Optical Wave Solutions Atangana Conformable Fractional Nonlinear Schrödinger Equation

Authors: Mostafa M. A. Khater; Mustafa Inc; Raghda A. M. Attia; Dianchen Lu
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

This research paper explores the Atangana conformable nonlinear fractional Schrödinger equation’s optical soliton wave solutions through three recently introduced computational schemes. The simplest expanded equation, the generalized Kudryashov method, and the sech-tanh expansion approaches are used for describing the structure of optical solitons by nonlinear optical fibers with the modern fractional operator. Several formulas such as hyperbolic, trigonometric, logical, dim, light, moon-bright hybrid, singular, combined singular, and regular wave solutions have been created. The employed methods are effective and worthy of being tested. The features of the Hamiltonian process were used to analyze the stability properties of the solutions obtained.


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## Body

## 1. Introduction

The nonlinear partial differential equation is one of the most exciting science types, as it can describe several nonlinear phenomena as nonlinear partial distinctiveness equations (NLPD) or a form of NLPD [1–3], including diffusion, heat, electrostatics, fluid dynamics, electrodynamic, elasticity, and quantum mechanics. Some equations include the uncertain functions of multivariable and partial byproducts, including integral and partial derivatives [4, 5]. Several investigators in various fields have been studying the fractional order in the NLPD equations [6]. This analysis is focused on the nonlocal properties that only occur in such a derivative, as they are not present in an integer derivative [7–9]. This property is generally used to describe a location, quantity, and nonlocal Lagrangian behavior at a distance [10]. According to this property, various forms of concepts are employed to transform the NLPD equation into a standard differential equation with an integer order, including Riemann–Liouville, the two-scale fractal derivative, He fractional derivative, Caputo, and compliant fractional derivatives [11–13]. The compliant fractional derivative of Atangana is used here on the grounds of its supremacy over other concepts of fractional derivatives [14, 15]. This superiority is apparent in its capacity to extend all organizational characteristics, such as a chain law, quotient law, semigroup property, and element rule, to the traditional first derivative [16, 17].Many analytical and semianalytical schemes are derived for the same purpose, such as Lie group method,G′/G-expansion method, tanh-expansion method, extended tanh-expansion method, improved Bernoulli subequation function, Riccati–Bernoulli sub-ODE method, sinh-cosh method, simplest equation method, generalized Kudryashov method, Adomian decomposition method, homotopy perturbation method, general homotopy analysis method, and so on [18–25].The rest of the paper’s sections are ordered as follows: Section2 studies the performance of the extended simplest equation [26, 27], the generalized Kudryashov [28, 29], and the sech-tanh expansion methods [30, 31] on the perturbed time-fractional nonlinear Schrödinger equation. Moreover, we study the stability property of obtained analytical wave solutions. Section 3 represents some of the obtained solutions in three & two-dimensional and contour plots. Section 4 gives the conclusion.


## 2. Application

In this section, we apply the Atangana conformable fractional operator to the perturbed time-fractional nonlinear Schrödinger equation that is given by [31–35].(1)i∂ϑℱ∂tϑ+ℱxx+γℱℱ2+iγ1ℱxxx+γ2ℱ2ℱx+γ3ℱ2xℱ=0,where t>0,0<ϑ<1,i=−1,whileα,γ are the arbitrary constants. Also, γ1 represents the dispersion term, γ2 is the nonlinear dispersion, γ3 is the nonlinear dispersion term and ℱ=ℱx,t describes the propagation of optical solitons in optical fibers that exhibits a Kerr law nonlinearity. During the evolution age of communications, many fundamentals’ applications are related to equation (1) such as plasma physics and optical fiber communications. This fiber is used to transmit light between the two ends of the fiber and the widespread use in fiber-optical links. They provide a broader and higher range of communication (data) than electrical cables. Typically crafted from drawing glass (silica) or plastic of a much thicker diameter than a human scalp.Handling equation (1) via the Atangana conformable fractional in the next wave transformation ℱx,t=eihx−ΩtGℨ,ℨ=x−c/ϑt+1/Γϑϑ where c is an arbitrary constant, yields(2)eihx−tΩGℨ−h2+Ω+h3γ1+G3ℨγ−hγ2−ic−2h+3h2γ1G′ℨ+iG2ℨγ2+2γ3G′ℨ+1−3hγ1G″ℨ+iγ1G‴ℨ=0.Separating the real and imaginary part of equation (2) and integrating the imaginary part once with zero constant of integration lead to(3)−h2+Ω+h3γ1G+γ−hγ2G3+1−3hγ1G″=0,(4)−c−2h+3h2γ1G+γ2+2γ33G3+γ1G″=0.Comparing the coefficients ofG,G″,G3 in equations (3) and (4) explains the equivalence between both equations under the following constraint(5)−h2+Ω+h3γ1−c−2h+3h2γ1=3γ−hγ2γ2+2γ3=1−3hγ1γ1.This relation leads toΩ=−c+2h+3chγ1−8h2γ1+8h3γ12/γ1,γ2=3γγ1−2γ3+6hγ1γ3. Applying the homogenous balance rule to equation (4) for determining the balance between the nonlinear and highest order derivative terms.


### 2.1. Extended Simplest Equation Method

Applying this scheme to equation (4) leads to the following general solution(6)Gℨ=∑i=−nnaifℨi=a−1fℨ+a0+fℨa1,where ai,i=−1,0,1 are arbitrary constants. Also, fℨ satisfies the next auxiliary equation [26, 27](7)f′ℨ=α+λfℨ+μfℨ2,where α,λ,μ are arbitrary constants. Substituting equations (6) along (7) into equation (4) and collecting all terms with the same power of fℨj,j=−3,−2,…,2,3 lead to a system of algebraic equations. Solving this system yields the following families.


#### 2.1.1. Family I

(8)a0=3/2λγ1−γ2−2γ3,a−1=0,a1=6μγ1−γ2−2γ3,c=124h−6h2γ1−λ2γ1+4αμγ1,where−γ2−2γ3>0,γ1>0.


#### 2.1.2. Family II

(9)a0=3/2λγ1−γ2−2γ3,a−1=6αγ1−γ2−2γ3,a1=0,c=124h−6h2γ1−λ2γ1+4αμγ1,where−γ2−2γ3>0,γ1>0.Thus, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on family I as follows:Whenλ=0,forαμ>0,(10)ℱ1x,t=6eihx−itΩαμγ1Tanαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ−γ2−2γ3,ℱ2x,t=6eihx−itΩαμCotαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑγ1−γ2−2γ3.Forαμ<0,(11)ℱ3x,t=6eihx−itΩ−αμγ1Tanh−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2−γ2−2γ3,(12)ℱ4x,t=6eihx−itΩ−αμCoth−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2γ1−γ2−2γ3.Whenα=0;Forλ>0(13)ℱ5x,t=−3/2eihx−itΩλγ1−1+eλx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑμ−γ2−2γ3−3/2eihx−itΩ+λx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑλμγ1−1+eλx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑμ−γ2−2γ3.Forλ<0(14)ℱ6x,t=3/2eihx−itΩλγ1−γ2−2γ3−6eihx−itΩμγ1−γ2−2γ3+6eihx−itΩμγ11+eλx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑμ−γ2−2γ3.When4αμ>λ2(15)ℱ7x,t=eihx−itΩ−3λ2/2+6αμγ1Tan1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑ−γ2−2γ3,ℱ8x,t=eihx−itΩ−3λ2/2+6αμCot1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑγ1−γ2−2γ3.Additionally, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on family II byWhenλ=0,forαμ>0,(16)ℱ9x,t=6eihx−itΩαμCotαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑγ1−γ2−2γ3,ℱ10x,t=6eihx−itΩαμγ1Tanαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ−γ2−2γ3.Forαμ<0,(17)ℱ11x,t=−6eihx−itΩ−αμCoth−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2γ1−γ2−2γ3,(18)ℱ12x,t=−6eihx−itΩ−αμγ1Tanh−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2−γ2−2γ3.When4αμ>λ2,(19)ℱ13x,t=3/2eihx−itΩλγ1−γ2−2γ3−26eihx−itΩαμγ1−γ2−2γ3λ−−λ2+4αμTan1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑ,ℱ14x,t=3/2eihx−itΩλγ1−γ2−2γ3−26eihx−itΩαμγ1λ−−λ2+4αμCot1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑ−γ2−2γ3.


### 2.2. Generalized Kudryashov Method

Applying this scheme to equation (4) leads to the following general solution(20)Gℨ=∑i=0maiQℨi∑j=0nbjQℨj=a0+Qℨa1+Qℨ2a2b0+Qℨb1,where aibj,i=0,1,2,j=0,1 are arbitrary constants. Also, Qℨ satisfies the next auxiliary equation [26, 27]:(21)Q′ℨ=Qℨ2−Qℨ.Substituting equation (21) along (22) into equation (4) and collecting all terms with the same power of Qℨj,j=0,1,…,5,6 lead to a system of algebraic equations. Solving this system yields the following families.


#### 2.2.1. Family I

(22)a0=a1b0−2b0+b1,a2=−2a1b1−2b0+b1,c=124h−γ1−6h2γ1,γ3=−12b02γ1+12b0b1γ1−3b12γ1−2a12γ24a12.


#### 2.2.2. Family II

(23)a0=−a12,a2=−a1,b1=−2b0,c=2h−2γ1−3h2γ1,γ3=−24b02γ1−a12γ22a12.


#### 2.2.3. Family III

(24)a0=−a14,a2=−a1,b1=−2b0,c=124h−γ1−6h2γ1,γ3=−24b02γ1−a12γ22a12.


#### 2.2.4. Family IV

(25)a1=0,a2=−4a0,b1=2b0,c=124h−γ1−6h2γ1,γ3=−3b02γ1−2a02γ24a02.Consequently, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on the above families respectively by(26)ℱ15x,t=eihx−tΩ−Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑa1Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑ2b0−b1,(27)ℱ16x,t=eihx−tΩA2e2x+e2t+1/Γϑϑ2h−2γ1−3h2γ1/ϑa12−A2e2x+e2t+1/Γϑϑ2h−2γ1−3h2γ1/ϑb0,(28)ℱ17x,t=eihx−tΩ−Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑa14Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑb0,(29)ℱ18x,t=−eihx−tΩ−Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑa0Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑb0.


### 2.3. Sech-Tanh Expansion Method

Applying this scheme to equation (4) leads to the following general solution:(30)Gℨ=a0+∑i=1nSechℨi−1aiSechℨ+biTanhℨ=a0+Sechℨa1+b1Tanhℨ,where aibj,i=0,1,j=1 are arbitrary constants. Substituting equation (30) into equation (4) and collecting all terms of Sechℨ,Sechℨ2,Sechℨ3,Tanhℨ,Sechℨ2Tanhℨ,SechℨTanhℨ lead to a system of algebraic equations. Solving this system yields the following families.


#### 2.3.1. Family I

(31)a0=0,a1=6γ1γ2+2γ3,b1=0,c=2h+γ1−3h2γ1,whereγ2+2γ3>0,γ1>0.


#### 2.3.2. Family II

(32)a0=0,a1=0,b1=6γ1−γ2−2γ3,c=2h−2γ1−3h2γ1,whereγ2+2γ3<0,γ1>0.Consequently, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on the above families respectively by(33)ℱ19x,t=6eihx−tΩSechx−t+1/Γϑϑ2h+γ1−3h2γ1/ϑγ1γ2+2γ3,ℱ20x,t=6eihx−tΩγ1Tanhx−t+1/Γϑϑ2h−2γ1−3h2γ1/ϑ−γ2−2γ3.


## 2.1. Extended Simplest Equation Method

Applying this scheme to equation (4) leads to the following general solution(6)Gℨ=∑i=−nnaifℨi=a−1fℨ+a0+fℨa1,where ai,i=−1,0,1 are arbitrary constants. Also, fℨ satisfies the next auxiliary equation [26, 27](7)f′ℨ=α+λfℨ+μfℨ2,where α,λ,μ are arbitrary constants. Substituting equations (6) along (7) into equation (4) and collecting all terms with the same power of fℨj,j=−3,−2,…,2,3 lead to a system of algebraic equations. Solving this system yields the following families.


### 2.1.1. Family I

(8)a0=3/2λγ1−γ2−2γ3,a−1=0,a1=6μγ1−γ2−2γ3,c=124h−6h2γ1−λ2γ1+4αμγ1,where−γ2−2γ3>0,γ1>0.


### 2.1.2. Family II

(9)a0=3/2λγ1−γ2−2γ3,a−1=6αγ1−γ2−2γ3,a1=0,c=124h−6h2γ1−λ2γ1+4αμγ1,where−γ2−2γ3>0,γ1>0.Thus, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on family I as follows:Whenλ=0,forαμ>0,(10)ℱ1x,t=6eihx−itΩαμγ1Tanαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ−γ2−2γ3,ℱ2x,t=6eihx−itΩαμCotαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑγ1−γ2−2γ3.Forαμ<0,(11)ℱ3x,t=6eihx−itΩ−αμγ1Tanh−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2−γ2−2γ3,(12)ℱ4x,t=6eihx−itΩ−αμCoth−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2γ1−γ2−2γ3.Whenα=0;Forλ>0(13)ℱ5x,t=−3/2eihx−itΩλγ1−1+eλx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑμ−γ2−2γ3−3/2eihx−itΩ+λx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑλμγ1−1+eλx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑμ−γ2−2γ3.Forλ<0(14)ℱ6x,t=3/2eihx−itΩλγ1−γ2−2γ3−6eihx−itΩμγ1−γ2−2γ3+6eihx−itΩμγ11+eλx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑμ−γ2−2γ3.When4αμ>λ2(15)ℱ7x,t=eihx−itΩ−3λ2/2+6αμγ1Tan1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑ−γ2−2γ3,ℱ8x,t=eihx−itΩ−3λ2/2+6αμCot1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑγ1−γ2−2γ3.Additionally, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on family II byWhenλ=0,forαμ>0,(16)ℱ9x,t=6eihx−itΩαμCotαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑγ1−γ2−2γ3,ℱ10x,t=6eihx−itΩαμγ1Tanαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ−γ2−2γ3.Forαμ<0,(17)ℱ11x,t=−6eihx−itΩ−αμCoth−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2γ1−γ2−2γ3,(18)ℱ12x,t=−6eihx−itΩ−αμγ1Tanh−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2−γ2−2γ3.When4αμ>λ2,(19)ℱ13x,t=3/2eihx−itΩλγ1−γ2−2γ3−26eihx−itΩαμγ1−γ2−2γ3λ−−λ2+4αμTan1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑ,ℱ14x,t=3/2eihx−itΩλγ1−γ2−2γ3−26eihx−itΩαμγ1λ−−λ2+4αμCot1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑ−γ2−2γ3.


## 2.1.1. Family I

(8)a0=3/2λγ1−γ2−2γ3,a−1=0,a1=6μγ1−γ2−2γ3,c=124h−6h2γ1−λ2γ1+4αμγ1,where−γ2−2γ3>0,γ1>0.


## 2.1.2. Family II

(9)a0=3/2λγ1−γ2−2γ3,a−1=6αγ1−γ2−2γ3,a1=0,c=124h−6h2γ1−λ2γ1+4αμγ1,where−γ2−2γ3>0,γ1>0.Thus, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on family I as follows:Whenλ=0,forαμ>0,(10)ℱ1x,t=6eihx−itΩαμγ1Tanαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ−γ2−2γ3,ℱ2x,t=6eihx−itΩαμCotαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑγ1−γ2−2γ3.Forαμ<0,(11)ℱ3x,t=6eihx−itΩ−αμγ1Tanh−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2−γ2−2γ3,(12)ℱ4x,t=6eihx−itΩ−αμCoth−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2γ1−γ2−2γ3.Whenα=0;Forλ>0(13)ℱ5x,t=−3/2eihx−itΩλγ1−1+eλx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑμ−γ2−2γ3−3/2eihx−itΩ+λx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑλμγ1−1+eλx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑμ−γ2−2γ3.Forλ<0(14)ℱ6x,t=3/2eihx−itΩλγ1−γ2−2γ3−6eihx−itΩμγ1−γ2−2γ3+6eihx−itΩμγ11+eλx+θ+t+1/Γϑϑ−4h+6h2+λ2γ1/2ϑμ−γ2−2γ3.When4αμ>λ2(15)ℱ7x,t=eihx−itΩ−3λ2/2+6αμγ1Tan1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑ−γ2−2γ3,ℱ8x,t=eihx−itΩ−3λ2/2+6αμCot1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑγ1−γ2−2γ3.Additionally, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on family II byWhenλ=0,forαμ>0,(16)ℱ9x,t=6eihx−itΩαμCotαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑγ1−γ2−2γ3,ℱ10x,t=6eihx−itΩαμγ1Tanαμx+θ+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ−γ2−2γ3.Forαμ<0,(17)ℱ11x,t=−6eihx−itΩ−αμCoth−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2γ1−γ2−2γ3,(18)ℱ12x,t=−6eihx−itΩ−αμγ1Tanh−αμx+t+1/Γϑϑ−4h+6h2−4αμγ1/2ϑ∓Logθ/2−γ2−2γ3.When4αμ>λ2,(19)ℱ13x,t=3/2eihx−itΩλγ1−γ2−2γ3−26eihx−itΩαμγ1−γ2−2γ3λ−−λ2+4αμTan1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑ,ℱ14x,t=3/2eihx−itΩλγ1−γ2−2γ3−26eihx−itΩαμγ1λ−−λ2+4αμCot1/2−λ2+4αμx+θ+t+1/Γϑϑ−4h+6h2+λ2−4αμγ1/2ϑ−γ2−2γ3.


## 2.2. Generalized Kudryashov Method

Applying this scheme to equation (4) leads to the following general solution(20)Gℨ=∑i=0maiQℨi∑j=0nbjQℨj=a0+Qℨa1+Qℨ2a2b0+Qℨb1,where aibj,i=0,1,2,j=0,1 are arbitrary constants. Also, Qℨ satisfies the next auxiliary equation [26, 27]:(21)Q′ℨ=Qℨ2−Qℨ.Substituting equation (21) along (22) into equation (4) and collecting all terms with the same power of Qℨj,j=0,1,…,5,6 lead to a system of algebraic equations. Solving this system yields the following families.


### 2.2.1. Family I

(22)a0=a1b0−2b0+b1,a2=−2a1b1−2b0+b1,c=124h−γ1−6h2γ1,γ3=−12b02γ1+12b0b1γ1−3b12γ1−2a12γ24a12.


### 2.2.2. Family II

(23)a0=−a12,a2=−a1,b1=−2b0,c=2h−2γ1−3h2γ1,γ3=−24b02γ1−a12γ22a12.


### 2.2.3. Family III

(24)a0=−a14,a2=−a1,b1=−2b0,c=124h−γ1−6h2γ1,γ3=−24b02γ1−a12γ22a12.


### 2.2.4. Family IV

(25)a1=0,a2=−4a0,b1=2b0,c=124h−γ1−6h2γ1,γ3=−3b02γ1−2a02γ24a02.Consequently, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on the above families respectively by(26)ℱ15x,t=eihx−tΩ−Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑa1Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑ2b0−b1,(27)ℱ16x,t=eihx−tΩA2e2x+e2t+1/Γϑϑ2h−2γ1−3h2γ1/ϑa12−A2e2x+e2t+1/Γϑϑ2h−2γ1−3h2γ1/ϑb0,(28)ℱ17x,t=eihx−tΩ−Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑa14Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑb0,(29)ℱ18x,t=−eihx−tΩ−Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑa0Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑb0.


## 2.2.1. Family I

(22)a0=a1b0−2b0+b1,a2=−2a1b1−2b0+b1,c=124h−γ1−6h2γ1,γ3=−12b02γ1+12b0b1γ1−3b12γ1−2a12γ24a12.


## 2.2.2. Family II

(23)a0=−a12,a2=−a1,b1=−2b0,c=2h−2γ1−3h2γ1,γ3=−24b02γ1−a12γ22a12.


## 2.2.3. Family III

(24)a0=−a14,a2=−a1,b1=−2b0,c=124h−γ1−6h2γ1,γ3=−24b02γ1−a12γ22a12.


## 2.2.4. Family IV

(25)a1=0,a2=−4a0,b1=2b0,c=124h−γ1−6h2γ1,γ3=−3b02γ1−2a02γ24a02.Consequently, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on the above families respectively by(26)ℱ15x,t=eihx−tΩ−Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑa1Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑ2b0−b1,(27)ℱ16x,t=eihx−tΩA2e2x+e2t+1/Γϑϑ2h−2γ1−3h2γ1/ϑa12−A2e2x+e2t+1/Γϑϑ2h−2γ1−3h2γ1/ϑb0,(28)ℱ17x,t=eihx−tΩ−Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑa14Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑb0,(29)ℱ18x,t=−eihx−tΩ−Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑa0Aex+et+1/Γϑϑ4h−γ1−6h2γ1/2ϑb0.


## 2.3. Sech-Tanh Expansion Method

Applying this scheme to equation (4) leads to the following general solution:(30)Gℨ=a0+∑i=1nSechℨi−1aiSechℨ+biTanhℨ=a0+Sechℨa1+b1Tanhℨ,where aibj,i=0,1,j=1 are arbitrary constants. Substituting equation (30) into equation (4) and collecting all terms of Sechℨ,Sechℨ2,Sechℨ3,Tanhℨ,Sechℨ2Tanhℨ,SechℨTanhℨ lead to a system of algebraic equations. Solving this system yields the following families.


### 2.3.1. Family I

(31)a0=0,a1=6γ1γ2+2γ3,b1=0,c=2h+γ1−3h2γ1,whereγ2+2γ3>0,γ1>0.


### 2.3.2. Family II

(32)a0=0,a1=0,b1=6γ1−γ2−2γ3,c=2h−2γ1−3h2γ1,whereγ2+2γ3<0,γ1>0.Consequently, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on the above families respectively by(33)ℱ19x,t=6eihx−tΩSechx−t+1/Γϑϑ2h+γ1−3h2γ1/ϑγ1γ2+2γ3,ℱ20x,t=6eihx−tΩγ1Tanhx−t+1/Γϑϑ2h−2γ1−3h2γ1/ϑ−γ2−2γ3.


## 2.3.1. Family I

(31)a0=0,a1=6γ1γ2+2γ3,b1=0,c=2h+γ1−3h2γ1,whereγ2+2γ3>0,γ1>0.


## 2.3.2. Family II

(32)a0=0,a1=0,b1=6γ1−γ2−2γ3,c=2h−2γ1−3h2γ1,whereγ2+2γ3<0,γ1>0.Consequently, the optical solitary wave solutions of the Atangana conformable fractional nonlinear Schrödinger equation are given based on the above families respectively by(33)ℱ19x,t=6eihx−tΩSechx−t+1/Γϑϑ2h+γ1−3h2γ1/ϑγ1γ2+2γ3,ℱ20x,t=6eihx−tΩγ1Tanhx−t+1/Γϑϑ2h−2γ1−3h2γ1/ϑ−γ2−2γ3.


## 3. Stability Property of Solutions

This part studies the stability property of each one of the obtained solutions by using the Hamiltonian system property since the momentumM in this system is given by [36, 37](34)M=12∫−llG2ℨdℨ,where Gℨ is the solution of the model. Consequently, the condition for stability of the solutions can be formulated as(35)∂M∂c>0,where c is the wave velocity. The momentum in the Hamiltonian system for equation (11) is given by(36)M=1c2c−1095.1320422182555+1337.3986588468201i+10799.999990891356+6.613092659621341×10−14ic+−67.49999939305316+10.602875127368723i−63.68904259247388−1.479114197289397×10−31icLog1.+2.061153687919397×10−9−2.524185266461253×10−25ie−18.87082757451449c+−67.49999939305316+10.602875127368723i+63.68904259247388−1.479114197289397×10−31icLog1.+4.851651799965657×108+5.941559847405437×10−8ie−18.87082757451449c+67.49999939305316−10.602875127368721i+3.482018748651115×10−15+67.49999939305316−10.602875127368721i+3.482018748651115×10−15−2.524185266461253×10−25ie−1.031709266497698×10−15−16.849090974083545ic+67.49999939305316−10.602875127368723iLog1.+4.851651799965657×108+5.941559847405437×10−8ie−1.031709266497698×10−15−16.849090974083545ic−3.482018748651115×10−15+56.8656816165352icLog1.+4.851651799965657×108+5.941559847405437×10−8ie−1.031709266497698×10−15−16.849090974083545ic−67.49999939305316−10.602875127368723iLogSin1.5707963267948966+9.999999984115489i−8.424545487041772−5.158546332488493×10−16ic+67.49999939305316−10.602875127368723iLog−1.Sin1.5707963267948966+9.999999984115489i−0.+9.435413787257245ic+67.49999939305316−10.602875127368723iLogSin1.5707963267948966+9.999999984115489i+0.+9.435413787257245ic−67.49999939305316−10.602875127368723iLog−1.Sin1.5707963267948966+9.999999984115489i+8.424545487041772−5.158546332488493×10−16ic−3.3749999750136808−3.081487911019577×10−33iPolyLog2.,−4.851651799965657×108−5.941559847405437×10−8ie−18.87082757451449c+3.3749999750136808−3.081487911019577×10−33iPolyLog2.,−2.061153687919397×10−9+2.524185266461253×10−25ie−18.87082757451449c+3.3749999750136808−3.081487911019577×10−33iPolyLog2.,−4.851651799965657×108−5.941559847405437×10−8ie−1.031709266497698×10−15−16.849090974083545ic−3.3749999750136808−3.081487911019577×10−33iPolyLog2.,−2.061153687919397×10−9+2.524185266461253×10−25ie−1.031709266497698×10−15−16.849090974083545icand, thus(37)∂M∂c|c=−176>0.We conclude that this solution is unstable on the intervalx∈−5,5,t∈−5,5 when θ=1,α=−1,μ=4,γ1=9,ϑ=0.5,γ2=−9,γ3=4,h=2,λ=0. This result shows the ability of the solutions for their application. Applying the previous steps to other obtained solutions leads to studying the stability property of all solutions. This process makes choosing suitable stable solutions to use in model applications is very easy and interesting.


## 4. Representation of Obtained Solutions

This part studies the physical interpretation of obtained solutions under a suitable choice for the values of parameters. It represents each one of the selected solutions by three- and two-dimensional plots. For each kind, we mean the shape of the solution in real and imaginary plots to show the similarities and differences between them in these cases. We have plotted each ofℱ3x,t,ℱ15x,t,andℱ20x,t, Figures 1–6 under the following conditions:Figure1 illustrates the optical periodic solitary wave solution of ℱ3x,t in three-dimensional view when θ=1,α=−1,μ=4,γ1=9,ϑ=0.5,γ2=−9,γ3=4,h=2 in the next interval x∈−5,5,t∈−5,5 to explain the perspective view of the solution.Figure2 describes the optical periodic solitary wave solution of ℱ3x,t in two-dimensional view when θ=1,α=−1,μ=4,γ1=9,ϑ=0.5,γ2=−9,γ3=4,h=2 in the next interval x∈−5,5,t∈−5,5 to represent the wave propagation pattern of the wave along the x-axis.Figure3 illustrates the optical periodic solitary wave solution of ℱ15x,t in three-dimensional view when A=3,γ1=1,A=−2,a1=4,b0=5,b1=−1,h=2 in the next interval x∈−5,5,t∈−5,5 to explain the perspective view of the solution.Figure4 presents the optical periodic solitary wave solution of ℱ15x,t in two-dimensional view when A=3,γ1=1,A=−2,a1=4,b0=5,b1=−1,h=2 in the next interval x∈−5,5,t∈−5,5 to represent the wave propagation pattern of the wave along the x-axis.Figure5 illustrates the optical periodic solitary wave solution of ℱ20x,t in three-dimensional view when γ1=4,ϑ=0.5,γ2=−1,γ3=−4,h=2 in the next interval x∈−5,5,t∈−5,5 to explain the perspective view of the solution.Figure6 illustrates the optical periodic solitary wave solution of ℱ20x,t in two-dimensional view when γ1=4,ϑ=0.5,γ2=−1,γ3=−4,h=2 in the next interval x∈−5,5,t∈−5,5 to represent the wave propagation pattern of the wave along the x-axis.Figure 1
Optical soliton wave solutions of the absolute (a), real (b), and imaginary (c) value of equation (8) in three-dimensional view when (θ=1,α=−1, μ=4, γ1=9, ϑ=0.5, γ2=−9, γ3=4, and h=2).
(a)(b)(c)Figure 2
Optical soliton wave solutions of the absolute (a), real (b), and imaginary (c) value of equation (8) in two-dimensional view when (θ=1,α=−1, μ=4, γ1=9, ϑ=0.5, γ2=−9, γ3=4, and h=2).
(a)(b)(c)Figure 3
Optical soliton wave solutions of the absolute (a), real (b), and imaginary (c) value of equation (18) in three-dimensional view when (A=3, γ1=1, A=−2, a1=4, b0=5, b1=−1, and h=2).
(a)(b)(c)Figure 4
Optical soliton wave solutions of the absolute (a), real (b), and imaginary (c) value of equation (18) in two-dimensional view when (A=3, γ1=1, A=−2, a1=4, b0=5, b1=−1, and h=2).
(a)(b)(c)Figure 5
Optical soliton wave solutions of the absolute, real, and imaginary value of equation (26) in three-dimensional view (γ1=4, ϑ=0.5, γ2=−1, γ3=−4, and h=2).
(a)(b)(c)Figure 6
Optical soliton wave solutions of the absolute, real, and imaginary value of equation (26) in two-dimensional view when (γ1=4, ϑ=0.5, γ2=−1, γ3=−4, and h=2).
(a)(b)(c)


## 5. Conclusion

The consistent fragmental derivative concept of Atangana successfully enables us to introduce three recent computational systems for the disturbed time-fractional nonlinear Schrödinger equation. This model has been given modern wave analytical solutions. We have researched the reliability of strategies to illustrate their potential to work in the sample implementations. The robust and efficient implementation of the computing schemes has been tested, demonstrating its capacity with an integer or fractional order to handle many nonlinear partial equations.



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*Source: 2196913-2022-02-10.xml*