# Some Nonlinear Fractional PDEs Involvingβ-Derivative by Using Rational exp−Ωη-Expansion Method

Authors: Haifa Bin Jebreen
Journal: Complexity
 (2020)

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## Abstract

In this article, some new nonlinear fractional partial differential equations (PDEs) (the space-time fractional order Boussinesq equation; the space-time (2 + 1)-dimensional breaking soliton equations; and the space-time fractional order SRLW equation) have been considered, in which the treatment of these equations in the diverse applications are described. Also, the fractional derivatives in the sense ofβ-derivative are defined. Some fractional PDEs will convert to consider ordinary differential equations (ODEs) with the help of transformation β-derivative. These equations are analyzed utilizing an integration scheme, namely, the rational exp−Ωη-expansion method. Different kinds of traveling wave solutions such as solitary, topological, dark soliton, periodic, kink, and rational are obtained as a by product of this scheme. Finally, the existence of the solutions for the constraint conditions is also shown. The outcome indicates that some fractional PDEs are used as a growing finding in the engineering sciences, mathematical physics, and so on.


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## Body

## 1. Introduction

Traveling wave and soliton solutions are one of the most interesting and fascinating areas of research in different fields of engineering and physical sciences. This type of solutions, basically structured by solving nonlinear partial differential equations (NLPDEs), arises in physical phenomena such as optic, acoustic wave, and mechanic sciences. Consequently, it is imperative to address the dynamics of these soliton solutions from a mathematical aspect. This will lead to a deeper understanding of the engineering perspective of these solutions [1–10]. Some of the authors investigated novel models of partial differential equations containing the Fokas–Lenells equation [11], the General Satisfaction Index [12], the multiasset option pricing problem [13], the affine jump-diffusion model [14], and the (1 + 1)-dimensional coupled nonlinear Schrödinger equation [15]. An important analysis in signal analysis including the transfer function, Bode diagram, Nyquist plot, and Nichols plot was obtained based on the Laplace transform by Atangana and Akgül [16]. As a result, in the past few years, several effective, rising, and realistic methods have been initiated and dilated to extract closed form solutions to the NLPDEs; for example, the semi-inverse principle method is one of the powerful methods many scholars studied including the fractional nonlinear model of the low-pass electrical transmission lines [17] and the higher-order nonlinear Schrödinger equation with the non-Kerr nonlinear term [18]. Xie et al. worked on nonlinear optical soliton containing a variable-coefficient AB system in the geophysical fluids or nonlinear optics [19], studied the turbulence for a coupled nonlinear Schrödinger system to find the rogue waves solutions [20], and described the propagation of ultrashort optical pulses in a birefringent fiber in the coupled Hirota equations [21]. Rizvi et al. investigated the optical solitons for the Biswas–Milovic equation by the new extended auxiliary equation method [22] and the modulated compressional dispersive alfven and heisenberg ferromagnetic spin chains [23]. Some authors studied the conformable fractional derivative on some of the conformable time-fractional variant bussinesq equations [24], the space-time fractional simplified MCH and SRLW equations [25], and the fractional and classical GEW-Burgers equations [26] by utilizing the analytical methods.In this paper, we will study the different kinds of traveling wave solutions in generalized nonlinear Schrödinger equation modeling which describe the model of few-cycle pulse propagation in metamaterials with parabolic law of nonlinearity from a purely mathematical viewpoint. Therefore, the importance of this paper will be to extract exact traveling wave solution for the nonlinear model. This model is described by the decoupled nonlinear Schrödinger equation (NLSE) with group velocity mismatch, group velocity dispersion, and spatio-temporal dispersion. There are several integration tools available to solve the model. Many nonlinear Schrödinger equations have been examined with regards to soliton theory, where complete integrability was emphasized by various analytical techniques. A few methods are improvedtanϕ/2-expansion method, the homotopy analysis method, the variation iteration principle, the exp-function method, the generalized (G′/G)-expansion approach, the Hirota bilinear method, simplest equation method, etc. [1–46]. The space-time fractional order Boussinesq equation (STFBE) with quadratic-cubic nonlinearity involving the β-derivative [47, 48] is presented as follows:(1)Dt2αΨ+bDx2αΨ+βDx2αΨ2+γDx4αΨ=0,t>0,0<α≤1,where Dt2α⋅=d2α/dt2α⋅Ψx,t is the vertical deflection, β is the coefficient controlling nonlinearity and the quadratic nonlinearity, and γ is the dispersion parameter depending on the rigidity characteristics of the material and compression [47, 48]. Furthermore, we take the space-time (2 + 1)-dimensional breaking soliton equations (STFBSEs) [48, 49] in the following:(2)Dt2αΨ+aDx2y3αΨ+4aΨDxαΩ+4aΩDxαΨ=0,t>0,0<α≤1,DyαΨ−DxαΩ=0.Mohyud-Din and Bibi [48] used the exponential rational function method to get exact solutions of nonlinear STFBSEs. Also, Wen and Zheng [49] implemented the new fractional subequation method to nonlinear STFBSEs. Finally, a few of authors including Mohyud-Din and Bibi [48], Xu [50], and Alzaidy [51] considered the space-time fractional order SRLW equation (STFSRLWE) as below:(3)Dt2αΨ+Dx2αΨ+ΨDtαDxαΨ+DtαΨDxαΨ+Dt2αDx2αΨ=0,t>0,0<α≤1.The rest of this paper is structured as follows. The initial definitions are given in Section2. In Section 3, an overview of the integration scheme is given along with the analysis of the model. In Sections 4, 5, and 6, respectively, the STFBE, STFBSEs, and STFSRLWE have been employed to obtain the rational dark, periodic, kink-singular, cupson, and travelling wave solutions with the considered method. In Section 7, discussion of mitigating internet bottleneck of parabolic law of nonlinearity has been offered. In Section 8, the conclusion has been given.


## 2. Initial Definitions

Definition 1 (definition ofβ-derivative).
Supposeχ:0;1⟶R, then the β derivative of χ of order α is defined as(4)Dtαχt=limϵ⟶0χt+ϵt+1/Γα1−α−χtϵ,α∈0,1,t>0.
The properties and new theorems will used as follows.Theorem 1.
Supposeα∈0,1; let χandψ be α-differentiable at a point t; therefore, (1)
Dtαaχt+bψt=aDtαχt+bDtαψt,fora,b∈R(2)
Dtαc=0,forc∈R(3)
Dtαχtψt=χtDtαψt+ψtDtαχt(4)
Dtαχt/ψt=χtDtαψt−ψtDtαχt/ψ2t(5)
Dtαχt=t+1/Γα1−αdχt/dtTheorem 2.
Supposeχ:0;1⟶R be a function such that χ is differentiable and also α-differentiable. Assume ψ be a differentiable function defined in the range of χ. Therefore, we have(5)Dtαχoψt=t+1/Γα1−αψ′tχ′ψt,where prime denotes the classical derivatives with respect to t.The proofs of the aboveβ-derivative properties were obviously presented in [52]. Also, improvement of fractional derivative has been made in works of Atangana and Baleanu in Refs. [53, 54].


## 3. The Rationalexp−Ωη-Expansion Method

This section elucidates a systematic explanation of rationalexp−Ωη-expansion method so that it can be further applied to optical metamaterials with parabolic nonlinearity in order to furnish its exact solutions.Step 1:.
the following NLPDE,(6)Nu,ux,ut,uxx,utt,…=0,can be transformed into an ordinary differential equation (ODE)(7)QU,U′,νU′,U″,ν2U″,…=0,by using the suitable transformation η=x+νt, where ν is the free parameter which would be calculated subsequently.Step 2:.
assume the solution of the ODE to be of the following form:(8)Uη=∑j=0NAjFjη∑j=0MBjFjη,where Fη=exp−Ωη and Aj0≤j≤NandBj0≤j≤M are constants to be determined, such that AN,BM≠0; Ω=Ωη satisfying the ODE is given below:(9)Ω′=μF−1η+Fη+λ.The special cases formed from the solutions of the ODE given in equation (9) are mentioned below.Solution 1.
Ifμ≠0 and λ2−4μ>0, then we have(10)Ωη=ln−λ2−4μ2μtanhλ2−4μ2η+E−λ2μ,where E is integral constant.Solution 2.
Ifμ≠0 and λ2−4μ<0, then we have(11)Ωη=ln−λ2+4μ2μtan−λ2+4μ2η+E−λ2μ.Solution 3.
Ifμ=0, λ≠0, and λ2−4μ>0, then we obtain(12)Ωη=−lnλexpλη+E−1.Solution 4.
Ifμ≠0, λ≠0, and λ2−4μ=0, then we obtain(13)Ωη=ln−2λη+E+4λ2η+E.Solution 5.
Ifμ=0, λ=0, and λ2−4μ=0, then we obtainΩη=lnη+E,(14)Ωη=lnη+E,where Aj0≤j≤NandBj0≤j≤M; λ and μ are the constants to be explored later. The values N and M are determined by equalizing the maximum order nonlinear term and the maximum order partial derivative term appearing in (7). If N and M are the rational, then the appropriate transformations can be applied to conquer these hurdles.Step 3.
Put (8) into equation (7), as well as the values of N and M determined in previous step, into (8). Gathering coefficients of all the powers of Fη and then equating every coefficient with zero, we derive a set of overdetermined nonlinear algebraic equations for A0,B0,A1,B1,…,AN,BM, λ, and μ.
Here, it is important to note thatE is the integration constant. We have the following relations as(15)Ψη≃δFN−M,(16)Ψ′η≃δFN−M−1F′=−δμFN−M−1+λFN−M+FN−M+1≃−δFN−M+1,(17)Ψ″η≃δFN−M+2,(18)Ψη2≃δ4F2N−2M,where δ=AN/BM. Balancing Ψ″ with Ψ2 yields(19)FN−M+2≃Ψ″ξ=Ψη2≃F2N−2M.
We can determine values ofN and M as follows:(20)N−M+2=2N−2M⟹N=M+2.


## 4. The STFBE

Consider the nonlinear space-time fractional order Boussinesq equation by the following variable change as(21)η=kx+1Γαα−ωt+1Γαα.Then, equation (1) is transformed to(22)ω2Ψ″+bk2Ψ″+βk2Ψ2″+γk4Ψ″″=0,in which Ψ′=dΨ/dη. By integrating equation (22) twice with respect to η reads as(23)ω2+bk2Ψ+βk2Ψ2+γk4Ψ″=0.With the help of relations (15)–(18), we balance the term of nonlinear and derivative of higher order in equation (23) and we attain N=M+2. We investigated two following cases as follows.


### 4.1. Case I:N=2,M=0

The IEFM allows us to recruit the substitutions:(24)Ψη=A0+A1Fη+A2F2ηB0.Plugging (24) along with (9) into equation (23) and equating all the coefficients of powers of Fη to be zero, one gains a system of algebraic equations. Solving the considered system by the help of Maple the following results will be obtained.Set I:(25)k=k,ω=ω,A0=A0,B0=B0,A1=2−9bγk2B02−6βγk2A0B0−9γω2B02β,A2=−6γk2B0β,λ=−13−9bγk2B02−6βγk2A0B0−9γω2B02γk2B0,μ=−16βA0γk2B0,λ2−4μ=−bk2+ω2γk4,where A0 and B0 are arbitrary values. Imposing the solution set (25) into (24) can be concluded in the following cases.Subcase IA:With the help of (10), the dark soliton solution to STFBE is deducted as(26)Ψ1x,t=A0B0−2/B0−9bγk2B02−6βγk2A0B0−9γω2B02/βλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ−6γk2βλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ2,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton exist for(27)γk4bk2+ω2<0.As seen in Figure1, we can observe dark behavior in the −20<x<20 range. Different interpretations and studies can be made for this situation. In Figure 1(c), for three cases α=0.5, α=0.75, and α=0.99, the fractional model of equation has been plotted and compared.Figure 1
The plot of (26) Ψ1x,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)Subcase IB:With the help of (11), the periodic wave solution to STFBE is concluded as(28)Ψ2x,t=A0B0+2/B0−9bγk2B02−6βγk2A0B0−9γω2B02/β−λ2+4μ/2μtan−λ2+4μ/2η−λ/2μ−6γk2β−λ2+4μ/2μtan−λ2+4μ/2η−λ/2μ2,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be understood that this soliton is available for(29)γk4bk2+ω2>0.As seen in Figure2, we can observe periodic behavior in the −20<x<20 range. Different interpretations and studies can be made for this situation. In Figure 2(c), for three cases α=0.5, α=0.75, and α=0.99 at the space 0<x<10 the fractional model of equation have been plotted and compared.Figure 2
The plot of (28) Ψ2x,t at A0=1,B0=2,a=1,b=2,k=1.5,ω=2,β=3,γ=−1,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)Subcase IC:With the help of (12), the kink-singular wave solution to STFBE is concluded as(30)Ψ3x,t=2λ−9bγk2−9γω2βexpλη−1−6γλ2k2βexpλη−12,λ=−−9bγk2−9γω23γk2,in which η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton exist for(31)γbk2+ω2<0.Subcase ID:With the help of (13), the rational cupson wave solution to STFBE is obtained as(32)Ψ4x,t=A0B0−2λ2η−6βγk2A0B0B0β2λη+4−6γk22λη+42βλ4η2,in which η=kx+1/Γαα+ωt+1/Γαα+E. Also, it should be noted that, for getting the soliton solution in the rational analysis of the function Ψ, we have the relation as follows:(33)b=−ω2k2.Set II:(34)k=k,ω=16−3γ12bγk2B02−24βγk2A0B0−β2A12γB0,A0=A0,B0=B0,A1=A1,A2=−6γk2B0β,λ=−βA16γk2B0,μ=−β36γk2A0B0+βA1272γ2k4B02,λ2−4μ=β24γk2A0B0+βA1212γ2k4B02,where A0,B0, and A1 are arbitrary constants. Imposing the solution set (34) into (24), one can conclude the following cases.Subcase IIA:With the help of (10), the hyperbolic function solution to the STFBE is deducted as(35)Ψ5x,t=A0B0−A1/B0λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ−6γk2/βλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ2,where η=kx+1/Γαα+1/6−3γ12bγk2B02−24βγk2A0B0−β2A12/γB0t+1/Γαα+E. It should be noted that this soliton is available for(36)β24γk2A0B0+βA12>0.Subcase IIB:With the help of (11), the periodic wave solution to the STFBE is obtained as(37)Ψ6x,t=A0B0+A1/B0−λ2+4μ/2μtan−λ2+4μ/2η−λ/2μ−6γk2/β−λ2+4μ/2μtan−λ2+4μ/2η−λ/2μ2,where η=kx+1/Γαα+1/6−3γ12bγk2B02−24βγk2A0B0−β2A12/γB0t+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(38)β24γk2A0B0+βA12<0.Subcase IIC:With the help of (12), the kink-singular solution to the STFBE is obtained as(39)Ψ7x,t=A1λB0expλη−1−6γλ2k2βexpλη−12,λ=−βA16γk2B0,in which η=kx+1/Γαα+ω=−3γ12bγk2B02−β2A12/6γB0t+1/Γαα+E and B0 is optional arbitrary. One can note that these solitons are available for(40)γ12bγk2B02−β2A12<0.Subcase IID:With the help of (13), the rational cupson solution to the STFBE is gained as(41)Ψ8x,t=A0B0−−24γk2A0B0/βλ2ηB02λη+4−6γk2λ4η2β2λη+42,in which η=kx+1/Γαα+−bk2t+1/Γαα+E. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(42)γA0B0β<0.


### 4.2. Case II:N=3,M=1

The IEFM allows us to recruit the substitutions:(43)Ψη=A0+A1Fη+A2F2η+A3F3ηB0+B1Fη.Plugging (43) along with (9) into equation (1) and equating all the coefficients of powers of Fkη,k=0,1,…,7 to be zero, one gains a system of algebraic equations. Solving the considered system with the help of Maple the following results will be obtained.Set I:(44)k=k,ω=ω,A0=−B03γk4λ2−bk2−ω22βk2,B0=B0,A1=−3γk4λ2B1+12γk4λB0−bk2B1−ω2B12βk2,B1=B1,A2=−6γk2λB1+B0β,A3=−6γk2B1β,λ=λ,μ=γk4λ2−bk2−ω24γk4,λ2−4μ=bk2+ω2γk4,where A0 and B0 are arbitrary constants. Imposing the solution set (44) into (24), one can conclude the following cases.Subcase IA:With the help of (10), the exact solution to the STFBE is deducted as(45)Ψ1x,t=−B03γk4λ2−bk2−ω2/2βk2−3γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton is available for(46)γbk2+ω2>0.Subcase IB:With the help of (11), the periodic wave solution to the STFBE is concluded as(47)Ψ2x,t=−B03γk4λ2−bk2−ω2/2βk2−3γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(48)γbk2+ω2<0.Subcase IC:With the help of (12), the rational kink-singular solution to the STFBE is obtained as(49)Ψ3x,t=−B03γk4λ2−bk2−ω2/2βk2−3γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt1/Γαα+E. It should be noted that this soliton is available for(50)λ=γbk2+ω2γk2,γbk2+ω2>0.Subcase ID:With the help of (13), the exact rational cupson solution to the STFBE is gained as(51)Ψ4x,t=−B03γk4λ2−bk2−ω2/2βk2−3γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−λ2η+E2λη+E+4,in which η=kx+1/Γαα+ωt+1/Γαα. It should be noted that these solitons exist for(52)b=−ω2k2.Set II:(53)k=k,ω=ω,A0=−3B03γk4λ2+bk2+ω22βk2,B0=B0,A1=−33γk4λ2B1+12γk4λB0+bk2B1+ω2B12βk2,A2=−6γk2λB1+B0β,A3=−6γk2B1β,λ=λ,μ=γk4λ2+bk2+ω24γk4,λ2−4μ=−bk2+ω2γk4,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (53) into (24), one can obtain the following cases.Subcase IIA:With the help of (10), the exact dark soliton solution to the STFBE is deducted as(54)Ψ5x,t=−3B03γk4λ2−bk2−ω2/2βk2−33γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(55)γbk2+ω2<0.Subcase IB:With the help of (11), the periodic wave solution to the STFBE is concluded as(56)Ψ6x,t=−3B03γk4λ2−bk2−ω2/2βk2−33γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(57)γbk2+ω2>0.Subcase IIC:With the help of (12), the kink-singular solution to the STFBE is obtained as(58)Ψ7x,t=−3B03γk4λ2−bk2−ω2/2βk2−33γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=exp−γbk2+ω2/γk2η+E−1−γbk2+ω2/γk2,where η=kx+1/Γαα+ωt+1/Γαα. It should be noted that this soliton is available for(59)λ=−γbk2+ω2γk2,γbk2+ω2<0.Subcase IID:With the help of (13), the exact rational solution to the STFBE is resulted as(60)Ψ8x,t=−3B03γk4λ2−bk2−ω2/2βk2−33γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−λ2η+E2λη+E+4,in which η=kx+1/Γαα+ωt+1/Γαα. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(61)b=−ω2k2.


## 4.1. Case I:N=2,M=0

The IEFM allows us to recruit the substitutions:(24)Ψη=A0+A1Fη+A2F2ηB0.Plugging (24) along with (9) into equation (23) and equating all the coefficients of powers of Fη to be zero, one gains a system of algebraic equations. Solving the considered system by the help of Maple the following results will be obtained.Set I:(25)k=k,ω=ω,A0=A0,B0=B0,A1=2−9bγk2B02−6βγk2A0B0−9γω2B02β,A2=−6γk2B0β,λ=−13−9bγk2B02−6βγk2A0B0−9γω2B02γk2B0,μ=−16βA0γk2B0,λ2−4μ=−bk2+ω2γk4,where A0 and B0 are arbitrary values. Imposing the solution set (25) into (24) can be concluded in the following cases.Subcase IA:With the help of (10), the dark soliton solution to STFBE is deducted as(26)Ψ1x,t=A0B0−2/B0−9bγk2B02−6βγk2A0B0−9γω2B02/βλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ−6γk2βλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ2,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton exist for(27)γk4bk2+ω2<0.As seen in Figure1, we can observe dark behavior in the −20<x<20 range. Different interpretations and studies can be made for this situation. In Figure 1(c), for three cases α=0.5, α=0.75, and α=0.99, the fractional model of equation has been plotted and compared.Figure 1
The plot of (26) Ψ1x,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)Subcase IB:With the help of (11), the periodic wave solution to STFBE is concluded as(28)Ψ2x,t=A0B0+2/B0−9bγk2B02−6βγk2A0B0−9γω2B02/β−λ2+4μ/2μtan−λ2+4μ/2η−λ/2μ−6γk2β−λ2+4μ/2μtan−λ2+4μ/2η−λ/2μ2,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be understood that this soliton is available for(29)γk4bk2+ω2>0.As seen in Figure2, we can observe periodic behavior in the −20<x<20 range. Different interpretations and studies can be made for this situation. In Figure 2(c), for three cases α=0.5, α=0.75, and α=0.99 at the space 0<x<10 the fractional model of equation have been plotted and compared.Figure 2
The plot of (28) Ψ2x,t at A0=1,B0=2,a=1,b=2,k=1.5,ω=2,β=3,γ=−1,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)Subcase IC:With the help of (12), the kink-singular wave solution to STFBE is concluded as(30)Ψ3x,t=2λ−9bγk2−9γω2βexpλη−1−6γλ2k2βexpλη−12,λ=−−9bγk2−9γω23γk2,in which η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton exist for(31)γbk2+ω2<0.Subcase ID:With the help of (13), the rational cupson wave solution to STFBE is obtained as(32)Ψ4x,t=A0B0−2λ2η−6βγk2A0B0B0β2λη+4−6γk22λη+42βλ4η2,in which η=kx+1/Γαα+ωt+1/Γαα+E. Also, it should be noted that, for getting the soliton solution in the rational analysis of the function Ψ, we have the relation as follows:(33)b=−ω2k2.Set II:(34)k=k,ω=16−3γ12bγk2B02−24βγk2A0B0−β2A12γB0,A0=A0,B0=B0,A1=A1,A2=−6γk2B0β,λ=−βA16γk2B0,μ=−β36γk2A0B0+βA1272γ2k4B02,λ2−4μ=β24γk2A0B0+βA1212γ2k4B02,where A0,B0, and A1 are arbitrary constants. Imposing the solution set (34) into (24), one can conclude the following cases.Subcase IIA:With the help of (10), the hyperbolic function solution to the STFBE is deducted as(35)Ψ5x,t=A0B0−A1/B0λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ−6γk2/βλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ2,where η=kx+1/Γαα+1/6−3γ12bγk2B02−24βγk2A0B0−β2A12/γB0t+1/Γαα+E. It should be noted that this soliton is available for(36)β24γk2A0B0+βA12>0.Subcase IIB:With the help of (11), the periodic wave solution to the STFBE is obtained as(37)Ψ6x,t=A0B0+A1/B0−λ2+4μ/2μtan−λ2+4μ/2η−λ/2μ−6γk2/β−λ2+4μ/2μtan−λ2+4μ/2η−λ/2μ2,where η=kx+1/Γαα+1/6−3γ12bγk2B02−24βγk2A0B0−β2A12/γB0t+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(38)β24γk2A0B0+βA12<0.Subcase IIC:With the help of (12), the kink-singular solution to the STFBE is obtained as(39)Ψ7x,t=A1λB0expλη−1−6γλ2k2βexpλη−12,λ=−βA16γk2B0,in which η=kx+1/Γαα+ω=−3γ12bγk2B02−β2A12/6γB0t+1/Γαα+E and B0 is optional arbitrary. One can note that these solitons are available for(40)γ12bγk2B02−β2A12<0.Subcase IID:With the help of (13), the rational cupson solution to the STFBE is gained as(41)Ψ8x,t=A0B0−−24γk2A0B0/βλ2ηB02λη+4−6γk2λ4η2β2λη+42,in which η=kx+1/Γαα+−bk2t+1/Γαα+E. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(42)γA0B0β<0.


## 4.2. Case II:N=3,M=1

The IEFM allows us to recruit the substitutions:(43)Ψη=A0+A1Fη+A2F2η+A3F3ηB0+B1Fη.Plugging (43) along with (9) into equation (1) and equating all the coefficients of powers of Fkη,k=0,1,…,7 to be zero, one gains a system of algebraic equations. Solving the considered system with the help of Maple the following results will be obtained.Set I:(44)k=k,ω=ω,A0=−B03γk4λ2−bk2−ω22βk2,B0=B0,A1=−3γk4λ2B1+12γk4λB0−bk2B1−ω2B12βk2,B1=B1,A2=−6γk2λB1+B0β,A3=−6γk2B1β,λ=λ,μ=γk4λ2−bk2−ω24γk4,λ2−4μ=bk2+ω2γk4,where A0 and B0 are arbitrary constants. Imposing the solution set (44) into (24), one can conclude the following cases.Subcase IA:With the help of (10), the exact solution to the STFBE is deducted as(45)Ψ1x,t=−B03γk4λ2−bk2−ω2/2βk2−3γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton is available for(46)γbk2+ω2>0.Subcase IB:With the help of (11), the periodic wave solution to the STFBE is concluded as(47)Ψ2x,t=−B03γk4λ2−bk2−ω2/2βk2−3γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(48)γbk2+ω2<0.Subcase IC:With the help of (12), the rational kink-singular solution to the STFBE is obtained as(49)Ψ3x,t=−B03γk4λ2−bk2−ω2/2βk2−3γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt1/Γαα+E. It should be noted that this soliton is available for(50)λ=γbk2+ω2γk2,γbk2+ω2>0.Subcase ID:With the help of (13), the exact rational cupson solution to the STFBE is gained as(51)Ψ4x,t=−B03γk4λ2−bk2−ω2/2βk2−3γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−λ2η+E2λη+E+4,in which η=kx+1/Γαα+ωt+1/Γαα. It should be noted that these solitons exist for(52)b=−ω2k2.Set II:(53)k=k,ω=ω,A0=−3B03γk4λ2+bk2+ω22βk2,B0=B0,A1=−33γk4λ2B1+12γk4λB0+bk2B1+ω2B12βk2,A2=−6γk2λB1+B0β,A3=−6γk2B1β,λ=λ,μ=γk4λ2+bk2+ω24γk4,λ2−4μ=−bk2+ω2γk4,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (53) into (24), one can obtain the following cases.Subcase IIA:With the help of (10), the exact dark soliton solution to the STFBE is deducted as(54)Ψ5x,t=−3B03γk4λ2−bk2−ω2/2βk2−33γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(55)γbk2+ω2<0.Subcase IB:With the help of (11), the periodic wave solution to the STFBE is concluded as(56)Ψ6x,t=−3B03γk4λ2−bk2−ω2/2βk2−33γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(57)γbk2+ω2>0.Subcase IIC:With the help of (12), the kink-singular solution to the STFBE is obtained as(58)Ψ7x,t=−3B03γk4λ2−bk2−ω2/2βk2−33γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=exp−γbk2+ω2/γk2η+E−1−γbk2+ω2/γk2,where η=kx+1/Γαα+ωt+1/Γαα. It should be noted that this soliton is available for(59)λ=−γbk2+ω2γk2,γbk2+ω2<0.Subcase IID:With the help of (13), the exact rational solution to the STFBE is resulted as(60)Ψ8x,t=−3B03γk4λ2−bk2−ω2/2βk2−33γk4λ2B1+12γk4λB0−bk2B1−ω2B1/2βk2Δη−6γk2λB1+B0/βΔη2−6γk2B1/βΔη3B0+B1Δη,Δη=−λ2η+E2λη+E+4,in which η=kx+1/Γαα+ωt+1/Γαα. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(61)b=−ω2k2.


## 5. The STFBSEs

Consider the nonlinear space-time (2 + 1)-dimensional breaking soliton equations by the following variable change as(62)η=kx+1Γαα+ωy+1Γαα−ct+1Γαα.Then, equation (2) is transformed to(63)−cΨ′+ak2ω‴+4akΨΩ′+4akΨ′Ω=0,(64)ωΨ′−kΩ′=0,in which Ψ′=dΨ/dη and Ω′=dΩ/dη. By integrating equation (64) once with respect to η, we obtain(65)ωΨ−kΩ=0.Inserting equation (65) into first equation of system (63) and integrating the obtained equation once with respect to η, we obtain(66)−cΨ+ak2ωΨ′+4aωΨ2=0.With the help of relations (15)–(18), we balance the term of nonlinear and derivative of higher order in equation (64), and we attain N=M+2. We investigate the following two cases.


### 5.1. Case I:N=2,M=0

The IEFM allows us to recruit the substitutions:(67)Ψη=A0+A1Fη+A2F2ηB0.Plugging (67) along with (9) into equation (2) and equating all the coefficients of powers of Fη to be zero, one gains a system of algebraic equations. Solving the considered system with the help of Maple the following results will be obtained.Set I:(68)λ=0,μ=μ,k=k,c=214aωk2μ,ω=ω,A0=−1+1414B0μk2,B0=B0,A1=0,A2=−14k2B0,where B0 is arbitrary constant. Imposing the solution set (68) into (24), one can conclude the following cases.Subcase IA:With the help of (10), the singular solution to the STFBSEs is deducted as(69)Ψ1x,y,t=−1+1414μk2+14k2μcoth2−μη+E,Ω1x,y,t=−1+1414μωk+14ωkμcoth2−μη+E,where η=kx+1/Γαα+ωy+1/Γαα−214aωk2μt+1/Γαα. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(70)μ<0.As seen in Figure3, we can observe dark behavior in the −10<x<20 range. Different interpretations and studies can be made for this situation. In Figure 3(b), for one case α=1, the fractional model of the equation has been plotted and compared for the two-dimensional plot.Figure 3
The plot of (69) Ψ1x,y,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,y=2,andα=1 for (a) 3D plot and (b) 2D plot.
(a)(b)Subcase IB:With the help of (11), the periodic solution to the STFBSEs is resulted as(71)Ψ2x,y,t=−1+1414μk2−14k2μcot2μη+E,Ω2x,y,t=−1+1414μωk−14ωkμcot2μη+E,where η=kx+1/Γαα+ωy+1/Γαα−214aωk2μt+1/Γαα. It should be noted that this soliton exists for(72)μ>0.As seen in Figure4, we can observe periodic behavior in the −10<x<10 range. Different interpretations and studies can be made for this situation. In Figure 4(c), for three cases α=0.5, α=0.75, and α=0.99, the fractional model of the equation has been plotted and compared.Set II:(73)λ=0,μ=83A12k4B02,k=k,c=163iaωA12k2B02,A0=23−1+iA12B0k2,B0=B0,A1=A1,A2=−14k2B0,where B0 is arbitrary constant. Imposing the solution set (73) into (24), one can conclude the following cases.Subcase IA:Figure 4
The plot of (69) Ψ2x,y,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,y=2,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)With the help of (10), the exact solution to the model is deducted as(74)Ψ3x,y,t=23−1+iA12B02k2+A1B0−μcoth−μη+E+14k2μcoth2−μη+E,Ω3x,y,t=23−1+iA12B02ωk+A1ωB0k−μcoth−μη+E+14k2μcoth2−μη+E,where η=kx+1/Γαα+ωy+1/Γαα−16/3iaωA12/k2B02t+1/Γαα. It should be noted that this soliton exist for(75)μ<0.Subcase IB:With the help of (11), the exact solution to the model is concluded as(76)Ψ4x,y,t=23−1+iA12B02k2+A1B0μcotμη+E−14k2μcot2μη+E,Ω4x,y,t=23−1+iA12B02ωk+A1ωB0kμcotμη+E−14ωkμcot2μη+E,where η=kx+1/Γαα+ωy+1/Γαα−16/3iaωA12/k2B02t+1/Γαα. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(77)μ>0.


### 5.2. Case II:N=3,M=1

The IEFM allows us to recruit the substitutions(78)vη=A0+A1Fη+A2F2η+A3F3ηB0+B1Fη.Plugging (78) along with (9) into equation (3) and equating all the coefficients of powers of Fkη,k=0,1,…,7 to be zero, one gains a system of algebraic equations. Solving the considered system, with the help of Maple, the following results can be obtained.Set I:(79)k=k,ω=ω,A0=−32k2μB0,B0=B0,A1=−32k2λB0+μB1,A2=−32k2λB1+B0,A3=−32k2B1,B1=B1,λ=λ,μ=μ,c=aωk2λ2−4μ,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (44) into (78), one can conclude the following cases.Subcase IA:With the help of (10), the exact solution to the model is obtained as(80)Ψ5x,y,t=−3/2k2μB0−3/2k2λB0+μB1Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Ω5x,y,t=−3/2ωkμB0−3/2ωkλB0+μB1Δη−3/2ωkλB1+B0Δη2−3/2ωkB1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωy+1/Γαα−aωk2λ2−4μt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(81)λ2−4μ>0.Subcase IB:With the help of (11), the exact solution to the model is concluded as(82)Ψ6x,y,t=−3/2k2μB0−3/2k2λB0+μB1Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Ω6x,y,t=−3/2ωkμB0−3/2ωkλB0+μB1Δη−3/2ωkλB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωy+1/Γαα−aωk2λ2−4μt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(83)λ2−4μ<0.Subcase IC:With the help of (12), the kink-singular solution to the STFBSEs is obtained as(84)Ψ7x,y,t=−3/2k2λB0Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Δη=expλη+E−1λ,Ω7x,y,t=−3/2ωkλB0Δη−3/2ωkλB1+B0Δη2−3/2ωkB1Δη3B0+B1Δη,Δη=expλη+E−1λ,where η=kx+1/Γαα+ωy+1/Γαα−aωk2λ2t+1/Γαα+E. It should be noted that this soliton is available for(85)λ2−4μ<0.Set II:(86)A0=−14B0k2λ2+2μ,B0=B0,A1=−14k2λ2B1+6λB0+2μB1,A2=−32k2λB1+B0,k=k,ω=ω,A3=−32k2B1,B1=B1,λ=λ,μ=μ,c=−aωk2λ2−4μ,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (86) into (24), one can conclude the following cases.Subcase IIA:With the help of (10), the exact dark solution to the STFBSEs is obtained as(87)Ψ8x,y,t=−1/4B0k2λ2+2μ−1/4k2λ2B1+6λB0+2μB1Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Ω8x,y,t=−1/4B0ωkλ2+2μ−1/4ωkλ2B1+6λB0+2μB1Δη−3/2ωkλB1+B0Δη2−3/2ωkB1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωy+1/Γαα+aωk2λ2−4μt+1/Γαα+E. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(88)λ2−4μ>0.Subcase IIB:With the help of (11), the periodic wave solution to the STFBSEs is obtained as(89)Ψ9x,y,t=−1/4B0k2λ2+2μ−1/4k2λ2B1+6λB0+2μB1Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωy+1/Γαα+aωk2λ2−4μt+1/Γαα+E and A0andB0 are optional arbitraries. It should be noted that these solitons exist for(90)λ2−4μ<0.Subcase IIC:With the help of (12), the exact solution to the model is obtained as(91)Ψ10x,y,t=−1/4B0k2λ2−1/4k2λ2B1+6λB0Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Δη=expλη+E−1λ,Ω10x,y,t=−1/4B0ωkλ2−1/4k2λ2B1+6λB0Δη−3/2ωkλB1+B0Δη2−3/2ωkB1Δη3B0+B1Δη,Δη=expλη+E−1λ,where η=kx+1/Γαα+ωy+1/Γαα+aωk2λ2t+1/Γαα+E.


## 5.1. Case I:N=2,M=0

The IEFM allows us to recruit the substitutions:(67)Ψη=A0+A1Fη+A2F2ηB0.Plugging (67) along with (9) into equation (2) and equating all the coefficients of powers of Fη to be zero, one gains a system of algebraic equations. Solving the considered system with the help of Maple the following results will be obtained.Set I:(68)λ=0,μ=μ,k=k,c=214aωk2μ,ω=ω,A0=−1+1414B0μk2,B0=B0,A1=0,A2=−14k2B0,where B0 is arbitrary constant. Imposing the solution set (68) into (24), one can conclude the following cases.Subcase IA:With the help of (10), the singular solution to the STFBSEs is deducted as(69)Ψ1x,y,t=−1+1414μk2+14k2μcoth2−μη+E,Ω1x,y,t=−1+1414μωk+14ωkμcoth2−μη+E,where η=kx+1/Γαα+ωy+1/Γαα−214aωk2μt+1/Γαα. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(70)μ<0.As seen in Figure3, we can observe dark behavior in the −10<x<20 range. Different interpretations and studies can be made for this situation. In Figure 3(b), for one case α=1, the fractional model of the equation has been plotted and compared for the two-dimensional plot.Figure 3
The plot of (69) Ψ1x,y,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,y=2,andα=1 for (a) 3D plot and (b) 2D plot.
(a)(b)Subcase IB:With the help of (11), the periodic solution to the STFBSEs is resulted as(71)Ψ2x,y,t=−1+1414μk2−14k2μcot2μη+E,Ω2x,y,t=−1+1414μωk−14ωkμcot2μη+E,where η=kx+1/Γαα+ωy+1/Γαα−214aωk2μt+1/Γαα. It should be noted that this soliton exists for(72)μ>0.As seen in Figure4, we can observe periodic behavior in the −10<x<10 range. Different interpretations and studies can be made for this situation. In Figure 4(c), for three cases α=0.5, α=0.75, and α=0.99, the fractional model of the equation has been plotted and compared.Set II:(73)λ=0,μ=83A12k4B02,k=k,c=163iaωA12k2B02,A0=23−1+iA12B0k2,B0=B0,A1=A1,A2=−14k2B0,where B0 is arbitrary constant. Imposing the solution set (73) into (24), one can conclude the following cases.Subcase IA:Figure 4
The plot of (69) Ψ2x,y,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,y=2,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)With the help of (10), the exact solution to the model is deducted as(74)Ψ3x,y,t=23−1+iA12B02k2+A1B0−μcoth−μη+E+14k2μcoth2−μη+E,Ω3x,y,t=23−1+iA12B02ωk+A1ωB0k−μcoth−μη+E+14k2μcoth2−μη+E,where η=kx+1/Γαα+ωy+1/Γαα−16/3iaωA12/k2B02t+1/Γαα. It should be noted that this soliton exist for(75)μ<0.Subcase IB:With the help of (11), the exact solution to the model is concluded as(76)Ψ4x,y,t=23−1+iA12B02k2+A1B0μcotμη+E−14k2μcot2μη+E,Ω4x,y,t=23−1+iA12B02ωk+A1ωB0kμcotμη+E−14ωkμcot2μη+E,where η=kx+1/Γαα+ωy+1/Γαα−16/3iaωA12/k2B02t+1/Γαα. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(77)μ>0.


## 5.2. Case II:N=3,M=1

The IEFM allows us to recruit the substitutions(78)vη=A0+A1Fη+A2F2η+A3F3ηB0+B1Fη.Plugging (78) along with (9) into equation (3) and equating all the coefficients of powers of Fkη,k=0,1,…,7 to be zero, one gains a system of algebraic equations. Solving the considered system, with the help of Maple, the following results can be obtained.Set I:(79)k=k,ω=ω,A0=−32k2μB0,B0=B0,A1=−32k2λB0+μB1,A2=−32k2λB1+B0,A3=−32k2B1,B1=B1,λ=λ,μ=μ,c=aωk2λ2−4μ,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (44) into (78), one can conclude the following cases.Subcase IA:With the help of (10), the exact solution to the model is obtained as(80)Ψ5x,y,t=−3/2k2μB0−3/2k2λB0+μB1Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Ω5x,y,t=−3/2ωkμB0−3/2ωkλB0+μB1Δη−3/2ωkλB1+B0Δη2−3/2ωkB1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωy+1/Γαα−aωk2λ2−4μt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(81)λ2−4μ>0.Subcase IB:With the help of (11), the exact solution to the model is concluded as(82)Ψ6x,y,t=−3/2k2μB0−3/2k2λB0+μB1Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Ω6x,y,t=−3/2ωkμB0−3/2ωkλB0+μB1Δη−3/2ωkλB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωy+1/Γαα−aωk2λ2−4μt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(83)λ2−4μ<0.Subcase IC:With the help of (12), the kink-singular solution to the STFBSEs is obtained as(84)Ψ7x,y,t=−3/2k2λB0Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Δη=expλη+E−1λ,Ω7x,y,t=−3/2ωkλB0Δη−3/2ωkλB1+B0Δη2−3/2ωkB1Δη3B0+B1Δη,Δη=expλη+E−1λ,where η=kx+1/Γαα+ωy+1/Γαα−aωk2λ2t+1/Γαα+E. It should be noted that this soliton is available for(85)λ2−4μ<0.Set II:(86)A0=−14B0k2λ2+2μ,B0=B0,A1=−14k2λ2B1+6λB0+2μB1,A2=−32k2λB1+B0,k=k,ω=ω,A3=−32k2B1,B1=B1,λ=λ,μ=μ,c=−aωk2λ2−4μ,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (86) into (24), one can conclude the following cases.Subcase IIA:With the help of (10), the exact dark solution to the STFBSEs is obtained as(87)Ψ8x,y,t=−1/4B0k2λ2+2μ−1/4k2λ2B1+6λB0+2μB1Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Ω8x,y,t=−1/4B0ωkλ2+2μ−1/4ωkλ2B1+6λB0+2μB1Δη−3/2ωkλB1+B0Δη2−3/2ωkB1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωy+1/Γαα+aωk2λ2−4μt+1/Γαα+E. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(88)λ2−4μ>0.Subcase IIB:With the help of (11), the periodic wave solution to the STFBSEs is obtained as(89)Ψ9x,y,t=−1/4B0k2λ2+2μ−1/4k2λ2B1+6λB0+2μB1Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωy+1/Γαα+aωk2λ2−4μt+1/Γαα+E and A0andB0 are optional arbitraries. It should be noted that these solitons exist for(90)λ2−4μ<0.Subcase IIC:With the help of (12), the exact solution to the model is obtained as(91)Ψ10x,y,t=−1/4B0k2λ2−1/4k2λ2B1+6λB0Δη−3/2k2λB1+B0Δη2−3/2k2B1Δη3B0+B1Δη,Δη=expλη+E−1λ,Ω10x,y,t=−1/4B0ωkλ2−1/4k2λ2B1+6λB0Δη−3/2ωkλB1+B0Δη2−3/2ωkB1Δη3B0+B1Δη,Δη=expλη+E−1λ,where η=kx+1/Γαα+ωy+1/Γαα+aωk2λ2t+1/Γαα+E.


## 6. The STFSRLWE

Take the nonlinear space-time fractional SRLW equation with the following variable change:(92)η=kx+1Γαα−ωt+1Γαα.Then, equation (3) is integrated respect to η and simplified to obtain(93)2ω2+k2Ψ+kωΨ2+2k2ω2Ψ″=0,in which Ψ′=dΨ/dη. With the help of relations (15)–(18), we balance the term of nonlinear and derivative of higher order in equation (93), and we attain N=M+2. We investigate the two following cases,


### 6.1. Case I:N=2,M=0

The IEFM allows us to recruit the substitutions:(94)Ψη=A0+A1Fη+A2F2ηB0.Plugging (94) along with (9) into equation (93) and equating all the coefficients of powers of Fη to be zero, one gains a system of algebraic equations. Solving the considered system with the help of Maple, the following results will be obtained.Set I:(95)k=k,ω=ω,A0=14848k2B02+48ω2B02−A12kωB0,B0=B0,A1=A1,A2=−12kωB0,λ=−112A1kωB0,μ=−144k2B02+144ω2B02−A12576k2ω2B02,λ2−4μ=k2+ω2k2ω2,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (95) into (24), one can conclude the following cases.Subcase IA:With the help of (10), the exact dark soliton solution to the STFSRLWE is deducted as(96)Ψ1x,t=14848k2B02+48ω2B02−A12kωB02−A1/B0λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ−12kωλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ2,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(97)k2+ω2>0.As seen in Figure5, we can observe dark behavior in the −20<x<20 range. Different interpretations and studies can be made for this situation. In Figure 5(c), for three cases α=0.5, α=0.75, and α=0.99, the fractional model of the equation has been plotted and compared.Subcase IB:Figure 5
The plot of (96) Ψ1x,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)With the help of (11), the kink-singular solution to the model is concluded as(98)Ψ2x,t=−2k2+ω2kω+12k2+ω2B0B0Δη−12kωΔ2η,Δη=λexpλη+E−1,in which η=kx+1/Γαα+ωt+1/Γαα+E. Likewise, indeed need to be assured to warrantee the rational analysis of the function Ψ as(99)λ=−k2+ω2kω,kω≠0.As seen in Figure6, we can observe periodic behavior in the −20<x<20 range. Different interpretations and studies can be made for this situation. In Figure 6(c), for three cases α=0.5, α=0.75, and α=0.99, the fractional model of the equation has been plotted and compared.Set II:(100)k=k,ω=ω,A0=−14848k2B02+48ω2B02+A12kωB0,B0=B0,A1=A1,A2=−12kωB0,λ=−112A1kωB0,μ=144k2B02+144ω2B02+A12576k2ω2B02,λ2−4μ=−k2+ω2k2ω2,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (100) into (24), one can obtain the following cases.Figure 6
The plot of (98) Ψ2x,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)Subcase IIA:With the help of (10), the exact periodic wave solution to the STFSRLWE is deducted as(101)Ψ3x,t=−14848k2B02+48ω2B02+A12kωB02+A1/B0λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ−12kωλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ2,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(102)k2+ω2>0.Subcase IB:With the help of (11), the exact solution to the STFSRLWE is obtained as(103)Ψ4x,t=−2k2+ω2kω+12−k2−ω2B0B0Δη−12kωΔ2η,Δη=λexpλη+E−1,where η=kx+1/Γαα+ωt+1/Γαα+E. Likewise, indeed need to be assured to warrantee the rational analysis of the function Ψ as(104)λ=−−k2−ω2kω,kω≠0.


### 6.2. Case II:N=3,M=1

The IEFM allows us to recruit the substitutions:(105)Ψη=A0+A1Fη+A2F2η+A3F3ηB0+B1Fη.Plugging (105) along with (9) into equation (93) and equating all the coefficients of powers of Fkη,k=0,1,…,7 to be zero, one gains a system of algebraic equations. Solving the considered system with the help of Maple, the following results will be obtained.Set I:(106)A0=−λ3k2λ2ω2B12+21k2λω2B0B1+24k2ω2B02+3k2B12−kωA1B1+3ω2B12+9k2B0B1−2kωA1B0+9ω2B0B1ωkB1,k=k,ω=ω,A1=−−2−λ2ω2k2−k2−ω2B1+λ2B1ω2k2+12k2λω2B0+k2+ω2B1kω,A2=−12λB1ωk−12B0ωk,λ=λ,μ=−14−λ2ω2k2−k2−ω2k2ω2,λ2−4μ=−k2+ω2k2ω2,A3=−12ωkB1,B0=B0,B1=B1,where B0 and B1 are arbitrary constants. Imposing the solution set (106) into (24), one can obtain the following cases.Subcase IIA:With the help of (11), the exact periodic wave solution to the STFSRLWE is deducted as(107)Ψ5x,t=A0+A1Δη−12kωλB1+12B0Δη2−12ωkB1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(108)kω≠0.Subcase IB:With the help of (12), the kink-singular solution to the STFSRLWE is obtained as(109)Ψ6x,t=A0+A1Δη−12kωλB1+12B0Δη2−12ωkB1Δη3B0+B1Δη,Δη=−k2−ω2kωexp−k2−ω2/kωη−1,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton is available for(110)kω≠0.Subcase IIC:With the help of (13), the exact rational solution to the STFSRLWE is gained as(111)Ψ7x,t=A0−3λ2B1ω4+12λB0ω4B1/−ωωΔη−12ω−ωλB1+B0Δη2−12ω−ωB1Δη3B0+B1Δη,A0=2k2λ3B12+4λ3B12ω2+12λ2ω2B0B12+27λ2B0B1ω2+24λω2B02B1+24B02k2λ+6B1B0−ωB1,Δη=−λ2η+E2λη+E+4,where η=−ωx+1/Γαα+ωt+1/Γαα. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(112)ω<0.Set II:(113)A0=−λ3k2λ2ω2B12+21k2λω2B0B1+24k2ω2B02+3k2B12−kωA1B1+3ω2B12+B09k2B1−2kωA1+9ω2B1ωkB1,k=k,ω=ω,A1=−−2−λ2ω2k2+k2+ω2B1+λ2B1ω2k2+12k2λω2B0+k2+ω2B1kω,A2=−12λB1ωk−12B0ωk,λ=λ,μ=−14−λ2ω2k2+k2+ω2k2ω2,λ2−4μ=k2+ω2k2ω2,A3=−12ωkB1,B0=B0,B1=B1,where B0 and B1 are arbitrary constants. Imposing the solution set (113) into (24), one can conclude the following cases.Subcase IIA:With the help of (11), the exact dark soliton solution to the STFSRLWE is deducted as(114)Ψ8x,t=A0+A1Δη−12kωλB1+12B0Δη2−12ωkB1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η−λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton is available for(115)kω≠0.Subcase IIB:With the help of (12), the exact kink-singular solution to the STFSRLWE is obtained as(116)Ψ9x,t=A0+A1Δη−12kωλB1+12B0Δη2−12ωkB1Δη3B0+B1Δη,Δη=k2+ω2kωexpk2+ω2/kωη−1,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(117)kω≠0.Subcase IIC:With the help of (13), the exact rational cupson solution to the STFSRLWE is gained as(118)Ψ10x,t=A0−3λ2B1ω4+12λB0ω4B1/−ωωΔη−12ω−ωλB1+B0Δη2−12ω−ωB1Δη3B0+B1Δη,A0=λ3λ2ω4B12+21λB0ω4B1+24ω4B02+6ω2B12−−ωωA1B1+B018ω2B1−2−ωωA1,Δη=−λ2η+E2λη+E+4,where η=−ωx+1/Γαα+ωt+1/Γαα. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(119)ω<0.Remark.
This paper finds many novel hyperbolic, trigonometric, kink, and kink-singular soliton solutions to govern the model. These mathematical properties come from trigonometric and hyperbolic function properties. In this sense, from the mathematical and physical points of views, these results play an important role in explaining wave propagation of considered models. We obtained more and better results compared with the results of [48] as Mohyud-din and Bibi used the exponential rational function method.


## 6.1. Case I:N=2,M=0

The IEFM allows us to recruit the substitutions:(94)Ψη=A0+A1Fη+A2F2ηB0.Plugging (94) along with (9) into equation (93) and equating all the coefficients of powers of Fη to be zero, one gains a system of algebraic equations. Solving the considered system with the help of Maple, the following results will be obtained.Set I:(95)k=k,ω=ω,A0=14848k2B02+48ω2B02−A12kωB0,B0=B0,A1=A1,A2=−12kωB0,λ=−112A1kωB0,μ=−144k2B02+144ω2B02−A12576k2ω2B02,λ2−4μ=k2+ω2k2ω2,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (95) into (24), one can conclude the following cases.Subcase IA:With the help of (10), the exact dark soliton solution to the STFSRLWE is deducted as(96)Ψ1x,t=14848k2B02+48ω2B02−A12kωB02−A1/B0λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ−12kωλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ2,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(97)k2+ω2>0.As seen in Figure5, we can observe dark behavior in the −20<x<20 range. Different interpretations and studies can be made for this situation. In Figure 5(c), for three cases α=0.5, α=0.75, and α=0.99, the fractional model of the equation has been plotted and compared.Subcase IB:Figure 5
The plot of (96) Ψ1x,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)With the help of (11), the kink-singular solution to the model is concluded as(98)Ψ2x,t=−2k2+ω2kω+12k2+ω2B0B0Δη−12kωΔ2η,Δη=λexpλη+E−1,in which η=kx+1/Γαα+ωt+1/Γαα+E. Likewise, indeed need to be assured to warrantee the rational analysis of the function Ψ as(99)λ=−k2+ω2kω,kω≠0.As seen in Figure6, we can observe periodic behavior in the −20<x<20 range. Different interpretations and studies can be made for this situation. In Figure 6(c), for three cases α=0.5, α=0.75, and α=0.99, the fractional model of the equation has been plotted and compared.Set II:(100)k=k,ω=ω,A0=−14848k2B02+48ω2B02+A12kωB0,B0=B0,A1=A1,A2=−12kωB0,λ=−112A1kωB0,μ=144k2B02+144ω2B02+A12576k2ω2B02,λ2−4μ=−k2+ω2k2ω2,where A0,B0, and B1 are arbitrary constants. Imposing the solution set (100) into (24), one can obtain the following cases.Figure 6
The plot of (98) Ψ2x,t at A0=1,B0=2,a=3,b=2,k=1.5,ω=2,β=3,γ=−1,andα=1 for (a) 3D plot and (b) and (c) 2D plot (red α=0.5, blue α=0.75, and green α=0.99).
(a)(b)(c)Subcase IIA:With the help of (10), the exact periodic wave solution to the STFSRLWE is deducted as(101)Ψ3x,t=−14848k2B02+48ω2B02+A12kωB02+A1/B0λ2−4μ/2μtanhλ2−4μ/2η+λ/2μ−12kωλ2−4μ/2μtanhλ2−4μ/2η+λ/2μ2,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(102)k2+ω2>0.Subcase IB:With the help of (11), the exact solution to the STFSRLWE is obtained as(103)Ψ4x,t=−2k2+ω2kω+12−k2−ω2B0B0Δη−12kωΔ2η,Δη=λexpλη+E−1,where η=kx+1/Γαα+ωt+1/Γαα+E. Likewise, indeed need to be assured to warrantee the rational analysis of the function Ψ as(104)λ=−−k2−ω2kω,kω≠0.


## 6.2. Case II:N=3,M=1

The IEFM allows us to recruit the substitutions:(105)Ψη=A0+A1Fη+A2F2η+A3F3ηB0+B1Fη.Plugging (105) along with (9) into equation (93) and equating all the coefficients of powers of Fkη,k=0,1,…,7 to be zero, one gains a system of algebraic equations. Solving the considered system with the help of Maple, the following results will be obtained.Set I:(106)A0=−λ3k2λ2ω2B12+21k2λω2B0B1+24k2ω2B02+3k2B12−kωA1B1+3ω2B12+9k2B0B1−2kωA1B0+9ω2B0B1ωkB1,k=k,ω=ω,A1=−−2−λ2ω2k2−k2−ω2B1+λ2B1ω2k2+12k2λω2B0+k2+ω2B1kω,A2=−12λB1ωk−12B0ωk,λ=λ,μ=−14−λ2ω2k2−k2−ω2k2ω2,λ2−4μ=−k2+ω2k2ω2,A3=−12ωkB1,B0=B0,B1=B1,where B0 and B1 are arbitrary constants. Imposing the solution set (106) into (24), one can obtain the following cases.Subcase IIA:With the help of (11), the exact periodic wave solution to the STFSRLWE is deducted as(107)Ψ5x,t=A0+A1Δη−12kωλB1+12B0Δη2−12ωkB1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanλ2−4μ/2η+λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(108)kω≠0.Subcase IB:With the help of (12), the kink-singular solution to the STFSRLWE is obtained as(109)Ψ6x,t=A0+A1Δη−12kωλB1+12B0Δη2−12ωkB1Δη3B0+B1Δη,Δη=−k2−ω2kωexp−k2−ω2/kωη−1,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton is available for(110)kω≠0.Subcase IIC:With the help of (13), the exact rational solution to the STFSRLWE is gained as(111)Ψ7x,t=A0−3λ2B1ω4+12λB0ω4B1/−ωωΔη−12ω−ωλB1+B0Δη2−12ω−ωB1Δη3B0+B1Δη,A0=2k2λ3B12+4λ3B12ω2+12λ2ω2B0B12+27λ2B0B1ω2+24λω2B02B1+24B02k2λ+6B1B0−ωB1,Δη=−λ2η+E2λη+E+4,where η=−ωx+1/Γαα+ωt+1/Γαα. Also, indeed need to be assured to warrantee the rational analysis of the function Ψ as(112)ω<0.Set II:(113)A0=−λ3k2λ2ω2B12+21k2λω2B0B1+24k2ω2B02+3k2B12−kωA1B1+3ω2B12+B09k2B1−2kωA1+9ω2B1ωkB1,k=k,ω=ω,A1=−−2−λ2ω2k2+k2+ω2B1+λ2B1ω2k2+12k2λω2B0+k2+ω2B1kω,A2=−12λB1ωk−12B0ωk,λ=λ,μ=−14−λ2ω2k2+k2+ω2k2ω2,λ2−4μ=k2+ω2k2ω2,A3=−12ωkB1,B0=B0,B1=B1,where B0 and B1 are arbitrary constants. Imposing the solution set (113) into (24), one can conclude the following cases.Subcase IIA:With the help of (11), the exact dark soliton solution to the STFSRLWE is deducted as(114)Ψ8x,t=A0+A1Δη−12kωλB1+12B0Δη2−12ωkB1Δη3B0+B1Δη,Δη=−1λ2−4μ/2μtanhλ2−4μ/2η−λ/2μ,where η=kx+1/Γαα+ωt+1/Γαα+E. It should be noted that this soliton is available for(115)kω≠0.Subcase IIB:With the help of (12), the exact kink-singular solution to the STFSRLWE is obtained as(116)Ψ9x,t=A0+A1Δη−12kωλB1+12B0Δη2−12ωkB1Δη3B0+B1Δη,Δη=k2+ω2kωexpk2+ω2/kωη−1,where η=kx+1/Γαα+ωt+1/Γαα+E. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(117)kω≠0.Subcase IIC:With the help of (13), the exact rational cupson solution to the STFSRLWE is gained as(118)Ψ10x,t=A0−3λ2B1ω4+12λB0ω4B1/−ωωΔη−12ω−ωλB1+B0Δη2−12ω−ωB1Δη3B0+B1Δη,A0=λ3λ2ω4B12+21λB0ω4B1+24ω4B02+6ω2B12−−ωωA1B1+B018ω2B1−2−ωωA1,Δη=−λ2η+E2λη+E+4,where η=−ωx+1/Γαα+ωt+1/Γαα. Moreover, indeed need to be assured to warrantee the rational analysis of the function Ψ as(119)ω<0.Remark.
This paper finds many novel hyperbolic, trigonometric, kink, and kink-singular soliton solutions to govern the model. These mathematical properties come from trigonometric and hyperbolic function properties. In this sense, from the mathematical and physical points of views, these results play an important role in explaining wave propagation of considered models. We obtained more and better results compared with the results of [48] as Mohyud-din and Bibi used the exponential rational function method.


## 7. Discussion and Remark

This paper finds many novel hyperbolic, trigonometric, kink, and kink-singular soliton solutions to govern model. With the help of some calculations, surfaces of results reported have been observed in Figures1–6. In Figures 1–6, two dimension plots used the spaces x=−2, x=0, and x=2, respectively. These figures are depended on the family conditions which are important physically. It has been investigated that all figures plotted have symbolized the space-time fractional order Boussinesq equation, the space-time (2 + 1)-dimensional breaking soliton equations, and the space-time fractional order SRLW equation. These mathematical properties come from trigonometric and hyperbolic function properties. In this sense, from the mathematical and physical points of views, these results play an important role in explaining waves propagation and nonlinear ocean problems. Hence, when we consider surfaces plotted in this paper have proved such physical meaning of the solutions.


## 8. Conclusion

In this paper, the traveling wave solutions of different kinds, which are solitary, topological, dark, kink, singular, periodic, and rational solutions to the model, for the some nonlinear fractional partial differential equations (the STFBE; the STFBSEs; and the STFSRLWE) involving theβ-derivative have been obtained. The integration mechanism, namely, the rational exp−Ωη-expansion scheme is used for the considered models. It is quite visible that this integration scheme has its limitation. Thus, this paper provides a lot of encouragement for future research with β-derivative for other models. Afterwards, other extra solution methods will be applied to obtain optical and singular soliton solutions to the nonlinear model. In addition, some fractional PDEs are converted to ordinary differential equations by with the help of transformation β-derivative. Finally, the existence of the solutions for the constraint conditions is also presented. The constructed results may be helpful in explaining the physical meaning of the studied models and other related as a growing finding in the engineering sciences, mathematical physics, and so forth. The results are beneficial to the study of the wave propagation. All calculations in this paper have been made quickly with the aid of the Maple.



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*Source: 9179826-2020-05-27.xml*