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The exact solutions of conformable time-fractional modified nonlinear Schrödinger equation by Direct algebraic method and Sine-Gordon expansion method

  • In this article, we used direct algebraic method (DAM) and sine-Gordon expansion method (SGEM), to find the analytical solutions of conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE) and finally, we present numerical results in tables and charts.

    Citation: Safoura Rezaei Aderyani, Reza Saadati, Javad Vahidi, Nabil Mlaiki, Thabet Abdeljawad. The exact solutions of conformable time-fractional modified nonlinear Schrödinger equation by Direct algebraic method and Sine-Gordon expansion method[J]. AIMS Mathematics, 2022, 7(6): 10807-10827. doi: 10.3934/math.2022604

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  • In this article, we used direct algebraic method (DAM) and sine-Gordon expansion method (SGEM), to find the analytical solutions of conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE) and finally, we present numerical results in tables and charts.



    The notion of fractional derivative (FD) is more than three thousand years old, the role of fractional calculus has been increasing due to its application zone in various domains including 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, nonlinear optics etc. the fractional differential equations (FDEs), have received a great deal of interest from scholars and researchers. Many mathematicians presented diverse types of FDs, such as that given in [1,2]. The most famous ones are Hadamard, Marchaud, Riemann-Liouville, Grunwald-Letnikov, Kober, Caputo, Riesz and Erdelyi.

    Lately, a novel FD has been presented by Khalil et al. [3] and others [4,5,6], called conformable FD. Due to the significance of the exact solutions of NLSEs, plenty of mathematicians solved them with conformable derivative, who used the different methods like first integral method (FIM) [7,8], functional variable method (FVM) [7,9], trial equation method (TEM) [10], modified trial equation method (MTEM) [11], direct algebraic method (DAM) [12] and sine-Gordon expansion method (SGEM) [13] to find the exact solutions to NLSEs.

    In [14], Younas et al. presented CTFMNLSE and studied the exact solutions of it by the generalized exponential rational function method. Also in [15,16], authors introduced some new solutions of CTFMNLSE via FIM, FVM, TEM and MTEM.

    Motivated by the work done in [14,15,16], we consider the following CTFMNLSE:

    iDατΨ+σ1ΨXX+σ2|Ψ|2Ψ=iδ1ΨXXX+iδ2Ψ2ΨXiδ3|Ψ|2ΨX+δ4Ψ0<α1, (1.1)

    where σ1=P08K20(3cos(Θ)+2), σ2=P0K202, δ1=P0cos(Θ)16K30(5cos2(Θ)6), δ2=P0K0cos(Θ)4, δ3=3P0K02,δ4=K0|Ψ|2X|X=0, P0 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.

    In this section, we present some properties and definitions of the conformal derivative and other Preliminaries.

    Definition 2.1. [3] Suppose Ω:(0,)R is a function. Therefore the conformal fractional derivative of Ω of order α is as follows

    Tα(τ)=limϵ0Ω(τ+ϵτ1α)Ω(τ)ϵ (2.1)

    for all 0<α<1,0<τ.

    Definition 2.2. [3] Suppose ι0 and τι. Also, suppose Ω is a function defined on (ι,τ] and αR. Therefore, the α-fractional integral of Ω is defined by,

    IαιΩ(τ)=τιΩ(ς)ς1αdς, (2.2)

    if the Riemann improper integral exists.

    Theorem 2.3. [3]Suppose 0<α1, and Ω and are αdifferentiable at a point τ, therefore

    (i) Tα(ϖ1Ω+ϖ2)=ϖ1Tα(Ω)+ϖ2Tα(),ϖ1,ϖ2R.

    (ii) Tα(tϖ)=ϖτϖα,ϖR.

    (iii) τTα(Ω)=ΩTα()+Tα(Ω).

    (iv) Tα(Ω)=Tα(Ω)ΩTα()2.

    Furthermore, if Ω is differentiable, then Tα(Ω)(τ)=τ1αdΩdτ.

    Theorem 2.4. [3] Suppose Ω:(0,)R is a function s.t. Ω is differentiable and also α-differentiable. Suppose is a function defined in the range of Ω and also differentiable; therefore, one has the following rule

    Tα(Ωo)(τ)=τ1α(τ)Ω((τ)).

    Remark 2.5. Let

    Q(ξ)=Ln(A)(P0+P1Q(ξ)+P2Q2(ξ)),A0,1. (2.3)

    The solutions of ODE (2.3) are:

    (1) When P214P0P2<0 and P20,

    Q1(ξ)=P12P2+(P214P0P2)2P2tanA((P214P0P2)2ξ),
    Q2(ξ)=P12P2(P214P0P2)2P2cotA((P214P0P2)2ξ),
    Q3(ξ)=P12P2+(P214P0P2)2P2tanA((P214P0P2)ξ)±sr(P214P0P2)2P2secA((P214P0P2)ξ),
    Q4(ξ)=P12P2(P214P0P2)2P2cotA((P214P0P2)ξ)±sr(P214P0P2)2P2cscA((P214P0P2)ξ),
    Q5(ξ)=P12P2+(P214P0P2)4P2tanA((P214P0P2)4ξ)(P214P0P2)4P2cotA((P214P0P2)4ξ),

    (2) When P214P0P2>0 and P20,

    Q6(ξ)=P12P2P214P0P22P2tanhA(P214P0P22ξ),
    Q7(ξ)=P12P2P214P0P22P2cothA(P214P0P22ξ),
    Q8(ξ)=P12P2P214P0P22P2tanhA(P214P0P2ξ)±isr(P214P0P2)2P2sechA(P214P0P2ξ),
    Q9(ξ)=P12P2P214P0P22P2cothA(P214P0P2ξ)±sr(P214P0P2)2P2cschA(P214P0P2ξ),
    Q10(ξ)=P12P2P214P0P24P2tanhA(P214P0P24ξ)P214P0P24P2cothA(P214P0P24ξ),

    where generalized hyperbolic and triangular functions are given by

    coshA(θ)=sAθ+rAθ2,sinhA(θ)=sAθrAθ2,
    cothA(θ)=sAθ+rAθseθreθ,tanhA(θ)=sAθrAθseθ+reθ,
    cschA(θ)=2sAθrAθ,sechA(θ)=2sAθ+rAθ,
    cosA(θ)=sAiθ+rAiθ2i,sinA(θ)=sAiθrAiθ2i,
    cotA(θ)=isAiθ+rAiθsAiθrAiθ,tanA(θ)=isAiθrAiθsAiθ+rAiθ,
    cscA(θ)=2isAiθrAiθ,secA(θ)=2sAiθ+rAiθ,

    where θ is an independent variable, A0,1, and s and r are arbitrary constants greater than zero and are called deformation parameters.

    In this section, we present the first step of the DAM and the SGEM, for finding analytical solutions of CTFMNLSE defined as (1.1). Suppose a CTFNLPDE,

    Γ(Φ,Φτ,ΦX,DατΦ,DβXΦ,D2ατ,D2βX,)=0,0<α,β1, (3.1)

    where Γ and Φ are a polynomial and an unknown function in its arguments, respectively. Using a fractional travelling wave transformation

    Φ(X,τ)=Λ(ξ),ξ=XVατα (3.2)

    where V is velocity and substituting (3.2) into (3.1), we have a NLODE given by

    Υ(Λ,Λ,Λ,Λ,)=0, (3.3)

    in which signifies the derivative with respect to ξ.

    Since Ψ=Ψ(X,τ) in (1.1) is a complex function, we begin with the following travelling wave assumption

    Ψ(X,τ)=Λ(ξ)eiψ (3.4)

    where ξ=η(XVατα) and ψ=KX+Pατα+ζ, and ζ,P and K are parameters, represent the phase constant, frequency and wave number respectively. Substitute (3.4) into (1.1), we get real and imaginary parts as follows

    η2(σ13δ1K)Λ+(σ2+(δ2+δ3)K)Λ3+(Pσ1K2+δ1K3δ4)Λ=0, (3.5)

    and

    (3δ1K2V2σ1K)Λδ1η2Λ+(δ3δ2)Λ2Λ=0. (3.6)

    Now integrating the imaginary part of the equation and taking constant equal to zero one may have

    3(3δ1K2V2σ1K)Λ3δ1η2Λ+(δ3δ2)Λ3=0. (3.7)

    From (3.5) and (3.7), it can be followed that

    K3δ1Pσ1K2δ43(3δ1K2V2σ1)K=σ13δ1K3δ1=Kδ2+Kδ3+σ2δ3δ2. (3.8)

    From above, it can be followed that

    V=δ1P+δ1δ4+2K(σ12δ1K)2σ13δ1K,
    K=σ1(δ2δ3)3σ2δ16δ1δ2.

    Rewrite (3.5) into following form

    Λ+λ1Λ3λ2Λ=0 (3.9)

    or

    Λ=λ2Λλ1Λ3 (3.10)

    where λ1=σ2+(δ2+δ3)Kη2(σ13δ1K) and λ2=Pσ1K2+δ1K3δ4η2(σ13δ1K).

    In the next two subsections, we investigate the primary steps for detecting the exact solution of (3.5) by using DAM and the SGEM. Similarly we can find the exact solution of (3.7).

    Firstly, CTFNLPDE (3.1) is reduced to NLODE (3.3) under the transformation (3.2). Secondly, let us consider that Eq (3.3) has a formal solution of the form

    Λ(ξ)=Nj=0bjQj(ξ),bN0, (3.11)

    where bj(j=0,,N) are constant coefficients to be detected later and Q(ξ) satisfies the ODE in the form (2.3). Now, we are able to determine the value N in (3.11) by balancing the highest order derivative term and the highest order nonlinear term in (3.3). Substitute (3.11) along with its required derivatives into (3.3) and compares the coefficients of powers of Q(ξ) in the resultant equation for getting the set of algebraic equation. In the end, we {solve} the set of algebraic equations and put the results generated in (3.11) to obtain the exact solutions of (3.1).

    Now, balancing the order of Λ and Λ3 in (3.10), we get N=1. Therefore, Eq (3.11) is presented by

    Λ(ξ)=b0+b1Q(ξ). (3.12)

    By substituting (3.12) into (3.10) and gathering all terms with the same order of Q(ξ) together, the left-hand side of (3.10) are converted into polynomial in Q(ξ). {Putting} every coefficient of every polynomial to zero, we get a set of algebraic equations for b0 and b1. Now, we have

    Λ3=b30+3b20b1Q+3b0b21Q2+b31Q3andΛ=b1Q, (3.13)

    where

    Q=Ln2(A)(P0+P1Q+P2Q2)[P1+2P2Q]. (3.14)

    Coefficients of Q(ξ) as follows:

    0:b1Ln2(A)P0P1+λ1b30λ2b0=01:b1Ln2(A)[P21+2P0P2]+3λ1b20b1λ2b1=02:3b1Ln2(A)P1P2+3λ1b0b21=03:2b1Ln2(A)P22+λ1b31=0.

    We earn the following values, by solving the above system of equations for b0 and b1:

    b0=±i2λ1Ln(A)P1,b1=±i2λ1Ln(A)P2. (3.15)

    The solutions of (1.1) corresponding to (3.4), (3.12) and (3.15) are:

    (1) When P214P0P2<0, and P20,

    Ψ1,2(X,τ)=(±i)ln(A)(P214P0P2)2λ1ei(KX+Pατα+ζ)tanA((P214P0P2)2(η(XVατα)+ξ0)),
    Ψ1,3(X,τ)=(±i)ln(A)(P214P0P2)2λ1ei(KX+Pατα+ζ)cotA((P214P0P2)2(η(XVατα)+ξ0)),
    Ψ1,4(X,τ)=(±i)ln(A)ei(KX+Pατα+ζ)[(P214P0P2)2λ1tanA((P214P0P2)(η(XVατα)+ξ0))±sr(P214P0P2)2λ1secA((P214P0P2)(η(XVατα)+ξ0))],
    \begin{eqnarray*} \Psi_{1, 5}(\mathfrak{X}, \tau)& = &(\pm\text{i})\text{ln}(A)\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[-\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{cot}_{A}\bigg( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\\ &\pm &\notag\dfrac{\sqrt{-sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{csc}_{A}\bigg( \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg), \end{eqnarray*}
    \begin{eqnarray*} \Psi_{1, 6}(\mathfrak{X}, \tau)& = &(\pm\text{i})\text{ln}(A)\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\dfrac{\sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{8\lambda_{1}}}\text{tan}_{A}\bigg( \dfrac{ \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{4}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\\ &- &\notag\dfrac{\sqrt{- ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{8\lambda_{1}}}\text{cot}_{A}\bigg( \dfrac{ \sqrt{-( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{4}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\bigg], \end{eqnarray*}

    (2) When \mathfrak{P}_{1}^{2}-4\mathfrak{P}_{0}\mathfrak{P}_{2} > 0, and \mathfrak{P}_{2}\neq 0,

    \begin{equation*} \Psi_{1, 7}(\mathfrak{X}, \tau) = (\pm\text{i})\text{ln}(A)\dfrac{\sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{2\lambda_{1}}}\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\text{tanh}_{A}\bigg( \dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg), \end{equation*}
    \begin{equation*} \Psi_{1, 8}(\mathfrak{X}, \tau) = (\pm\text{i})\text{ln}(A)\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{2\lambda_{1}}}\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\text{coth}_{A}\bigg( \dfrac{\sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{2}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg), \end{equation*}
    \begin{eqnarray*} \Psi_{1, 9}(\mathfrak{X}, \tau)& = &(\pm\text{i})\text{ln}(A)\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[-\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{2\lambda_{1}}}\text{tanh}_{A}\bigg( \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\bigg)\\ &\pm &\notag\text{i}\dfrac{\sqrt{sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{sech}_{A}\bigg( \sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\bigg], \end{eqnarray*}
    \begin{eqnarray*} \Psi_{1, 10}(\mathfrak{X}, \tau)& = &\pm\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[-\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{2\lambda_{1}}}\text{coth}_{A}\bigg( \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\\ &\pm &\notag\dfrac{\sqrt{sr ( \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2})}}{\sqrt{2\lambda_{1}}}\text{csch}_{A}\bigg( \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\bigg], \end{eqnarray*}
    \begin{eqnarray*} \Psi_{1, 11}(\mathfrak{X}, \tau)& = &(\pm\text{i})\text{ln}(A)\text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\dfrac{\sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{8\lambda_{1}}}\text{tanh}_{A}\bigg( \dfrac{ \sqrt{ \mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{4}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\bigg)\\ &+ &\notag\dfrac{\sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{\sqrt{8\lambda_{1}}}\text{coth}_{A}\bigg( \dfrac{ \sqrt{\mathfrak{P}^{2}_{1} - 4\mathfrak{P}_{0} \mathfrak{P}_{2}}}{4}\bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg) \bigg)\bigg], \end{eqnarray*}

    where \xi_{0} is an arbitrary constant. For more details see [17,18].

    Consider the sine-Gordon equation from [19,20]

    \begin{equation} \Psi_{\mathfrak{X}\mathfrak{X}}-\Psi_{\tau\tau} = m^{2}\sin(\Psi), \end{equation} (3.16)

    where \Psi = \Psi (\mathfrak{X}, \tau) and m is a constant. To solve the equation through sine-Gordon expansion algorithm, first we use the transformation \Psi (\mathfrak{X}, \tau) = \Lambda(\xi) where \xi = \mu (\mathfrak{X}-c\tau) which reduce (3.16) to the following NLODE:

    \begin{equation} \Lambda^{\prime\prime} = \dfrac{m^{2}}{\mu^{2}(1-c^{2})}\sin(\Lambda). \end{equation} (3.17)

    After that, we multiply \Lambda^{\prime} on both sides of (3.17) and integrate it once which gives

    \begin{equation} \bigg[ \bigg( \dfrac{\Lambda}{2} \bigg) \bigg]^{2} = \dfrac{m^{2}}{\mu^{2}(1-c^{2})}\sin^{2}\bigg( \dfrac{\Lambda}{2} \bigg) +k, \end{equation} (3.18)

    in which k is an integration constant. Therefore by putting k = 0, \: \Lambda_{2} = w(\xi), and \dfrac{m^{2}}{\mu^{2}(1-c^{2})} = a^{2} in (3.18), we get

    \begin{equation} w^{\prime} = a\sin (w), \end{equation} (3.19)

    which by setting a = 1 in (3.19), we have

    \begin{equation} w^{\prime} = \sin (w). \end{equation} (3.20)

    Equation (3.20) is a simplified form of the sine-Gordon Eq (3.16). Thus, it has the following solutions:

    \begin{equation} \sin(w) = \text{sech}(\xi), \:\:\:\:\:\:\:\cos(w) = \text{tanh}(\xi), \end{equation} (3.21)

    and

    \begin{equation} \sin(w) = \text{i} \text{csch}(\xi), \:\:\:\:\:\:\:\cos(w) = \text{coth}(\xi). \end{equation} (3.22)

    Here, firstly, CTFNLPDE (3.1) is reduced to NLODE (3.3) under the transformation (3.2). Secondly, we apply the following transformation

    \begin{equation} \Lambda (w) = \sum\limits_{j = 1}^{N}\cos^{j-1}(w)[ B_{j} \sin(w)+A_{j}\cos(w)]+A_{0}. \end{equation} (3.23)

    It is supposed that the solution \Lambda (\xi) of the nonlinear (3.3) along with (3.21) and (3.22) can be demonstrated as

    \begin{equation} \Lambda (\xi) = \sum\limits_{j = 1}^{N}\text{tanh}^{j-1}(\xi)[ B_{j} \text{sech}(\xi)+A_{j}\text{tanh}(\xi)]+A_{0}. \end{equation} (3.24)

    and

    \begin{equation} \Lambda (\xi) = \sum\limits_{j = 1}^{N}\text{coth}^{j-1}(\xi)[ B_{j} \text{csch}(\xi)+A_{j}\text{coth}(\xi)]+A_{0}. \end{equation} (3.25)

    After detecting the value of N by means of using the homogeneous balance principle, substituting its value into (3.23) and setting the result into the reduced ODE (3.20) give a nonlinear algebraic system. Equating the coefficients of \sin^{j}(w) and \cos^{j}(w) equal to zero and solving the acquired system yield the values of A_{j} and B_{j} . Finally, after substituting the values of A_{j} and B_{j} into (3.24) and (3.25), we are able to earn the solitary wave solutions for (3.1).

    We used the balancing technique to Eq (3.10) by considering the highest derivative \Lambda^{\prime\prime} and the highest power nonlinear term \Lambda^{3} , which the value of N is gained as N+2 = 3N or N = 1 . Thus, we have the following equations

    \begin{equation} \Lambda(w) = B_{1}\sin(w)+A_{1}\cos(w)+A_{0}, \end{equation} (3.26)
    \begin{equation} \Lambda^{\prime}(w) = B_{1}\cos(w)\sin(w)-A_{1}\sin^{2}(w), \end{equation} (3.27)

    and

    \begin{equation} \Lambda^{\prime\prime}(w) = B_{1}[ \cos^{2}(w)\sin(w)-\sin^{3}(w)]-2A_{1}\sin^{2}(w)\cos(w). \end{equation} (3.28)

    Also we have

    \begin{eqnarray} \Lambda^{3}& = & B_{1}^{3}\sin^{3}(w)+A_{1}^{3}\cos^{3}(w)\\ &+& 3A_{0}B_{1}^{2}\sin^{2}(w)+3A_{1}^{2}A_{0}\cos^{2}(w)\\ &+& 3A_{0}^{2}B_{1}\sin(w)+3 A_{0}^{2}A_{1}\cos(w)\\ &+& 3A_{1}B_{1}^{2}\sin^{2}(w)\cos(w) +3A_{1}^{2}B_{1}\cos^{2}(w)\sin(w)\\ &+& 6A_{0}A_{1}B_{1}\sin(w)\cos(w)\\ &+& A_{0}^{3}. \end{eqnarray} (3.29)

    By substituting (3.26)–(3.29) into (3.10), we obtain the following nonlinear algebraic system:

    \begin{eqnarray*} && \sin^{3}(w):\:\:\: \:\:\: -B_{1}+\lambda_{1}B_{1}^{3} = 0, \\ && \cos^{3}(w):\:\:\:\:\:\: \lambda_{1}A_{1}^{3} = 0, \\ && \sin^{2}(w):\:\:\: \:\:\: 3\lambda_{1}A_{0}B_{1}^{2} = 0, \\ && \cos^{2}(w)\:\:\: \:\:\: 3\lambda_{1}A_{1}A_{0} = 0, \\ &&\sin(w):\:\:\: \:\:\: 3\lambda_{1}A_{0}^{2}B_{1}-\lambda_{2}B_{1} = 0\\ &&\cos(w):\:\:\:\: \:\:\: 3\lambda_{1}A_{0}^{2}A_{1}-\lambda_{2}A_{1} = 0, \\ &&\sin^{2}(w)\cos(w):\:\:\: \:\:\: -2A_{1}+3\lambda_{1}A_{1}B_{1}^{2} = 0, \\ &&\cos^{2}(w)\sin(w):\:\:\: \:\:\: B_{1}+3\lambda_{1}A_{1}^{2}B_{1} = 0, \\ &&\sin(w)\cos(w):\:\:\:\:\:\: 6\lambda_{1}A_{0}A_{1}B_{1} = 0, \\ &&\sin^{0}(w)\cos^{0}(w): \:\:\:\:\:\: \lambda_{1}A_{0}^{3}-\lambda_{2}A_{0} = 0. \end{eqnarray*}

    Using (3.21) and (3.26), we earn the following traveling wave solutions:

    Case 1: A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = \pm \sqrt{-\dfrac{1}{3\lambda_{1}}}, \: B_{1} = 0.

    \begin{equation} \Psi_{2, 1} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm \sqrt{-\dfrac{1}{3\lambda_{1}}} \text{tanh} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation} (3.30)
    \begin{equation} \Psi_{2, 2} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm \sqrt{-\dfrac{1}{3\lambda_{1}}} \text{coth} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg]. \end{equation} (3.31)

    Case 2: A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = 0, \: B_{1} = \pm \sqrt{\dfrac{2}{3\lambda_{1}}}.

    \begin{equation} \Psi_{2, 3} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm \sqrt{\dfrac{2}{3\lambda_{1}}} \text{sech} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation} (3.32)
    \begin{equation} \Psi_{2, 4} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm\text{i} \sqrt{\dfrac{2}{3\lambda_{1}}} \text{csch} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg]. \end{equation} (3.33)

    Case 3: A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = 0, \: B_{1} = \pm \sqrt{\dfrac{1}{\lambda_{1}}}.

    \begin{equation} \Psi_{2, 5} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm \sqrt{\dfrac{1}{\lambda_{1}}} \text{sech} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation} (3.34)
    \begin{equation} \Psi_{2, 6} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[\pm\text{i} \sqrt{\dfrac{1}{\lambda_{1}}}\text{csch} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg]. \end{equation} (3.35)

    Case 4: A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = \pm \sqrt{-\dfrac{1}{3\lambda_{1}}}, \: B_{1} = \pm \sqrt{\dfrac{1}{\lambda_{1}}}.

    \begin{equation} \Psi_{2, 7} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[ \pm \sqrt{\dfrac{1}{\lambda_{1}}} \text{sech} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\pm \sqrt{-\dfrac{1}{3\lambda_{1}}}\text{tanh} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation} (3.36)
    \begin{equation} \Psi_{2, 8} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[ \pm \text{i}\sqrt{\dfrac{1}{\lambda_{1}}} \text{csch} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\pm \sqrt{-\dfrac{1}{3\lambda_{1}}}\text{coth} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg]. \end{equation} (3.37)

    Case 5: A_{0} = \pm \sqrt{\dfrac{\lambda_{2}}{\lambda_{1}}}\: \:or \:\:\pm \sqrt{\dfrac{\lambda_{2}}{3\lambda_{1}}}\:\: or \:\: 0, \: A_{1} = \pm \sqrt{-\dfrac{1}{3\lambda_{1}}}, \: B_{1} = \pm \sqrt{\dfrac{2}{3\lambda_{1}}}.

    \begin{equation} \Psi_{2, 9} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[ \pm \sqrt{\dfrac{2}{3\lambda_{1}}} \text{sech} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\pm \sqrt{-\dfrac{1}{3\lambda_{1}}}\text{tanh} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation} (3.38)
    \begin{equation} \Psi_{2, 10} = \text{e}^{\text{i}(-\mathcal{K}\mathfrak{X} + \dfrac{\mathcal{P}}{\alpha}\tau^{\alpha}+\zeta)}\bigg[ \pm \text{i}\sqrt{\dfrac{2}{3\lambda_{1}}} \text{csch} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)\pm \sqrt{-\dfrac{1}{3\lambda_{1}}}\text{coth} \bigg(\eta (\mathfrak{X} - \dfrac{\mathcal{V}}{\alpha}\tau^{\alpha})+\xi_{0}\bigg)+A_{0}\bigg], \end{equation} (3.39)

    where \xi_{0} is an arbitrary constant. For more details, see [21,22].

    Let \delta_{1} = \delta_{3} = \eta = s = r = \mathfrak{P}_{0} = \mathfrak{P}_{1} = \mathfrak{P}_{2} = 1, \: \mathcal{P} = \delta_{2} = \delta_{4} = \sigma_{2} = A = 2, \: \zeta = \mathcal{V} = 0.5, \: \mathcal{K} = 0.25, \: \sigma_{1} = -1, \: \alpha = 0.90, \: \xi_{0} = A_{0} = 0. Therefore, we have \lambda_{1} = -1.57143 and \lambda_{2} = -2.24107. We now present numerical results in tables and charts.

    Figure 1 (a) and (e) show the 3D with the both real and imaginary parts of the solution \Psi_{1, 1} and \Psi_{1, 3} for different values of \mathfrak{X} and \tau and also, Figure 1 (b), (f) and (c), (d) show the 3D and 2D with the both real and imaginary part of the solution \Psi_{1, 1} and \Psi_{1, 3} for fixed \mathfrak{X} and different values of \tau through DAM, using the above values. Now, Figure 2 (a) and (c) show the 3D with the both real and imaginary parts of the solution \Psi_{2, 1} and \Psi_{2, 3} for different values of \mathfrak{X} and \tau and also Figure 2 (b) and (d) show the 3D with the both real and imaginary part of the solution \Psi_{2, 1} and \Psi_{2, 3} for fixed \mathfrak{X} and different values of \tau through SGEM, using the above values.

    Figure 1.  The figures (a) and (e) show the 3D with the both real and imaginary parts of the solution \Psi_{1, 1} and \Psi_{1, 3} for different values of \mathfrak{X} and \tau and also the figures (b), (f) and (c), (d) show 3D and 2D with the both real and imaginary parts of the solution \Psi_{1, 1} and \Psi_{1, 3} for fixed \mathfrak{X} and different values of \tau through DAM, under the values presented in Section 4.
    Figure 2.  The figures (a) and (c) show the 3D with the both real and imaginary parts of the solution \Psi_{2, 1} and \Psi_{2, 3} for different values of \mathfrak{X} and \tau . The figures (b) and (d) show 3D with the both real and imaginary parts of the solution \Psi_{2, 1} and \Psi_{2, 3} for fixed \mathfrak{X} and different values of \tau through SGEM, under the values presented in Section 4.

    Moreover, Figure 3 displays the 3D with the real and imaginary part of solution \Psi_{2, 1} for fixed \mathfrak{X} and different values of \alpha \:( = 0.50-0.09), obtained via SGEM.

    Figure 3.  The 3D with the real and imaginary parts of solution \Psi_{2, 1} for fixed \mathfrak{X} and different values of \alpha \:( = 0.50-0.09), obtained via SGEM.

    Figure 4 (a) shows the 3D with the differences between the real and also the imaginary part of solutions \Psi_{1, 1} and \Psi_{2, 1} , and also Figure 4 (b) shows the 3D with the differences between the real and also the imaginary part of solutions \Psi_{1, 3} and \Psi_{2, 3} , for fixed \mathfrak{X} and fixed \alpha . Note that, these differences are minor in a wide range of domains. which implies that both methods leads to similar results except for some values.

    Figure 4.  (a) shows the 3D with the differences between the real and also the imaginary part of solutions \Psi_{1, 1} and \Psi_{2, 1} , and also (b) shows the 3D with the differences between the real and also the imaginary part of the solutions \Psi_{1, 3} and \Psi_{2, 3} , for fixed \mathfrak{X} and fixed \alpha .

    Tables 14 present the numerical results of the solutions of CTFMNLSE (1.1) obtained by DAM and SGEM with several point sources trough arbitrary.

    Table 1.  The real part of exact solutions of CTFMNLSE (1.1) obtained by DAM, with several point sources trough arbitrary.
    \mathfrak{X} \tau \Psi_{1, 1}^{\Re}(\mathfrak{X}, \tau) \Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau) \Psi_{1, 3}^{\Re}(\mathfrak{X}, \tau) \Psi_{1, 4}^{\Re}(\mathfrak{X}, \tau)
    0.012 0.00058 -0.00058 587.19354 -587.19354
    0.012 0.037 -0.00550 0.00550 -55.91444 55.91444
    0.062 -0.01056 0.01056 -26.29075 26.29075
    0.012 0.00933 -0.00933 36.46445 -36.46445
    0.037 0.037 0.00284 -0.00284 109.34814 -109.34814
    0.062 -0.00266 0.00266 -105.25186 105.25186
    0.012 0.01816 -0.01816 18.87874 -18.87874
    0.062 0.037 0.01124 -0.01124 27.81915 -27.81915
    0.062 0.00531 -0.00531 53.34247 -53.34247

     | Show Table
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    Table 2.  The imaginary part of exact solutions of CTFMNLSE (1.1) obtained by DAM, with several point sources trough arbitrary.
    \mathfrak{X} \tau \Psi_{1, 1}^{\Im}(\mathfrak{X}, \tau) \Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau) \Psi_{1, 3}^{\Im}(\mathfrak{X}, \tau) \Psi_{1, 4}^{\Im}(\mathfrak{X}, \tau)
    0.012 0.00034 -0.00034 350.78189 -350.78189
    0.012 0.037 -0.00385 0.00385 -39.18969 39.18969
    0.062 -0.00852 0.00852 -21.21368 21.21368
    0.012 0.00549 -0.00549 21.47530 -21.47530
    0.037 0.037 0.00196 -0.00196 75.62594 -75.62594
    0.062 -0.00212 0.00212 -83.84577 83.84577
    0.012 0.01054 -0.01054 10.96007 -10.96007
    0.062 0.037 0.00767 -0.00767 18.98398 -18.98398
    0.062 0.00417 -0.00417 41.95143 -41.95143

     | Show Table
    DownLoad: CSV
    Table 3.  The real part of exact solutions of CTFMNLSE (1.1) obtained by SGEM, with several point sources trough arbitrary.
    \mathfrak{X} \tau \Psi_{2, 1}^{\Re}(\mathfrak{X}, \tau) \Psi_{2, 2}^{\Re}(\mathfrak{X}, \tau) \Psi_{2, 3}^{\Re}(\mathfrak{X}, \tau) \Psi_{2, 4}^{\Re}(\mathfrak{X}, \tau)
    0.012 0.00065 -0.00065 241.48348 -241.48348
    0.012 0.037 -0.00624 0.00624 -22.75660 22.75660
    0.062 -0.01197 0.01197 -10.70930 10.70930
    0.012 0.01055 -0.01055 14.88481 -14.88481
    0.037 0.037 0.00320 -0.00320 44.79798 -44.79798
    0.062 -0.00304 0.00304 -42.62661 42.62661
    0.012 0.02052 -0.02052 7.71125 -7.71125
    0.062 0.037 0.01270 -0.01270 11.36482 -11.36482
    0.062 0.00599 -0.00599 21.83731 -21.83731

     | Show Table
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    Table 4.  The imaginary part of exact solutions of CTFMNLSE (1.1) obtained by SGEM, with several point sources trough arbitrary.
    \mathfrak{X} \tau \Psi_{2, 1}^{\Im}(\mathfrak{X}, \tau) \Psi_{2, 2}^{\Im}(\mathfrak{X}, \tau) \Psi_{2, 3}^{\Im}(\mathfrak{X}, \tau) \Psi_{2, 4}^{\Im}(\mathfrak{X}, \tau)
    0.012 0.00038 -0.00038 144.25913 -144.25913
    0.012 0.037 -0.00437 0.00437 -15.94980 15.94980
    0.062 -0.00965 0.00965 -8.64120 8.64120
    0.012 0.00621 -0.00621 8.76623 -8.76623
    0.037 0.037 0.00221 -0.00221 30.98259 -30.98259
    0.062 -0.00241 0.00241 -33.95722 33.95722
    0.012 0.01191 -0.01191 4.47677 -4.47677
    0.062 0.037 0.00866 -0.00866 7.75543 -7.75543
    0.062 0.00471 -0.00471 17.17404 -17.17404

     | Show Table
    DownLoad: CSV

    In Table 5, based on Tables 14, separately, we calculate the differences between solutions \Psi_{1, 1}, \Psi_{1, 3}, \Psi_{2, 1}, and \Psi_{2, 3}, represented by \Delta \Psi_{1, 1}, \Delta \Psi_{2, 1}, \Delta \Psi_{2, 1}, and \Delta \Psi_{2, 3}, for fixed \mathfrak{X} = 0.012 and different values of \tau . Note that \Re and \Im show the real and imaginary part of solutions. As we can observe, for fixed \mathfrak{X} by changing the value of \tau , both DAM and SGEM result major changes for solutions \Psi_{1, 3} and \Psi_{2, 3}, and here we are not dealing with an advantageous result. Nevertheless, SGEM results more minor changes than DAM.

    Table 5.  According to Tables 1-4, separately, we calculate the differences between solutions \Psi_{1, 1}, \Psi_{1, 3}, \Psi_{2, 1}, and \Psi_{2, 3}, represented by \Delta \Psi_{1, 1}, \Delta \Psi_{2, 1}, \Delta \Psi_{2, 1}, and \Delta \Psi_{2, 3}, for fixed \mathfrak{X} = 0.012 and different values of \tau . Note that \Re and \Im show the real and imaginary part of solutions.
    \mathfrak{X}=0.012
    \tau=0.012, 0.037, 0.062
    DAM SGEM
    \Delta\Psi_{1, 1}^{\Re} \Delta\Psi_{1, 1}^{\Im} \Delta\Psi_{1, 3}^{\Re} \Delta\Psi_{1, 3}^{\Im} \Delta\Psi_{2, 1}^{\Re} \Delta\Psi_{2, 1}^{\Im} \Delta\Psi_{2, 3}^{\Re} \Delta\Psi_{2, 3}^{\Im}
    0.00608 0.00419 643.10798 389.97158 0.00689 0.00475 264.24008 160.20893
    0.01114 0.00886 613.48429 771.99557 0.01262 0.01003 252.19278 152.90033
    0.00506 0.00467 29.62369 17.97601 0.00573 0.00528 12.04730 7.30860

     | Show Table
    DownLoad: CSV

    Tables 69 propose the real and imaginary part of exact solutions of CTFMNLSE (1.1) obtained by six different methods: FIM, FVM, TEM, MTEM [15,16] and also DAM and SGEM, with several point sources trough arbitrary. For some values the results obtained through DAM and SGEM, are near to the results obtained in four other methods.

    Table 6.  The real part of exact solutions of CTFMNLSE (1.1) obtained by 6 different methods, with several point sources trough arbitrary.
    \mathfrak{X} \tau \overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{FIM}} \overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{FVM}} \overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{TEM}} \overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{MTEM}} \overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{DAM}} \overbrace{\Psi_{1, 2}^{\Re}(\mathfrak{X}, \tau)}^{\text{SGEM}}
    0.012 0.012 \mp 0.00099 \mp 0.86524 \pm 1.44911 \mp 0.00083 \pm 0.00057 \pm 0.00064
    0.037 \pm 0.01209 \mp 0.96849 \pm 1.38186 \pm 0.00941 \mp 0.00549 \mp 0.00623
    0.062 \pm 0.02666 \mp 1.06066 \pm 1.31204 \pm 0.02081 \mp 0.01055 \mp 0.01196
    0.087 \pm 0.04264 \mp 1.14551 \pm 1.23880 \pm 0.03333 \mp 0.01482 \mp 0.01679
    0.037 0.012 \mp 0.01689 \mp 0.85415 \pm 1.45334 \mp 0.01342 \pm 0.00933 \pm 0.01055
    0.037 \mp 0.00583 \mp 0.95628 \pm 1.38820 \mp 0.00478 \pm 0.00283 \pm 0.00319
    0.062 \pm 0.00691 \mp 1.04758 \pm 1.32017 \pm 0.00518 \mp 0.00266 \mp 0.00303
    0.087 \pm 0.02124 \mp 1.13177 \pm 1.24849 \pm 0.01639 \mp 0.00738 \mp 0.00838
    0.062 0.012 \mp 0.03247 \mp 0.84187 \pm 1.45547 \mp 0.02575 \pm 0.01815 \pm 0.02052
    0.037 \mp 0.02346 \mp 0.94274 \pm 1.39254 \mp 0.01873 \pm 0.01124 \pm 0.01270
    0.062 \mp 0.01251 \mp 1.03303 \pm 1.32641 \mp 0.01019 \pm 0.00530 \pm 0.00598
    0.087 \pm 0.00016 \mp 1.11640 \pm 1.25641 \mp 0.00028 \pm 0.00013 \pm 0.00012
    0.087 0.012 \mp 0.04776 \mp 0.82848 \pm 1.45550 \mp 0.03784 \pm 0.02705 \pm 0.03053
    0.037 \mp 0.04079 \mp 0.92793 \pm 1.39486 \mp 0.03244 \pm 0.01972 \pm 0.02227
    0.062 \mp 0.03165 \mp 1.01707 \pm 1.33074 \mp 0.02533 \pm 0.01336 \pm 0.01509
    0.087 \mp 0.02062 \mp 1.09949 \pm 1.26251 \mp 0.01673 \pm 0.00773 \pm 0.00872

     | Show Table
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    Table 7.  The real part of exact solutions of CTFMNLSE (1.1) obtained by 6 different methods, with several point sources through arbitrary.
    \mathfrak{X} \tau \overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{FIM}} \overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{FVM}} \overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{TEM}} \overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{MTEM}} \overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{DAM}} \overbrace{\Psi_{3, 4}^{\Re}(\mathfrak{X}, \tau)}^{\text{SGEM}}
    0.012 0.012 \mp 378.21073 \mp 70.07373 \pm 351.38054 \mp 351.20376 \pm 587.19354 \pm 241.48348
    0.037 \pm 38.81779 \mp 30.84863 \mp 39.15773 \pm 39.17920 \mp 55.91443 \mp 22.75659
    0.062 \pm 21.08640 \mp 19.36377 \mp 21.19424 \pm 21.20629 \mp 26.29074 \mp 10.70930
    0.087 \pm 15.32460 \mp 14.48859 \mp 15.38516 \pm 15.39533 \mp 16.74259 \mp 6.82515
    0.037 0.012 \mp 21.73603 \mp 28.35492 \pm 21.46931 \mp 21.47600 \pm 36.46444 \pm 14.88481
    0.037 \mp 78.99292 \mp 24.09697 \pm 75.68598 \mp 75.67674 \pm 109.34813 \pm 44.79797
    0.062 \pm 80.05677 \mp 18.81347 \mp 83.68725 \pm 83.77139 \mp 105.25185 \mp 42.62660
    0.087 \pm 30.34363 \mp 15.03660 \mp 30.84459 \pm 30.86541 \mp 33.98710 \mp 13.82036
    0.062 0.012 \mp 11.06619 \mp 16.76490 \pm 10.95095 \mp 10.95634 \pm 18.87873 \pm 7.71124
    0.037 \mp 19.30540 \mp 16.27571 \pm 18.97941 \mp 18.98481 \pm 27.8191 \pm 11.36481
    0.062 \mp 43.52313 \mp 14.65664 \pm 41.97445 \mp 41.97341 \pm 53.34247 \pm 21.83730
    0.087 \pm 3894.75067 \mp 12.88518 \pm 1783.25721 \mp 1751.00331 \pm 1903.01800 \pm 950.20622
    0.087 0.012 \mp 7.36271 \mp 11.81796 \pm 7.29069 \mp 7.29620 \pm 12.76218 \pm 5.21999
    0.037 \mp 10.90080 \mp 11.82677 \pm 10.75601 \mp 10.76155 \pm 15.99035 \pm 6.53613
    0.062 \mp 16.93629 \mp 11.28924 \pm 16.60991 \mp 16.61494 \pm 21.40072 \pm 8.74788
    0.087 \mp 30.39447 \mp 10.50579 \pm 29.40686 \mp 29.40818 \pm 33.17835 \pm 13.57924

     | Show Table
    DownLoad: CSV
    Table 8.  The imaginary part of exact solutions of CTFMNLSE (1.1) obtained by 6 different methods, with several point sources through arbitrary.
    \mathfrak{X} \tau \overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{FIM}} \overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{FVM}} \overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{TEM}} \overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{MTEM}} \overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{DAM}} \overbrace{\Psi_{1, 2}^{\Im}(\mathfrak{X}, \tau)}^{\text{SGEM}}
    0.012 0.012 \pm 0.00165 \pm 1.44929 \pm 0.86568 \pm 0.00140 \pm 0.00034 \pm 0.00038
    0.037 \mp 0.01726 \pm 1.38406 \pm 0.96853 \mp 0.01342 \mp 0.00385 \mp 0.00437
    0.062 \mp 0.033041 \pm 1.31780 \pm 1.05867 \mp 0.02579 \mp 0.00851 \mp 0.00965
    0.087 \mp 0.04635 \pm 1.24922 \pm 1.13976 \mp 0.03622 \mp 0.01364 \mp 0.01544
    0.037 0.012 \pm 0.02868 \pm 1.45317 \pm 0.85592 \pm 0.02279 \pm 0.00549 \pm 0.00621
    0.037 \pm 0.00844 \pm 1.38971 \pm 0.96009 \pm 0.00691 \pm 0.00196 \pm 0.00220
    0.062 \mp 0.00867 \pm 1.32525 \pm 1.05167 \mp 0.00651 \mp 0.00212 \mp 0.00241
    0.087 \mp 0.02338 \pm 1.25853 \pm 1.13434 \mp 0.01804 \mp 0.00671 \mp 0.00762
    0.062 0.012 \pm 0.05593 \pm 1.45494 \pm 0.84497 \pm 0.04436 \pm 0.01054 \pm 0.01191
    0.037 \pm 0.03438 \pm 1.39330 \pm 0.95027 \pm 0.02745 \pm 0.00767 \pm 0.00866
    0.062 \pm 0.01591 \pm 1.33069 \pm 1.04316 \pm 0.01295 \pm 0.00417 \pm 0.00470
    0.087 \mp 0.00018 \pm 1.26590 \pm 1.12728 \pm 0.00031 \pm 0.00011 \pm 0.00011
    0.087 0.012 \pm 0.08346 \pm 1.45459 \pm 0.83287 \pm 0.06613 \pm 0.01548 \pm 0.01747
    0.037 \pm 0.06058 \pm 1.39481 \pm 0.93913 \pm 0.04818 \pm 0.01327 \pm 0.01499
    0.062 \pm 0.04077 \pm 1.33411 \pm 1.03317 \pm 0.03263 \pm 0.01037 \pm 0.01171
    0.087 \pm 0.02328 \pm 1.27129 \pm 1.11860 \pm 0.01889 \pm 0.00685 \pm 0.00773

     | Show Table
    DownLoad: CSV
    Table 9.  The imaginary part of exact solutions of CTFMNLSE (1.1) obtained by 6 different methods, with several point sources through arbitrary.
    \mathfrak{X} \tau \overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{FIM}} \overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{FVM}} \overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{TEM}} \overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{MTEM}} \overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{DAM}} \overbrace{\Psi_{3, 4}^{\Im}(\mathfrak{X}, \tau)}^{\text{SGEM}}
    0.012 0.012 \pm 633.10822 \pm 12.28177 \mp 588.19565 \pm 587.89974 \pm 350.78189 \pm 144.25913
    0.037 \mp 55.38382 \pm 19.40644 \pm 55.86884 \mp 55.89947 \mp 39.18969 \mp 15.94980
    0.062 \mp 26.13300 \pm 14.23268 \pm 26.26665 \mp 26.28158 \mp 21.21368 \mp 8.64120
    0.087 \mp 16.65633 \pm 10.67754 \pm 16.72216 \mp 16.73321 \mp 15.40396 \mp 6.27945
    0.037 0.012 \pm 36.90715 \mp 7.51554 \mp 36.45427 \pm 36.46563 \pm 21.47530 \pm 8.76623
    0.037 \pm 114.21647 \pm 1.26633 \mp 109.43494 \pm 109.42158 \pm 75.62594 \pm 30.98259
    0.062 \mp 100.49552 \pm 4.11852 \pm 105.05287 \mp 105.15849 \mp 83.84576 \mp 33.95722
    0.087 \mp 33.39707 \pm 4.55124 \pm 33.94843 \mp 33.97135 \mp 30.87972 \mp 12.55679
    0.062 0.012 \pm 19.06153 \mp 6.31111 \mp 18.86303 \pm 18.87231 \pm 10.96007 \pm 4.47677
    0.037 \pm 28.29015 \mp 2.73162 \mp 27.81244 \pm 27.82036 \pm 18.98398 \pm 7.75543
    0.062 \pm 55.34093 \mp 0.47397 \mp 53.37174 \pm 53.37043 \pm 41.95143 \pm 17.17404
    0.087 \mp 4340.87665 \pm 0.69068 \mp 1987.52122 \pm 1951.57277 \pm 1707.43866 \pm 852.55044
    0.087 0.012 \pm 12.86682 \mp 5.00514 \mp 12.74097 \pm 12.75059 \pm 7.30283 \pm 2.98701
    0.037 \pm 16.19046 \mp 3.32263 \mp 15.97541 \pm 15.98364 \pm 10.76607 \pm 4.40068
    0.062 \pm 21.81420 \mp 2.01126 \mp 21.39382 \pm 21.40030 \pm 16.61527 \pm 6.79175
    0.087 \pm 34.30495 \mp 1.10820 \mp 33.19028 \pm 33.19177 \pm 29.39629 \pm 12.03132

     | Show Table
    DownLoad: CSV

    Using the DAM and SGEM, firstly we found the exact solutions of CTFMNLSE (1.1) and finally, we presented numerical results in tables and charts. Also, we compared the real and imaginary parts of the exact solutions of CTFMNLSE (1.1) obtained by DAM and SGEM with four other different methods: FIM, FVM, TEM, MTEM. For some values the results obtained through DAM and SGEM, were near to the results obtained in four other methods. Overall, the performance of the proposed methods (DAM and SGEM) is reliable and effective and gives more solutions. These methods are direct and concise. Therefore, we conclude these methods can be extended to solve many nonlinear conformable fractional PDEs which are arising in the theory of solitons and other areas.

    The authors T. Abdeljawad and N. Mlaiki would like to thank Prince Sultan University for paying the APC and for the support through the TAS research lab.

    The authors declare no conflicts of interest.



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