# Obtaining the Soliton Type Solutions of the Conformable Time-Fractional Complex Ginzburg–Landau Equation with Kerr Law Nonlinearity by Using Two Kinds of Kudryashov Methods

Authors: Arzu Akbulut
Journal: Journal of Mathematics
 (2023)

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## Abstract

The main idea of this study is to obtain the soliton-type solutions of the conformable time-fractional complex Ginzburg–Landau equation with Kerr law nonlinearity. For this aim, the generalized and modified Kudryashov methods are applied to the given model. The reason for using a conformable derivative is that the chain rule can be applied to this derivative. Thus, using the suitable wave transform, the given equation is converted into an ordinary differential equation. Then, the proposed methods are applied to the reduced equation. According to our results, both of the used methods are effective and powerful. Finally, 3D and contour plots are given for some results with suitable variables. Our findings in this paper are critical for explaining a wide range of scientific and physical applications. According to our knowledge, our results are new in the literature.


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## Body

## 1. Introduction

The exact solutions of the nonlinear partial differential equations (NLPDEs) have an important place in different fields of science, such as fluid mechanics, plasma physics, solid-state physics, and optical fibers. This being the case, many methods were discovered to solve nonlinear partial differential equations, for example, the method of undetermined coefficients [1], the Riccati equation mapping approach [2], the trial equation method [3], the finite element method [4], the extended trial approach [5], the Petrov–Galerkin method [6], the unified and expa function methods [7], the modified extended tanh expansion method [8], the modified simple procedure [9], the exponential rational function procedure [10], the Kudryashov method [11], the ansatz method [12], and so on.In this study, the following equation, called the conformable time-fractional complex Ginzburg–Landau equation, will be considered [13]:(1)iqtδ+ϵqxx+λFq2q−q2q∗−1ρq2q2xx−σq2x2−εq=0,where δ∈0,1 represents the conformable derivative, qx,t is a complex-valued function, the spatial coordinate is represented by x and the temporal coordinate is represented by t. The group velocity dispersions are represented by ϵ and λ, the perturbation effects are represented by ρ, σ, and ε. Fq2 is a function of q2 and F is a real-valued algebraic function that must have the smoothness of the function Fq2q:ℂ⟶ℂ. When the complex plane ℂ is assumed as 2D linear space ℝ2, the Fq2q is k times continuously differentiable, namely, Fq2q∈∪a,b=1∞ℂk−b,b×−a,a;ℝ2.In literature, lots of researchers obtained the exact solutions of the given model with the different types of nonlinearity. Some researchers obtained the exact solutions of the generalized derivative of the given model for example Kudryashov applied the first integral method to the equation in [14], Das et al. applied the F-expansion to the model in [15], the modified G′/G-expansion method is applied to the model by Wang et al. in [16], the modified Jacobi elliptic expansion method is applied by Hosseini et al. in [17], Hosseini et al. implemented Kudryashov and exponential methods to the model including the parabolic nonlinearity in [18]. Some researchers obtained the exact solutions of equation (1) with different kinds of fractional derivatives, for example, Tozar obtained the analytical solutions of the conformable time-fractional complex Ginzburg–Landau equation with the help of the 1/G′ method in [19], optical solutions were discovered with the help of the generalized exponential rational function method in [20], Sulaiman et al. explored the optical solitons with the help of the extended sinh-Gordon equation expansion method in [21], the form of the space-time conformable fractional complex Ginzburg–Landau equation is handled in [22], Sadaf et al. applied the wξ/2 method to the model with the different types of senses as the conformable, beta, truncated derivatives in [23].


### 1.1. The Conformable Derivative

In literature, fractional derivatives have an essential role, so many definitions of fractional derivatives are discovered, for example, Riemann–Liouville, Grunwald–Letnikov, the Caputo, Atangana–Baleanu, and modified Riemann–Liouville derivatives [24, 25]. In this study, the conformable derivative will be used, which is developed by Khalil et al. [26]. An important feature of this derivative is that we can apply the chain rule so we can reduce nonlinear differential equations to ordinary differential equations with the help of wave transforms. Basic definitions of the conformable derivative are given as follows:Whenψ:0,∞⟶ℝ, the conformable derivative of ψ of order δ,0<δ<1, is defined as follows [27, 28]:(2)Tδψt=limε⟶0ψt+ϵt1−δ−ψtϵ,for all t>0. The basic properties of the conformable derivative are given as follows [29–31]:(1)
Tδaψ+bφ=aTδψ+bTδφ, for all a,b∈ℝ(2)
Tδtα=αtα−δ, for all α∈ℝ(3)
Tδψφ=ψTδφ+φTδψ(4)
Tδψ/φ=φTδψ−ψTδφ/φ2(5)
Ifψ is differentiable, then Tδψt=t1−δdψ/dt(6)
ψt=λ, Tδλ=0, for all constant functions(7)
Chain rule: Letψ,φ:0,∞⟶ℝ be a differentiable and δ− differentiable function then the chain rule is given by the following:(3)Tδψ∘φt=t1−αφ′tψ′φt.In this paper, the conformable time-fractional complex Ginzburg–Landau equation with Kerr law was solved by two procedures, namely, the generalized Kudryashov and the modified Kudryashov procedures. For this aim, the main ideas of generalized Kudryashov and the modified Kudryashov procedures were in Section2. Then, these procedures were applied to the given model, and 3D and contour plots of obtained solutions were given in Section 3. Finally, conclusions were given.


## 1.1. The Conformable Derivative

In literature, fractional derivatives have an essential role, so many definitions of fractional derivatives are discovered, for example, Riemann–Liouville, Grunwald–Letnikov, the Caputo, Atangana–Baleanu, and modified Riemann–Liouville derivatives [24, 25]. In this study, the conformable derivative will be used, which is developed by Khalil et al. [26]. An important feature of this derivative is that we can apply the chain rule so we can reduce nonlinear differential equations to ordinary differential equations with the help of wave transforms. Basic definitions of the conformable derivative are given as follows:Whenψ:0,∞⟶ℝ, the conformable derivative of ψ of order δ,0<δ<1, is defined as follows [27, 28]:(2)Tδψt=limε⟶0ψt+ϵt1−δ−ψtϵ,for all t>0. The basic properties of the conformable derivative are given as follows [29–31]:(1)
Tδaψ+bφ=aTδψ+bTδφ, for all a,b∈ℝ(2)
Tδtα=αtα−δ, for all α∈ℝ(3)
Tδψφ=ψTδφ+φTδψ(4)
Tδψ/φ=φTδψ−ψTδφ/φ2(5)
Ifψ is differentiable, then Tδψt=t1−δdψ/dt(6)
ψt=λ, Tδλ=0, for all constant functions(7)
Chain rule: Letψ,φ:0,∞⟶ℝ be a differentiable and δ− differentiable function then the chain rule is given by the following:(3)Tδψ∘φt=t1−αφ′tψ′φt.In this paper, the conformable time-fractional complex Ginzburg–Landau equation with Kerr law was solved by two procedures, namely, the generalized Kudryashov and the modified Kudryashov procedures. For this aim, the main ideas of generalized Kudryashov and the modified Kudryashov procedures were in Section2. Then, these procedures were applied to the given model, and 3D and contour plots of obtained solutions were given in Section 3. Finally, conclusions were given.


## 2. The Procedures

In this section, the used procedures will be given. We take into consideration a general nonlinear differential equation in the following form:(4)Φq,∂δq∂tδ,∂q∂x,∂2δq∂t2δ,∂2u∂x2,…=0,where q=qx,t is a complex-valued function and δ represents a conformable derivative. If we apply the following wave transformation to equation (4):(5)qx,t=uζeiφ,where ζ=x−vtδ/δ and φ=−κx+ωtδ/δ+η, the following ordinary differential equation (ODE) is obtained:(6)ϕu,u′,u″,…,=0,here prime represents the differentiation of u with respect to ζ.


### 2.1. The Generalized Kudryashov Procedure

According to the method, we assumeuζ as follows (32, 33):(7)uζ=∑n=0Nanψnζ∑m=0Mbmψmζ,where an,bmn=0,1,…N,m=0,1,…M are constants and they should be aN≠0, bM≠0 and the following ODE is satisfied by ψζ:(8)dψdζ=ψ2ζ−ψζ,and ψζ is given as follows:(9)ψξ=11+χeζ,χis integration constant,N and M are calculated by the homogeneous balance principle at (6). We can calculate a polynomial of ψ by substituting equation (7) into equation (6) without ignoring equation (8). Then, all the coefficients of the polynomial ψ are set to zero. If the obtained system is solved, the values of the an,bm,κ,v,ω are obtained. Finally, the soliton-type solutions of the given model are obtained.


### 2.2. The Modified Kudryashov Procedure

According to the method, the solutions of equation (6) are assumed as follows [34–36]:(10)uζ=∑m=0Mϖmψζm,ϖM≠0,where ϖmm=0,1,…,M are constants that will be determined later, M is calculated by the homogeneous balance principle, and the function ψζ is given by the following:(11)ψζ=11+χaζ,where (11) satisfies the following ODE:(12)ψ′ζ=ψ2ζ−ψζlna.Substituting equation (10) into equation (6) without ignoring equation (12), a set of algebraic equations is obtained for ϖm,a,χ,κ,v and ω. Finally, solving this obtained system, the exact solutions of equation (2) are calculated.


## 2.1. The Generalized Kudryashov Procedure

According to the method, we assumeuζ as follows (32, 33):(7)uζ=∑n=0Nanψnζ∑m=0Mbmψmζ,where an,bmn=0,1,…N,m=0,1,…M are constants and they should be aN≠0, bM≠0 and the following ODE is satisfied by ψζ:(8)dψdζ=ψ2ζ−ψζ,and ψζ is given as follows:(9)ψξ=11+χeζ,χis integration constant,N and M are calculated by the homogeneous balance principle at (6). We can calculate a polynomial of ψ by substituting equation (7) into equation (6) without ignoring equation (8). Then, all the coefficients of the polynomial ψ are set to zero. If the obtained system is solved, the values of the an,bm,κ,v,ω are obtained. Finally, the soliton-type solutions of the given model are obtained.


## 2.2. The Modified Kudryashov Procedure

According to the method, the solutions of equation (6) are assumed as follows [34–36]:(10)uζ=∑m=0Mϖmψζm,ϖM≠0,where ϖmm=0,1,…,M are constants that will be determined later, M is calculated by the homogeneous balance principle, and the function ψζ is given by the following:(11)ψζ=11+χaζ,where (11) satisfies the following ODE:(12)ψ′ζ=ψ2ζ−ψζlna.Substituting equation (10) into equation (6) without ignoring equation (12), a set of algebraic equations is obtained for ϖm,a,χ,κ,v and ω. Finally, solving this obtained system, the exact solutions of equation (2) are calculated.


## 3. The Applications

In this section, the used procedures will be applied to the given model. For this aim, the given model will be reduced to the nonlinear differential equation by the wave transformation. If we implement the wave transformation, namely, equation (5) to equation (1) then separate the real and imaginary parts, we get the following ODE:(13)−ωu+ϵu″−κ2u+λFu2u−2ρ−2σu′2u−2ρu″−εu=0,(14)v=−2ϵκ.If we takeρ=2σ, equation (13) reduces to the following ODE:(15)ϵ−4σu″−ω+ϵκ2+εu+λFu2u=0.If we takeFu2=u for the Kerr law nonlinearity, equation (15) reduces to the following ODE:(16)ϵ−4σu″−ω+ϵκ2+εu+λu3=0.If we balanceu″ and u3 in equation (16), the balance number is obtained as 1.


### 3.1. First Method

In this subsection, the generalized Kudryashov procedure will be applied to the equation (16). According to the method, we assume(17)u=a0+a1ψ+a2ψ2b0+b1ψ.If we substitute the solution (17) without ignoring the (8) in equation (16), we obtain an overdetermining equation system. If the obtained system is solved, four solution families are obtained as follows.


#### 3.1.1. First Family

The values of the arbitrary constants are obtained as follows:(18)a0=0,a1=±b1−ϵ−4σ2λ,a2=±b1ϵ−4σλ−ϵ−4σ/2λ,b0=0,b1=b1,ω=−ϵκ2−ϵ2+2σ−ε.Then, the solutions of the given model are obtained as follows:(19)q1,2x,t=∓χex−vtδ/δ−1ϵ−4σ−2ϵ+8σ/λ1+χex−vtδ/δλei−κx+ωtδ/δ+η.


#### 3.1.2. Second Family

The values of the arbitrary constants are obtained as follows:(20)a0=0,a1=±b0−8ϵ−32σλ,a2=∓b0−8ϵ−32σλ,b0=b0,b1=−2b0,ω=−ϵκ2+ɛ−4σ−ε,and the solutions are given by the following:(21)q3,4x,t=±2χex−vtδ/δ−2ϵ+8σ/λχ2e2x−vtδ/δ−1ei−κx+ωtδ/δ+η.


#### 3.1.3. Third Family

The values of the arbitrary constants are obtained as follows:(22)a0=±b0ϵ−4σλ−2ϵ−8σ/λ,a1=∓2b0−b1ϵ−4σλ−2ϵ−8σ/λ,a2=±b1−2ϵ−8σλ,b0=b0,b1=b1,ω=−ϵκ2−ϵ2+2σ−ε,and the solutions are given as follows:(23)q5,6x,t=±ϵ−4σχex−vtδ/δ−1λ1+χex−vtδ/δ−2ϵ−8σ/λei−κx+ωtδ/δ+η.


#### 3.1.4. Fourth Family

The values of the arbitrary constants are obtained as follows:(24)a0=∓b1ϵ−4σλ−2ϵ−8σ/λ,a1=±2b1ϵ−4σλ−2ϵ−8σ/λ,a2=±b1−2ϵ−8σ/λ,b0=−b12,b1=b1,ω=−ϵκ2−2ϵ+8σ−ε,and the solutions are given as follows:(25)q7,8x,t=±2ϵ−4σχ2e2x−vtδ/δ+1λχ2e2x−vtδ/δ−1−2ϵ−8σ/λei−κx+ωtδ/δ+η.The 3D and contour plots were given for (25) in Figure 1.Figure 1
The plot of the |(25)|forϵ=1,λ=2,σ=4,κ=−2,η=1,ε=2,χ=2,δ=0.1: (a) 3D plot and (b) contour plot.
(a)(b)


### 3.2. Second Method

In this subsection, the modified Kudryashov procedure will be applied to the equation (16). According to the method, we assume(26)uζ=ϖ0+ϖ1ψζ.If we substitute the solution (26) without ignoring the (12) in equation (16) and collect the polynomial of ψζ, we get an overdetermining equation system as follows:(27)ψ3:2lna2ϖ1ϵ−8lna2ϖ1σ+ϖ13λ,ψ2:−3lna2ϖ1ϵ+12lna2ϖ1σ+3ϖ0ϖ12λ,ψ1:lna2ϖ1ϵ−4lna2ϖ1σ+3ϖ02ϖ1λ−ϖ1ϵκ2−ϖ1ε−ϖ1ω,ψ0:λϖ03−ϖ0ϵκ2−ϖ0ε−ϖ0ω.If the above system is solved, the values of the arbitrary constants are obtained as follows:(28)ϖ0=±lna−ϵ−4σ2λ,ϖ1=±lnaϵ−4σλ−ϵ−4σ/2λ,ω=−lna2ϵ2+2lna2σ−κ2ϵ−ε.Then, the exact solutions are given by(29)q9,10x,t=±4σ−ϵlnaχax−vtδ/δ−1λ−2ϵ+8σ/λ1+χax−vtδ/δ.The 3D and contour plots were given for (29) in Figure 2.Figure 2
The plot of the |(29)| for ϵ  = 1, λ=2,σ=4,κ=−2,η=1,ε=2,χ=2,δ=0.9,a=3.: (a) 3D plot and (b) contour plot.
(a)(b)


## 3.1. First Method

In this subsection, the generalized Kudryashov procedure will be applied to the equation (16). According to the method, we assume(17)u=a0+a1ψ+a2ψ2b0+b1ψ.If we substitute the solution (17) without ignoring the (8) in equation (16), we obtain an overdetermining equation system. If the obtained system is solved, four solution families are obtained as follows.


### 3.1.1. First Family

The values of the arbitrary constants are obtained as follows:(18)a0=0,a1=±b1−ϵ−4σ2λ,a2=±b1ϵ−4σλ−ϵ−4σ/2λ,b0=0,b1=b1,ω=−ϵκ2−ϵ2+2σ−ε.Then, the solutions of the given model are obtained as follows:(19)q1,2x,t=∓χex−vtδ/δ−1ϵ−4σ−2ϵ+8σ/λ1+χex−vtδ/δλei−κx+ωtδ/δ+η.


### 3.1.2. Second Family

The values of the arbitrary constants are obtained as follows:(20)a0=0,a1=±b0−8ϵ−32σλ,a2=∓b0−8ϵ−32σλ,b0=b0,b1=−2b0,ω=−ϵκ2+ɛ−4σ−ε,and the solutions are given by the following:(21)q3,4x,t=±2χex−vtδ/δ−2ϵ+8σ/λχ2e2x−vtδ/δ−1ei−κx+ωtδ/δ+η.


### 3.1.3. Third Family

The values of the arbitrary constants are obtained as follows:(22)a0=±b0ϵ−4σλ−2ϵ−8σ/λ,a1=∓2b0−b1ϵ−4σλ−2ϵ−8σ/λ,a2=±b1−2ϵ−8σλ,b0=b0,b1=b1,ω=−ϵκ2−ϵ2+2σ−ε,and the solutions are given as follows:(23)q5,6x,t=±ϵ−4σχex−vtδ/δ−1λ1+χex−vtδ/δ−2ϵ−8σ/λei−κx+ωtδ/δ+η.


### 3.1.4. Fourth Family

The values of the arbitrary constants are obtained as follows:(24)a0=∓b1ϵ−4σλ−2ϵ−8σ/λ,a1=±2b1ϵ−4σλ−2ϵ−8σ/λ,a2=±b1−2ϵ−8σ/λ,b0=−b12,b1=b1,ω=−ϵκ2−2ϵ+8σ−ε,and the solutions are given as follows:(25)q7,8x,t=±2ϵ−4σχ2e2x−vtδ/δ+1λχ2e2x−vtδ/δ−1−2ϵ−8σ/λei−κx+ωtδ/δ+η.The 3D and contour plots were given for (25) in Figure 1.Figure 1
The plot of the |(25)|forϵ=1,λ=2,σ=4,κ=−2,η=1,ε=2,χ=2,δ=0.1: (a) 3D plot and (b) contour plot.
(a)(b)


## 3.1.1. First Family

The values of the arbitrary constants are obtained as follows:(18)a0=0,a1=±b1−ϵ−4σ2λ,a2=±b1ϵ−4σλ−ϵ−4σ/2λ,b0=0,b1=b1,ω=−ϵκ2−ϵ2+2σ−ε.Then, the solutions of the given model are obtained as follows:(19)q1,2x,t=∓χex−vtδ/δ−1ϵ−4σ−2ϵ+8σ/λ1+χex−vtδ/δλei−κx+ωtδ/δ+η.


## 3.1.2. Second Family

The values of the arbitrary constants are obtained as follows:(20)a0=0,a1=±b0−8ϵ−32σλ,a2=∓b0−8ϵ−32σλ,b0=b0,b1=−2b0,ω=−ϵκ2+ɛ−4σ−ε,and the solutions are given by the following:(21)q3,4x,t=±2χex−vtδ/δ−2ϵ+8σ/λχ2e2x−vtδ/δ−1ei−κx+ωtδ/δ+η.


## 3.1.3. Third Family

The values of the arbitrary constants are obtained as follows:(22)a0=±b0ϵ−4σλ−2ϵ−8σ/λ,a1=∓2b0−b1ϵ−4σλ−2ϵ−8σ/λ,a2=±b1−2ϵ−8σλ,b0=b0,b1=b1,ω=−ϵκ2−ϵ2+2σ−ε,and the solutions are given as follows:(23)q5,6x,t=±ϵ−4σχex−vtδ/δ−1λ1+χex−vtδ/δ−2ϵ−8σ/λei−κx+ωtδ/δ+η.


## 3.1.4. Fourth Family

The values of the arbitrary constants are obtained as follows:(24)a0=∓b1ϵ−4σλ−2ϵ−8σ/λ,a1=±2b1ϵ−4σλ−2ϵ−8σ/λ,a2=±b1−2ϵ−8σ/λ,b0=−b12,b1=b1,ω=−ϵκ2−2ϵ+8σ−ε,and the solutions are given as follows:(25)q7,8x,t=±2ϵ−4σχ2e2x−vtδ/δ+1λχ2e2x−vtδ/δ−1−2ϵ−8σ/λei−κx+ωtδ/δ+η.The 3D and contour plots were given for (25) in Figure 1.Figure 1
The plot of the |(25)|forϵ=1,λ=2,σ=4,κ=−2,η=1,ε=2,χ=2,δ=0.1: (a) 3D plot and (b) contour plot.
(a)(b)


## 3.2. Second Method

In this subsection, the modified Kudryashov procedure will be applied to the equation (16). According to the method, we assume(26)uζ=ϖ0+ϖ1ψζ.If we substitute the solution (26) without ignoring the (12) in equation (16) and collect the polynomial of ψζ, we get an overdetermining equation system as follows:(27)ψ3:2lna2ϖ1ϵ−8lna2ϖ1σ+ϖ13λ,ψ2:−3lna2ϖ1ϵ+12lna2ϖ1σ+3ϖ0ϖ12λ,ψ1:lna2ϖ1ϵ−4lna2ϖ1σ+3ϖ02ϖ1λ−ϖ1ϵκ2−ϖ1ε−ϖ1ω,ψ0:λϖ03−ϖ0ϵκ2−ϖ0ε−ϖ0ω.If the above system is solved, the values of the arbitrary constants are obtained as follows:(28)ϖ0=±lna−ϵ−4σ2λ,ϖ1=±lnaϵ−4σλ−ϵ−4σ/2λ,ω=−lna2ϵ2+2lna2σ−κ2ϵ−ε.Then, the exact solutions are given by(29)q9,10x,t=±4σ−ϵlnaχax−vtδ/δ−1λ−2ϵ+8σ/λ1+χax−vtδ/δ.The 3D and contour plots were given for (29) in Figure 2.Figure 2
The plot of the |(29)| for ϵ  = 1, λ=2,σ=4,κ=−2,η=1,ε=2,χ=2,δ=0.9,a=3.: (a) 3D plot and (b) contour plot.
(a)(b)


## 4. Conclusions

In this study, the new soliton-type solutions of the conformable time-fractional complex Ginzburg–Landau equation with Kerr law nonlinearity were obtained with the help of generalized and modified Kudryashov methods. Firstly, the given model was reduced to the nonlinear differential equation with the help of the wave transformation. Then, the balance number was calculated by the balance method. We calculate the balance number for the generalized Kudryashov method in a different way than usual. The generalized Kudryashov method was applied to the given model. Four solution families were obtained. The 3D and contour plots were plotted for the latest family. Then, another method was applied to the given model. Also, the results of the modified Kudryashov method include the logarithmic solutions. The 3D and contour plots were given the obtain solutions. The Maple software program was used for all obtained results and figures. According to our knowledge, our results are new in the literature. If we can calculate the balance number, the given methods provide soliton solutions for the nonlinear partial differential equations. All obtained results were checked by Maple and they are different from each other. Our findings in this paper are critical for explaining a wide range of scientific and physical applications. Thanks to this implementation, we contributed to the physical motions of the waves and other related areas. The proposed methods are effective and powerful for finding the soliton solutions of the nonlinear differential equations.In new studies, the given equation can be solved with a different kind of derivative and compared with our results, or the used methods can be applied to the different nonlinear partial differential equations.



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