The analytical and semi-analytical solutions to the quadratic–cubic fractional nonlinear Schrödinger equation are discussed in this research article. The model’s fractional formula is transformed into an integer-order model by using a new fractional operator. The theoretical and computational approaches can now be applied to fractional models, thanks to this transition. The application of two separate computing schemes yields a large number of novel analytical strategies. The obtained solutions secure the original and boundary conditions, which are used to create semi-analytical solutions using the Adomian decomposition process, which is often used to verify the precision of the two computational methods. All the solutions obtained are used to describe the shifts in a physical structure over time in cases where the quantum effect is present, such as wave-particle duality. The precision of all analytical results is tested by re-entering them into the initial model using Mathematica software 12.

Topics
Functional analysis, Nonlinear Schrodinger equation, Soliton solutions, Computational methods, Operator theory, Partial differential equations, Complex functions, Wave particle duality, Quantum effects, Surface waves

Many natural phenomena have been represented by nonlinear partial differential equations (NLPDEs). Based on these models, specific studies are applied to find the approximate and exact traveling wave solutions. These solutions help discover new characteristics of these models since the physical properties of each model play an essential role in its applications. These applications extend to many fields (nuclear science, atomic science, engineering, biological science, chemistry, and so on). The nonlinear partial differential equation has two types of formulas: the first type is a nonlinear partial differential equation with an integer derivative order, while the second type uses the fractional derivative order where the order of the derivative is a fractional number. The second type of the nonlinear partial differential equation was recently discussed. There exist many fractional definitions that investigate and study this kind of nonlinear partial differential equations such as the conformable fractional derivative, fractional Riemann–Liouville derivatives, Caputo fractional derivative, Caputo–Fabrizio definition, and, recently, fractional derivative.1–12 

The Schrödinger equation is one of these models that have many formulas, and there exist many researchers interested in mathematics or even in physics who tried and did their best to get the closed-form of solutions for this vital model.

In 2010, Jianke Yang investigated the nonlinear wave in integrable and nonintegrable systems in his book. Moreover, he applied some numerical methods to this equation to get approximate solutions of these models. In 2011, Fujioka et al. investigated the chaotic solitons of our model,13 and in 2006, Galaktionov and Svirshchevskii applied some methods to this model to get exact and solitary traveling wave solutions.14 In 2017, Triki et al. obtained optical soliton solutions of our model by utilizing the method of undetermined coefficients.15 In 2009, Khare, Avinash, Avadh Saxena, and Kody JH Law tried to study the mapping between generalized nonlinear Schrödinger (NLS) equations and neutral scalar and obtained exact traveling wave solutions of this model.

Through the last five decades, these researchers, in the meantime, who succeeded in their research’s purpose, formulated powerful techniques to obtain a closed form of solutions and solitary traveling wave solutions, regarded by many as one-of-a-kind types of nonlinear partial differential equations.16–20 

In this paper, we use two methods that are considered novel in this field: the generalized exp−ϕ(ϑ)-expansion method and the modified Khater method. The generalized exp−ϕ(ϑ)-expansion method was discovered by Hafez and Lu21 while the modified Khater method was discovered by Khater.22,23 We can see in Refs. 24–28 that the MK method is not only a natural extension of many methods in this field but also one of the most productive methods for different forms of solitary wave solutions. This feature gives strength to the method. The number of solutions enables researchers to be interested in the physical properties of these models to discover more about the properties and applications of these models.

The approach of this paper is systematized as follows: The fractional NLS equation is solved using the modified Khater method and the generalized exp-−ϕξ-expansion method in Sec. II. Section III explains our ideas and the differences between our findings and those achieved through other approaches, as well as what is fresh about this paper that qualifies it for release. Our paper’s conclusion is included in Sec. IV.

This part implements two different analytical methods and one semi-analytical scheme to obtain novel forms of the exact traveling wave solutions and approximate solutions of the quadratic–cubic fractional NLS equation, which can be written in the following form:

iDαYDtα+aD2αYDx2α−h1YY+h2YY2=0,
(1)

where i=−1,0<α<1⁠, h1 and h2 are the arbitrary constants. In addition, Y = Y(x, t) is the dependent variable, where t and x are the independent variables representing the temporal and spatial variables, respectively. While the real-valued constant a represents group velocity dispersion (GVD), as well as b1 and b2 being real-valued constants, the non-fractional form of Eq. (1) adopts the same from the equation when α = 1.29–32 The chaotic phenomenon of the equation was studied in Ref. 33. In Ref. 34, the analytical self-similar wave solutions of the equation were constructed. In Ref. 35, the method of undetermined clients was adopted to extract the soliton solutions, and the conservation laws of the equation were reported. In Ref. 36, He’s semi-inverse variational principle was adopted to study the equation. The Atangana–Baleanu fractional derivative that has the following definition is applied:4,6,8–10

Da+αABRf(t)=B(α)1−αddt∫atf(x)Eα−α(t−α)α1−αdx,
(2)

where Eα is the Mittag–Leffler function, which is defined by

Eα−α(t−α)α1−α=∑n=0∞−α1−αn(t−x)αnΓ(αn+1),

and B(α) is a normalization function. Thus,

Da+αABRf(t)=B(α)1−α∑n=0∞−α1−αnIaαnRLf(t),
(3)

where a > 0, 0 < α < 1, and α are the order of derivatives for the function Y(x, t). Implementation of the wave transformation on (1) results in

Y(x,t)=eiφu(ϑ),
φ=(1−α)B(α)∑n=0∞−α1−αnΓ(1−αn)−ρx−αn+ωt−αn,
ϑ=η(1−α)B(α)∑n=0∞−α1−αnΓ(1−αn)x−αn+ϱt−αn,

where the term φ = φ(x, t) represents the phase component, η is the frequency, ω represents the wave number, c represents the phase constant, k is the velocity that represents the width of the traveling wave and separates the result into real and imaginary components. A pair of equations is acquired where the imaginary part yields ϱ = 2 ρ a, leading to a real part of the following form:

aη2u″−(aρ2+ω)u−h1u2+h2u3=0.
(4)

Balancing the highest order derivative term and the nonlinear term in Eq. (4), one gets N = 1.

Based on the MK method, the general solution of Eq. (4) is given by

u(ϑ)=a0+a1Kϒ(ϑ)+d1K−ϒ(ϑ),
(5)

where a0, a1, d1, and K are the arbitrary constants. In addition, ϒ(ϑ) is the solution function of the equation ϒ′(ϑ)=χ+δK−ϒ(ϑ)+ςKϒ(ϑ)ln(K), where χ, δ, and ς are the arbitrary constants. Substituting Eq. (5) and its derivative into Eq. (4) leads to obtaining a system of algebraic equations. Equating the coefficient of Kiϒ(ϑ), where {i = 3, 2, 1, 0}, to zero and solving the obtained system by Maple 16 yield the following:

Family I:

a1→−a02χ2−4δa02ς−a0χ2δ,b1→0,ω→aη2χ2−4δς−ρ2,h1→−3aη2χa02χ2−4δς+a0χ2−4δς2a02,h2→−aη2χa02χ2−4δς+a0χ2−2δςa03.

Consequently, the solitary traveling wave solutions of Eq. (1) are given as follows:

When χ2 − 4 δ ς < 0 and ς ≠ 0,

Y1(x,t)=expi(1−δ)at−δη2χ2−4δς−ρ2−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0+a02χ2−4δς−a0χ4δςχ−4δς−χ2tan(1−δ)η4δς−χ2ϱt−δ+x−δ2B(δ)∑n=0∞−δ1−δnΓ(1−δn),
(6)
Y2(x,t)=expi(1−δ)at−δη2χ2−4δς−ρ2−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0+a02χ2−4δς−a0χ4δςχ−4δς−χ2cot(1−δ)η4δς−χ2ϱt−δ+x−δ2B(δ)∑n=0∞−δ1−δnΓ(1−δn).
(7)

When χ2 − 4 δ ς > 0 and ς ≠ 0,

Y3(x,t)=expi(1−δ)at−δη2χ2−4δς−ρ2−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0+a02χ2−4δς−a0χ4δςχ2−4δςtanh(1−δ)ηχ2−4δςϱt−δ+x−δ2B(δ)∑n=0∞−δ1−δnΓ(1−δn)+χ,
(8)
Y4(x,t)=expi(1−δ)at−δη2χ2−4δς−ρ2−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0+a02χ2−4δς−a0χ4δςχ2−4δςcoth(1−δ)ηχ2−4δςϱt−δ+x−δ2B(δ)∑n=0∞−δ1−δnΓ(1−δn)+χ.
(9)

When χ2 + 4 δ2 < 0 and δ = −ς,

Y5(x,t)=expi(δ−1)t−δx−δaxδ4δη2ς+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0−−δa02ςtan(1−δ)ηδςϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)δς,
(10)
Y6(x,t)=expi(δ−1)t−δx−δaxδ4δη2ς+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)×−δa02ςcot(1−δ)ηδςϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)δς+a0.
(11)

When χ2 + 4δ2 > 0 and δ = −ς,

Y7(x,t)=expi(δ−1)t−δx−δaxδ4δη2ς+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0−−δa02ςtanh(1−δ)η−δςϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)−δς,
(12)
Y8(x,t)=expi(δ−1)t−δx−δaxδ4δη2ς+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0−−δa02ςcoth(1−δ)η−δςϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)−δς.
(13)

When χ = 0 and δ = −ς,

Y9(x,t)=expi(1−δ)at−δ4δ2η2−ρ2−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)δa0−δ2a02coth(1−δ)δηϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)δ.
(14)

When χ=δ2=κandς=0⁠,

Y10(x,t)=expi(1−2κ)at−2κ(ηκ−ρ)(ηκ+ρ)−ρx−2κB(2κ)∑n=0∞2n−κ1−2κnΓ(1−2κn)a0+a0κ−a02κ24κexpη(1−2κ)κϱt−2κ+x−2κB(2κ)∑n=0∞2n−κ1−2κnΓ(1−2κn)−2.
(15)

When χ = 0 and δ = ς,

Y11(x,t)=expi(δ−1)t−δx−δaxδ4δ2η2+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)a0−−δ2a02δtan(1−δ)δηϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)+C.
(16)

When ς = 0, χ ≠ 0, and δ ≠ 0,

Y12(x,t)=expi(1−δ)at−δ(χη−ρ)(χη+ρ)−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)a0−a02χ2−a0χ2δexp(1−δ)χηϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)−δχ.
(17)

When χ2 − 4δς = 0,

Y13(x,t)=2a0B(δ)tδxδ∑n=0∞−δ1−δnΓ(1−δn)expi(δ−1)ρt−δx−δaρxδ+tδB(δ)∑n=0∞−δ1−δnΓ(1−δn)(δ−1)χηtδ+ϱxδ.
(18)

Family II:

a1→0,b1→−a02χ2−4δa02ς−a0χ2ς,ω→aη2χ2−4δς−ρ2,h1→−3aη2χa02χ2−4δς+a0χ2−4δς2a02,h2→−aη2χa02χ2−4δς+a0χ2−2δςa03.

Consequently, the solitary traveling wave solutions of Eq. (1) are given as follows:

When χ2 − 4 δ ς < 0, ς ≠ 0,

Y14(x,t)=exp(i(1−δ))at−δη2χ2−(4δ)ς−ρ2−x−δρB(δ)∑n=0∞−δ1−δnΓ(1−nδ)a0+χ2−(4δ)ςa02−χa0χ−(4δ)ς−χ2tan((1−δ)η)ϱt−δ+x−δ(4δ)ς−χ22B(δ)∑n=0∞−δ1−δnΓ(1−nδ),
(19)
Y15(x,t)=expi(1−δ)at−δη2χ2−4δς−ρ2−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0+a02χ2−4δς−a0χχ−4δς−χ2cot(1−δ)η4δς−χ2ϱt−δ+x−δ2B(δ)∑n=0∞−δ1−δnΓ(1−δn).
(20)

When χ2 − 4 δ ς > 0, ς ≠ 0,

Y16(x,t)=expi(1−δ)at−δη2χ2−4δς−ρ2−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0+a02χ2−4δς−a0χχ2−4δςtanh(1−δ)ηχ2−4δςϱt−δ+x−δ2B(δ)∑n=0∞−δ1−δnΓ(1−δn)+χ,
(21)
Y17(x,t)=expi(1−δ)at−δη2χ2−4δς−ρ2−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0+a02χ2−4δς−a0χχ2−4δςcoth(1−δ)ηχ2−4δςϱt−δ+x−δ2B(δ)∑n=0∞−δ1−δnΓ(1−δn)+χ.
(22)

When δς > 0 and ς ≠ 0 and δ ≠ 0 and χ = 0,

Y18(x,t)=expi(δ−1)t−δx−δaxδ4δη2ς+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0−−δa02ςcot(1−δ)ηδςϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)δς,
(23)
Y19(x,t)=expi(δ−1)t−δx−δaxδ4δη2ς+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0+−δa02ςtan(1−δ)ηδςϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)δς.
(24)

When δς < 0 and ς ≠ 0 and δ ≠ 0 and χ = 0,

Y20(x,t)=expi(δ−1)t−δx−δaxδ4δη2ς+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)×a0+−δa02ςcoth(1−δ)η−δςϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)−δς,
(25)
Y21(x,t)=expi(δ−1)t−δx−δaxδ4δη2ς+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)a0+−δa02ςtanh(1−δ)η−δςϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)−δς.
(26)

When χ = 0 and δ = −ς,

Y22(x,t)=expi(1−δ)at−δ4δ2η2−ρ2−ρx−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)δ2a02tanh(1−δ)δηϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)+δa0δ.
(27)

When χ = 0 and δ = ς,

Y23(x,t)=expi(δ−1)t−δx−δaxδ4δ2η2+ρ2+ρtδB(δ)∑n=0∞−δ1−δnΓ(1−δn)a0−−δ2a02cot(1−δ)δηϱt−δ+x−δB(δ)∑n=0∞−δ1−δnΓ(1−δn)+Cδ.
(28)

When χ2 − 4δς = 0,

Y24(x,t)=expi(δ−1)ρt−δx−δaρxδ+tδB(δ)∑n=0∞−δ1−δnΓ(1−δn)a0−(δ−1)a0χ3ηtδ+ϱxδ4δς(δ−1)χηtδ+ϱxδ−2B(δ)tδxδ∑n=0∞−δ1−δnΓ(1−δn).
(29)

Based on the generalized exp−ϕϑ-expansion method, the general solution of Eq. (4) takes the following form:

u(φ)=a0+a1e−ϕϑ,
(30)

where a0, a1 are the arbitrary constants. In addition, ϕ(φ) is the solution function of the equation ϕ′(ϑ)=L1+L2eϕ(ϑ)+L3eϕ(ϑ), where L1,L2,L3 are the arbitrary constants. Substituting Eq. (30) and its derivative into Eq. (4), obtaining the coefficient of e−iϕϑ⁠, where {i = 0, 1, 2, 3}, and equating them by zero give a system of algebraic equations. Solving this system of equations by any computer software yields

a1→a0L1−a0L12−4L2L32L2,ω→aη2L12−4L2L3−aρ2,h1→−3aη2L1L12−4L2L3+L1−4L2L32a0,h2→−aη2L1L12−4L2L3+L1−2L2L3a02.

Consequently, the solitary traveling wave solutions of Eq. (1) have the following form:

Case I. ⇒(L3=1)

When L12−4L2>0andL2≠0⁠,

Y25(x,t)=a0⁡expi(1−α)t−αaη2L12−4L2−aρ2−ρx−αB(α)∑n=0∞−α1−αnΓ(1−αn)L12−4L2−L1L1−L12−4L2tanh12L12−4L2(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ+1,
(31)
Y26(x,t)=a0⁡expi(1−α)t−αaη2L12−4L2−aρ2−ρx−αB(α)∑n=0∞−α1−αnΓ(1−αn)L12−4L2−L1L1−L12−4L2coth12L12−4L2(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ+1.
(32)

When L12−4L2=0andL2≠0⁠,

Y27(x,t)=12a0⁡expi(1−α)t−αaη2L12−4L2−aρ2−ρx−αB(α)∑n=0∞−α1−αnΓ(1−αn)L1−L12−4L2L1L2expL1(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ−1+2.
(33)

When L12−4L2=0&L1≠0andL2≠0⁠,

Y28(x,t)=14a0L12−4L2−L1L12(α−1)ηtα+ϱxα−φB(α)tαxα∑n=0∞−α1−αnΓ(1−αn)L2xα(α−1)ηL1ϱ−B(α)tαL1φ+2∑n=0∞−α1−αnΓ(1−αn)+(α−1)ηL1tα+4×expi(1−α)t−αaη2L12−4L2−aρ2−ρx−αB(α)∑n=0∞−α1−αnΓ(1−αn).
(34)

When L12−4L2=0&L1=0andL2=0⁠,

Y29(x,t)=12a0L1−L12−4L2L2(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ+2expi(1−α)t−αaη2L12−4L2−aρ2−ρx−αB(α)∑n=0∞−α1−αnΓ(1−αn).
(35)

When L12−4L2<0andL2≠0⁠,

Y30(x,t)=a0⁡expi(1−α)t−αaη2L12−4L2−aρ2−ρx−αB(α)∑n=0∞−α1−αnΓ(1−αn)L12−4L2−L1L1−4L2−L12tan124L2−L12(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ+1,
(36)
Y31(x,t)=a0⁡expi(1−α)t−αaη2L12−4L2−aρ2−ρx−αB(α)∑n=0∞−α1−αnΓ(1−αn)L12−4L2−L1L1−4L2−L12cot124L2−L12(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ+1.
(37)

Case II. ⇒(L1=0)

When L3andL2>0⁠,

Y32(x,t)=1L3a0⁡expi(α−1)t−αx−αaxαρ2+4η2L2L3+ρtαB(α)∑n=0∞−α1−αnΓ(1−αn)×L3−L3L2−L2L3cotL2L3(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ,
(38)
Y33(x,t)=1L3a0⁡expi(α−1)t−αx−αaxαρ2+4η2L2L3+ρtαB(α)∑n=0∞−α1−αnΓ(1−αn)×L3−L3L2−L2L3tanL2L3(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ.
(39)

When L2L3<0⁠,

Y34(x,t)=1L2a0⁡expi(α−1)t−αx−αaxαρ2+4η2L2L3+ρtαB(α)∑n=0∞−α1−αnΓ(1−αn)×−L2L3−L2L3tanh−L2L3(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ+L2.
(40)

When δχ<0⁠,

Y35(x,t)=1L2a0⁡expi(α−1)t−αx−αaxαρ2+4η2L2L3+ρtαB(α)∑n=0∞−α1−αnΓ(1−αn)×−L2L3−L2L3coth−L2L3(1−α)ηϱt−α+x−αB(α)∑n=0∞−α1−αnΓ(1−αn)+φ+L2.
(41)

Applying the Adomian decomposition method to Eq. (4) makes it take the following form:

Lu(ϑ)+Ru(ϑ)+Nu(ϑ)=0,
(42)

where (L,R,N) represent a differential operator, a linear operator and a nonlinear term, respectively. Using the inverse operator L−1 on (42) gets

∑i=0∞ui(ϑ)=u(0)+u′(0)ϑ+ak2+ωaη2L−1∑i=0∞ui+b1aη2L−1∑i=0∞Ai−b2aη2L−1∑i=0∞Ai.
(43)

Using the above-mentioned conditions and applying the Adomian decomposition method on Eq. (4) lead to the data shown in Table I.

TABLE I.

Absolute error between semi-analytical and exact solutions, which were obtained by using the modified Khater method and the generalized exp−ϕϑ-expansion method.

Value of ϑAbs. error of the MK methodAbs. error of the generalized exp−ϕϑ-expansion method
0.001 2.347 93 × 10−8 4.447 99 × 10−6 
0.002 9.379 49 × 10−8 1.780 61 × 10−5 
0.003 2.107 64 × 10−7 4.009 55 × 10−5 
0.004 3.742 03 × 10−7 7.133 72 × 10−5 
0.005 5.839 29 × 10−7 0.000 111 552 
0.006 8.397 6 × 10−7 0.000 160 76 
0.007 1.141 51 × 10−6 0.000 218 983 
0.008 1.489 01 × 10−6 0.000 286 241 
0.009 1.882 06 × 10−6 0.000 362 554 
0.01 2.320 48 × 10−6 0.000 447 941 
Value of ϑAbs. error of the MK methodAbs. error of the generalized exp−ϕϑ-expansion method
0.001 2.347 93 × 10−8 4.447 99 × 10−6 
0.002 9.379 49 × 10−8 1.780 61 × 10−5 
0.003 2.107 64 × 10−7 4.009 55 × 10−5 
0.004 3.742 03 × 10−7 7.133 72 × 10−5 
0.005 5.839 29 × 10−7 0.000 111 552 
0.006 8.397 6 × 10−7 0.000 160 76 
0.007 1.141 51 × 10−6 0.000 218 983 
0.008 1.489 01 × 10−6 0.000 286 241 
0.009 1.882 06 × 10−6 0.000 362 554 
0.01 2.320 48 × 10−6 0.000 447 941 
View Large

The necessary steps of the methods and the relation between these two methods and the Riccati equation are as follows:

  • Basic steps of the methods:

    In this paper, two analytical techniques were employed to find novel traveling wave solution formulas of the quadratic–cubic fractional NLS equation. The main idea of these methods [the modified Khater method and the generalized exp−ϕϑ-expansion method] is to convert the nonlinear partial differential equations to nonlinear ordinary differential equations by using the traveling wave transformation; then, using the homogeneous balance rule between the highest derivative term and nonlinear term, the nonlinear ODE is obtained; then, using the general solution, the suggestions by the methods are as follows:
    Y(ϑ)=∑i=0Naikiϒ(ϑ)⇒The modified Khater method,∑i=0Naiexp−iϕ(ϑ)⇒ Generalized exp −ϕϑ expansion method.
    These solutions depend on the following auxiliary equations, respectively:
    f′(ϑ)=1ln(K)δk−ϒ(ϑ)+χ+ςkϒ(ϑ),ϕ′(ϑ)=δexp(−ϕ(ϑ))+χexp(ϕ(ϑ))+ς.
    Using the solutions of these auxiliary equations under specific conditions and submitting these solutions into the exact traveling wave solution lead to the solitary traveling wave solutions of the suggested model.

  • The similarity between both methods:

    The solitary solutions of both methods depend on the solutions of auxiliary equations for each of them. By carefully looking into both equations, we can find that they are the same when K=e=2.7183,ϒ(ϑ)=ϕ(ϑ) that leads to the same solutions. The exp-function properties are used for both methods to get many forms of solutions that help many researchers who do not have a background in mathematics. This similarity is not limited to these three ways, but it applies to most of the schemes in this area.33,37

  • Comparison between our solutions and that obtained in previous work:

    We show a comparison between our solutions and that obtained by Aslan and Inc in Ref. 38 as follows:

    Aslan and Inc applied the Jacobi elliptic functions to the quadratic–cubic NLS equation when δ = 1. They obtained bright and dark optical soliton solutions (15) and (28). We obtain many different forms of solutions that are completely different from that obtained in Ref. 38 that makes our solutions novel and considerable for publication.

  • Numerical solutions of the quadratic–cubic fractional NLS equation:

    The Adomian decomposition method is applied to this model under specific boundary conditions obtained by using the resulting solutions of the analytically used methods. Figure 1 and Table II show the difference between the absolute error for both analytical schemes used [the modified Khater method and the generalized exp−ϕϑ-expansion method]. This shows that the accuracy of solutions obtained by the modified Khater method is more than that obtained by the second method.

FIG. 1.
FIG. 1. Relation of the absolute error between the Adomian decomposition method and the two used analytical schemes based on Table I, which shows the superiority of the modified Khater method over the other used method.
View largeDownload slide

Relation of the absolute error between the Adomian decomposition method and the two used analytical schemes based on Table I, which shows the superiority of the modified Khater method over the other used method.

FIG. 1.
FIG. 1. Relation of the absolute error between the Adomian decomposition method and the two used analytical schemes based on Table I, which shows the superiority of the modified Khater method over the other used method.
View largeDownload slide

Relation of the absolute error between the Adomian decomposition method and the two used analytical schemes based on Table I, which shows the superiority of the modified Khater method over the other used method.

Close modal
TABLE II.

Initial conditions for both analytical methods.

Modified Khater methodGeneralized exp−ϕϑ-expansion method
Sol. Num. Equation (8) Equation (40) 
Arbitrary constants a0=−1&χ=5&δ=6&ς=1 a0=−1&L2=−4&L3=1&L1=0 
Exact solution −124+524tanhϑ2 tanh(2ϑ)+1 
Modified Khater methodGeneralized exp−ϕϑ-expansion method
Sol. Num. Equation (8) Equation (40) 
Arbitrary constants a0=−1&χ=5&δ=6&ς=1 a0=−1&L2=−4&L3=1&L1=0 
Exact solution −124+524tanhϑ2 tanh(2ϑ)+1 
View Large

In this research, we succeeded in the implementation of the modified Khater method and the generalized exp(−ϕ(ξ))-expansion method for the quadratic–cubic fractional NLS equation. We obtained different formulas of solitary traveling wave solutions of this model. The modified Khater method has been considered as one of the few generalization methods to get the exact and solitary solution as it can cover most of the solitary traveling wave solutions obtained by some of the other methods. We gave a comparison between our solutions and that obtained by another researcher who used a different method.38 We also gave a numerical study of our obtained solutions to show the accuracy of our exact solutions. We show this convergence between exact and numerical solutions in Fig. 1.

The authors would like to thank Taif University Researchers Supporting Project (No. TURSP-2020/159), Taif University, Saudi Arabia.

There is no conflict of interest.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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2021
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