# The Exact Solutions of the Conformable Time-Fractional Modified Nonlinear Schrödinger Equation by the Trial Equation Method and Modified Trial Equation Method

Authors: Safoura Rezaei Aderyani; Reza Saadati; Javad Vahidi; Tofigh Allahviranloo
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

Using the trial equation method (TEM) and modified trial equation method (MTEM), firstly, we find the analytical solutions of the conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE), and finally, we present numerical results in tables and charts.


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## Body

## 1. Introduction

In the past ten years, NLSEs have received a great deal of interest from scholars and researchers. Actually, simplified versions of Zakharov’s system give rise to certain NLSHs. For more details, see [1]. NLSEs have applications in different subjects, e.g., quantum mechanics, biology, semiconductor industry, optical communication, energy quantization, quantum chemistry, wave propagation, protein folding and bending, condensed matter physics, solid-state physics, nanotechnology and industry, laser propagation, and nonlinear optics.Lately, the investigation of the CTFNLSE in the form(1)iDταΨ+12ΨXX+Ψ2Ψ+iℏΨXXX=0,α≤1,τ≥0,including numerics, analysis, and applications becomes a significant issue in applied mathematics, and diverse computational methods have been designed to discuss the exact solutions of it. Every method has its own proportion formulas and merits for the application to the administering equation for exploring the exact solutions.In [2–4], the exact solution of CTFNLSE (1) is obtained through different methods such as the first integral method, functional variable method, sine-Gordon method, and direct algebraic method. In [5], Younas et al. presented the CTFMNLSE and studied the exact solutions of it by the generalized exponential rational function method.Motivated by Younas et al. [5], we consider the following CTFMNLSE:(2)iDταΨ+σ1ΨXX+σ2Ψ2Ψ=iδ1ΨXXX+iδ2Ψ2ΨX∗−iδ3Ψ2ΨX+δ4Ψ,0<α≤1,where σ1=X0/8K02−3cosΘ+2,σ2=−X0K02/2,δ1=X0cosΘ/16K03−5cos2Θ−6,δ2=X0K0cosΘ/4,δ3=3X0K0/2,δ4=K0ΨX2|X=0, X0 and K0 are the frequency and the wave number of the carrier wave, respectively, and the operator Dα of order α, where α∈0,1, represents the conformable fractional derivative.Here, using the TEM and MTEM, we investigate the exact solutions for the above CTFMNLSE. For more details, you can see [6–11].


## 2. Preliminaries

Here, it would be helpful to present some properties and definitions of the conformal derivative and other preliminaries.Definition 1 (see [12]).
SupposeΩ:0,∞⟶ℝ is a function. Then, the conformal fractional derivative of Ω of order α is presented as(3)Tατ=limϵ⟶0Ωτ+ϵτ1−α−Ωτϵ,for all 0<α≤1,0<τ.Definition 2 (fractional integral; see [12]).
Supposeι≥0 and τ≥ι. Also, suppose Ω is a function defined on ι,τ and 0<α<1. Then, the α−fractional integral of Ω is defined by(4)IιαΩτ=∫ιτΩςς1−αdς,if the Riemann improper integral exists.Theorem 1 (see [12]).
Suppose0<α≤1 and Ω and ℧ are α−differentiable at a point τ; then,(i)
Tαϖ1Ω+ϖ2℧=ϖ1TαΩ+ϖ2Tα℧,∀ϖ1,ϖ2∈ℝ(ii)
Tαtϖ=ϖτϖ−α,∀ϖ∈ℝ(iii)
TταΩ℧=ΩTα℧+℧TαΩ(iv)
TαΩ/℧=℧TαΩ−ΩTα℧/℧2
Furthermore, ifΩ is differentiable, then TαΩτ=τ1−αdΩ/dτ.Theorem 2 (see [12]).
SupposeΩ:0,∞⟶ℝ is a function such that Ω is differentiable and also α−differentiable. Suppose ℧ is a function defined in the range of Ω and also differentiable; then, one has the following rule:(5)TαΩo℧τ=τ1−α℧τΩ′℧τ.


## 3. Methods and Applications

In this section, we present the first step of the TEM and the MTEM for finding analytical solutions of the CTFMNLSE defined as (2). For more details, see [5].Suppose a CTFNLPDE(6)ΓΦ,Φτ,ΦX,DταΦ,DXβΦ,Dτ2α,DX2β,…=0,0<α≤1,0<β<1,where Φ and Γ are an unknown function and a polynomial in its arguments, respectively.Using a fractional travelling wave transformation(7)ΦX,τ=Λξ,ξ=X−Vατα,where V is the velocity and substituting (7) into (6), we have a NLODE given by(8)ΥΛ,Λ′,Λ″,Λ″′,…=0,where ′ denotes the derivative with respect to ξ.Here, sinceΨ=ΨX,τ in (2) is a complex function, for proceeding, we begin with the following travelling wave assumption:(9)ΨX,τ=Λξeiψ,where ξ=ηX−V/ατα and ψ=−KX+X/ατα+ζ, and ζ,X, and K are parameters, representing the phase constant, frequency, and wave number, respectively. Substituting (9) into (2), we get real and imaginary parts as follows:(10)η2σ1−3δ1KΛ″+σ2+δ2+δ3KΛ3+−P−σ1K2+δ1K3−δ4Λ=0,(11)3δ1K2−V−2σ1KΛ′−δ1η2Λ′′′+δ3−δ2Λ2Λ′=0.Now, integrating the imaginary part of the equation and taking constant equal to zero, one may have(12)33δ1K2−V−2σ1KΛ−3δ1η2Λ′′+δ3−δ2Λ3=0.From (10) and (12), it can be followed that(13)K3δ1−X−σ1K2−δ433δ1K2−V−2σ1K=σ1−3δ1K−3δ1=Kδ2+Kδ3+σ2δ3−δ2.From the above, it can be followed that(14)V=−δ1X+δ1δ4+2Kσ1−2δ1K2σ1−3δ1K,K=σ1δ2−δ3−3σ2δ16δ1δ2.Rewrite (10) into the following form:(15)Λ′′+λ1Λ3−λ2Λ=0,(16)orΛ″=λ2Λ−λ1Λ3,where λ1=σ2+δ2+δ3K/η2σ1−3δ1 and λ2=−−X−σ1K2+δ1K3−δ4/η2σ1−3δ1K.In the next two sections, we investigate the primary steps for detecting the exact solution of (10) by using the TEM and the MTEM.The exact solution of (12) can be found in a similar way.


### 3.1. Trial Equation Method (TEM)

Firstly, CTFNLPDE (6) is reduced to NLODE (8) under transformation (7). Secondly, consider the trial equation(17)Λ′2=∑i=0naiΛi,where ai and n are constants, which are derived from the solution of the system and the balancing principle, respectively. Finally, the solution of (17) can be given by the integral form:(18)±ξ−ξ0=∫1∑i=0naiΛidΛ.Note 1.
If(19)Λ″=a0+a1Λ+⋯+anΛn,then integrating (19) with respect to ξ once, we get(20)Λ′2=2ann+1Λn+1+⋯+a1Λ2+2a0Λ+d,where n,ai, and integration constant d are to be determined.Now, considering (16) and balancing Λ″ and Λ3, we obtain n=4. So, the trial equation is(21)Λ′2=a0+a1Λ+a2Λ2+a3Λ3+a4Λ4,in which(22)Λ′2=−λ12Λ4+λ2Λ2+d,so we have(23)a0=d,a1=0,a2=λ2,a3=0,a4=−λ12.Now, (18) is rewritten with (23):(24)±ξ−ξ0=∫1a0+λ2Λ2−λ1/2Λ4dΛ.If we seta0=0 in (24) and integrate this equation, the exact solution of (2) is obtained:Ifλ2>0, then we have bright and singular solutions, respectively:(25)ΨX,τ=±2λ2/λ1ei−XK+X/ατα+ζsechλ2ηX−Vατα+ξ0,(26)ΨX,τ=±−2λ2λ1ei−XK+X/ατα+ζcschλ2ηX−Vατα+ξ0.Ifλ2<0, the singular periodic solutions appear as(27)ΨX,τ=±2λ2λ1ei−XK+X/ατα+ζsec−λ2ηX−Vατα+ξ0,(28)ΨX,τ=±2λ2λ1ei−KX+X/ατα+ζcsc−λ2ηX−Vατα+ξ0,where ξ0 is an arbitrary constant. For more details, see [13–15].


### 3.2. Modified Trial Equation Method (MTEM)

In this section, instead of using (17), the trial equation can be chosen as(29)Λ′=FΛGΛ=aNΛN+⋯+a1Λ+a0bMΛM+⋯+b1Λ+b0,which is seen as the modified trial equation method when the same procedure is applied with the integral form solution of (29) as(30)±ξ−ξ0=∫FΛGΛdΛ.Similarly, one can solve integral (30) and find the exact travelling wave solution of equation (6).Note 3.2.
If equation (29) holds, then we have(31)Λ″=FΛF′ΛGΛ−FΛG′ΛG3Λ.If we now use trial (29) in (16) for balancing the procedure, we get N−M=2. Choosing N=2 and M=0, the trial equation is(32)Λ′=a0+a1Λ+a2Λ2b0,and we have(33)Λ″=a1+2a2Λa0+a1Λ+a2Λ2b02,where a2≠0 and b0≠0.The corresponding system is(34)a0a1b02=0,2a0a2+a12b02=λ2,3a1a2b02=0,2a22b02=−λ1.Solving the corresponding system, we obtain(35)a0=±i12b02λ12λ2,a1=0,a2=±ib0λ12,b0=b0.Using these coefficients in (30), we get(36)±ξ−ξ0=∫b0±i1/2b02λ1/2λ2±ib0λ1/2Λ2dΛ.When we integrate this equation and use the wave transformation, the exact travelling wave solutions of (2) are obtained as follows.Ifb0λ1λ2<0, then we have(37)ΨX,τ=±−b0λ22ei−KX+X/ατα+ζtanh12−b0λ1λ2ηX−Vατα+ξ0,(38)ΨX,τ=±−b0λ22ei−KX+X/ατα+ζcoth12−b0λ1λ2ηX−Vατα+ξ0,and if b0λ1λ2>0, then we have(39)ΨX,τ=±b0λ22ei−KX+X/ατα+ζtan12b0λ1λ2ηX−Vατα+ξ0,(40)ΨX,τ=±b0λ22ei−KX+X/ατα+ζcot12b0λ1λ2ηX−Vατα+ξ0,where ξ0 is an arbitrary constant. For more details, see [16–18].


## 3.1. Trial Equation Method (TEM)

Firstly, CTFNLPDE (6) is reduced to NLODE (8) under transformation (7). Secondly, consider the trial equation(17)Λ′2=∑i=0naiΛi,where ai and n are constants, which are derived from the solution of the system and the balancing principle, respectively. Finally, the solution of (17) can be given by the integral form:(18)±ξ−ξ0=∫1∑i=0naiΛidΛ.Note 1.
If(19)Λ″=a0+a1Λ+⋯+anΛn,then integrating (19) with respect to ξ once, we get(20)Λ′2=2ann+1Λn+1+⋯+a1Λ2+2a0Λ+d,where n,ai, and integration constant d are to be determined.Now, considering (16) and balancing Λ″ and Λ3, we obtain n=4. So, the trial equation is(21)Λ′2=a0+a1Λ+a2Λ2+a3Λ3+a4Λ4,in which(22)Λ′2=−λ12Λ4+λ2Λ2+d,so we have(23)a0=d,a1=0,a2=λ2,a3=0,a4=−λ12.Now, (18) is rewritten with (23):(24)±ξ−ξ0=∫1a0+λ2Λ2−λ1/2Λ4dΛ.If we seta0=0 in (24) and integrate this equation, the exact solution of (2) is obtained:Ifλ2>0, then we have bright and singular solutions, respectively:(25)ΨX,τ=±2λ2/λ1ei−XK+X/ατα+ζsechλ2ηX−Vατα+ξ0,(26)ΨX,τ=±−2λ2λ1ei−XK+X/ατα+ζcschλ2ηX−Vατα+ξ0.Ifλ2<0, the singular periodic solutions appear as(27)ΨX,τ=±2λ2λ1ei−XK+X/ατα+ζsec−λ2ηX−Vατα+ξ0,(28)ΨX,τ=±2λ2λ1ei−KX+X/ατα+ζcsc−λ2ηX−Vατα+ξ0,where ξ0 is an arbitrary constant. For more details, see [13–15].


## 3.2. Modified Trial Equation Method (MTEM)

In this section, instead of using (17), the trial equation can be chosen as(29)Λ′=FΛGΛ=aNΛN+⋯+a1Λ+a0bMΛM+⋯+b1Λ+b0,which is seen as the modified trial equation method when the same procedure is applied with the integral form solution of (29) as(30)±ξ−ξ0=∫FΛGΛdΛ.Similarly, one can solve integral (30) and find the exact travelling wave solution of equation (6).Note 3.2.
If equation (29) holds, then we have(31)Λ″=FΛF′ΛGΛ−FΛG′ΛG3Λ.If we now use trial (29) in (16) for balancing the procedure, we get N−M=2. Choosing N=2 and M=0, the trial equation is(32)Λ′=a0+a1Λ+a2Λ2b0,and we have(33)Λ″=a1+2a2Λa0+a1Λ+a2Λ2b02,where a2≠0 and b0≠0.The corresponding system is(34)a0a1b02=0,2a0a2+a12b02=λ2,3a1a2b02=0,2a22b02=−λ1.Solving the corresponding system, we obtain(35)a0=±i12b02λ12λ2,a1=0,a2=±ib0λ12,b0=b0.Using these coefficients in (30), we get(36)±ξ−ξ0=∫b0±i1/2b02λ1/2λ2±ib0λ1/2Λ2dΛ.When we integrate this equation and use the wave transformation, the exact travelling wave solutions of (2) are obtained as follows.Ifb0λ1λ2<0, then we have(37)ΨX,τ=±−b0λ22ei−KX+X/ατα+ζtanh12−b0λ1λ2ηX−Vατα+ξ0,(38)ΨX,τ=±−b0λ22ei−KX+X/ατα+ζcoth12−b0λ1λ2ηX−Vατα+ξ0,and if b0λ1λ2>0, then we have(39)ΨX,τ=±b0λ22ei−KX+X/ατα+ζtan12b0λ1λ2ηX−Vατα+ξ0,(40)ΨX,τ=±b0λ22ei−KX+X/ατα+ζcot12b0λ1λ2ηX−Vατα+ξ0,where ξ0 is an arbitrary constant. For more details, see [16–18].


## 4. Numerical Results in Tables and Charts

Here, we letδ1=δ3=η=1,X=δ2=δ4=σ2=2,ζ=V=0.5,K=0.25,σ1=−1,α=0.90,andξ0=0. Therefore, we have λ1=−1.57143 and λ2=−2.24107. Now, in the following sections, we calculate the exact solutions of CTFMNLSE (2) obtained through the two different methods presented in Section 3.Figures1(a), 1(c), 2(a), and 2(c) and also Figures 1(b), 1(d), 2(b), and 2(d) display 3D and 2D with the real part of solutions Ψ1,1,Ψ1,3,Ψ2,1, and Ψ2,3 obtained by the TEM and MTEM under the values presented in the above paragraph. Tables 1–4 present the numerical results of analytical solutions of CTFMNLSE (2) obtained by the TEM and MTEM with several point sources through arbitrary. In addition, Figures 2–5 display 2D with the real part of solutions Ψ1,1 and Ψ2,1 for fixed X=0.062 and different values of α. Figure 3 displays 3D with the imaginary and the real part of solutions Ψ1,1 and Ψ1,3. Also, Figure 6 displays 3D with the imaginary and real part of solution Ψ2,1 and also 2D with the imaginary part of the solution Ψ2,1 for fixed X=0.062 and different values of α.Figure 1
(a, b) 3D and 2D with the real part of solutionΨ1,1. (c, d) 3D and 2D with the real part of solution Ψ1,3 obtained via the TEM under the values presented in Section 4.
(a)(b)(c)(d)Figure 2
2D with the real part of solutionΨ1,1 for fixed X=0.062 and different values of α.Table 1
The real part of exact solutions of CTFMNLSE (1) obtained by the TEM with several point sources through arbitrary.
XτΨ1,1X,τΨ1,2X,τΨ1,3X,τΨ1,4X,τ0.0120.0121.44911−1.44911351.38054−351.380540.0371.38186−1.38186−39.1577439.157740.0621.31204−1.31204−21.1942421.194240.0370.0121.45334−1.4533421.46932−21.469320.0371.38820−1.3882075.68599−75.685990.0621.32018−1.32018−83.6872683.687260.0620.0121.45547−1.4554710.95095−10.950950.0371.39254−1.3925418.97941−18.979410.0621.32642−1.3264241.97445−41.97445Table 2
The imaginary part of exact solutions of CTFMNLSE (1) obtained by the TEM with several point sources through arbitrary.
XτΨ1,1X,τΨ1,2X,τΨ1,3X,τΨ1,4X,τ0.0120.0120.86568−0.86568−588.19565588.195650.0370.96853−0.9685355.86884−55.868840.0621.05867−1.0586726.26665−26.266650.0370.0120.85592−0.85592−36.4542736.454270.0370.96009−0.96009−109.43494109.434940.0621.05167−1.05167105.05287−105.052870.0620.0120.84497−0.84497−18.8630318.863030.0370.95027−0.95027−27.8124427.812440.0621.04316−1.04316−53.3717453.37174Table 3
The real part of exact solutions of CTFMNLSE (1) obtained by the MTEM with several point sources through arbitrary.
XτΨ2,1X,τΨ2,2X,τΨ2,3X,τΨ2,4X,τ0.0120.012−0.000840.00084−351.20377351.203770.0370.00941−0.0094139.17921−39.179210.0620.02081−0.0208121.20630−21.206300.0370.012−0.013420.01342−21.4760021.476000.037−0.004790.00479−75.6767475.676740.0620.00519−0.0051983.77139−83.771390.0620.012−0.025750.02575−10.9563410.956340.037−0.018730.01873−18.9848218.984820.062−0.010190.01019−41.9734241.97342Table 4
The imaginary part of exact solutions of CTFMNLSE (1) obtained by the MTEM with several point sources through arbitrary.
XτΨ2,1X,τΨ2,2X,τΨ2,3X,τΨ2,4X,τ0.0120.0120.00140−0.00140587.89974−587.899740.037−0.013420.01342−55.8994755.899470.062−0.025790.02579−26.2815826.281580.0370.0120.02279−0.0227936.46563−36.465630.0370.00691−0.00691109.42158−109.421580.062−0.006510.00651−105.15849105.158490.0620.0120.04436−0.0443618.87231−18.872310.0370.02745−0.0274527.82036−27.820360.0620.01295−0.0129553.37043−53.37043Figure 3
3D with the imaginary and the real part of solutionsΨ1,1 and Ψ1,3.
(a)(b)Figure 4
(a, b) 3D and 2D with the real part of solutionΨ2,1. (c, d) 3D and 2D with the real part of solution Ψ2,3 obtained via the MTEM under the values presented in Section 4.
(a)(b)(c)(d)Figure 5
2D with the real part of the solutionΨ2,1 for fixed X=0.062 and different values of α.Figure 6
3D with the imaginary and the real part of solutionsΨ2,1 and also 2D with the imaginary part of the solution Ψ2,1 for fixed X=0.062 and different values of α.
(a)(b)


### 4.1. The Exact Solutions of CTFMNLSE (2) Obtained through the TEM

Considering the given values in Section4, we get Tables 1 and 2.


### 4.2. The Exact Solutions of CTFMNLSE (2) Obtained through the MTEM

Considering the given values in Section4, we get Tables 3 and 4.


## 4.1. The Exact Solutions of CTFMNLSE (2) Obtained through the TEM

Considering the given values in Section4, we get Tables 1 and 2.


## 4.2. The Exact Solutions of CTFMNLSE (2) Obtained through the MTEM

Considering the given values in Section4, we get Tables 3 and 4.


## 5. Concluding Remarks

Using the TEM and MTEM, firstly, we found the exact solutions of CTFMNLSE (2), and finally, we presented numerical results in tables and charts.According to Table5, we can observe that the differences between solutions Ψ1,1 and also Ψ2,1 for fixed X and different values of τ are considerable. It is clear that these differences obtained through the MTEM are less than those of the TEM. In other words, for fixed X, by changing the value of τ, the MTEM results in more minor changes than the TEM. Also, based on Figure 7, we can observe that the results gained by the MTEM have higher accuracy than the TEM.Table 5
Calculation of the difference between solutionsΨ1,1 and also Ψ2,1, represented by ΔTEM and ΔMTEM, for different values of X and τ.
τ = 0.012, 0.037, and 0.062X = 0.012X = 0.037X = 0.062ΔTEM0.336950.137070.069820.065140.133160.068020.062930.129050.06612ΔMTEM0.010250.021650.011400.008630.018610.009980.007920.015560.00854Figure 7
(a, e, b) 3D and 2D with the real part of solutionsΨ1,1 obtained via the TEM under the values presented in Section 4. (c, f, d) 3D and 2D with the real part of solutions Ψ2,1 obtained via the MTEM under the values presented in Section 4.
(a)(b)(c)(d)(e)(f)Figure8 displays the differences between the real part of solutions obtained by the TEM and MTEM for fixed X=0.062 and α=0.90. As you can see, the difference obtained among Ψ1,3,Ψ1,4 and Ψ2,3,Ψ2,4 is more minor than the difference obtained among Ψ1,1,Ψ1,2 and Ψ2,1,Ψ2,2.Figure 8
2D with the differences between the real part of solutions obtained through the TEM and MTEM for fixedX=0.062 and α=0.90.



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