# The New Solitary Solutions to the Time-Fractional Coupled Jaulent–Miodek Equation

Authors: Guiping Shen; Jalil Manafian; Syed Maqsood Zia; Dinh Tran Ngoc Huy; Trung-Hieu Le
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Here, two applicable methods, namely, thetanθ/2-expansion technique and modified exp−θξ-expansion technique are being applied on the time-fractional coupled Jaulent–Miodek equation. Materials such as photovoltaic-photorefractive, polymer, and organic contain spatial solitons and optical nonlinearities, which can be identified by seeking from energy-dependent Schrödinger potential. Plentiful exact traveling wave solutions containing unknown values are constructed in the sense of trigonometric, hyperbolic, exponential, and rational functions. Different arbitrary constants acquired in the solutions help us to discuss the dynamical behavior of solutions. Moreover, the graphical representation of solutions is shown vigorously in order to visualize the behavior of the solutions acquired for the mentioned equation. We obtain some periodic, dark soliton, and singular-kink wave solutions which have considerably fortified the existing literature on the time-fractional coupled Jaulent–Miodek equation. Via three-dimensional plot, density plot, and two-dimensional plot by utilizing Maple software, the physical properties of these waves are explained very well.


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## Body

## 1. Introduction

The main idea of this article focused on how we can get the new exact and numerical solutions for time-fractional coupled Jaulent–Miodek equation by using the new analytical methods, that is to say, thetanθ/2-expansion method and the modified exp−θξ-expansion method. Several authors studied the various nonlinear models through two different methods which may be applied to recent results such as the Biswas–Milovic equation for Kerr law nonlinearity [1], the Kerr law and dual power law Schrödinger equations [2], the fractional Bogoyavlenskii equations with conformable derivative [3], the Vakhnenko–Parkes equation [4], the Caudrey–Dodd–Gibbon and Pochhammer–Chree equations [5], some nonlinear fractional physical model [6], the time-fractional Kuramoto–Sivashinsky equation [7], new double-chain model of DNA, and a diffusive predator-prey system [8].Authors usually use the fractional differential equations to explain a few attractive physical phenomena. Many terms we use naturedly or randomly in daily life often have a fractional structure. Verbal and numerical expressions that we use when describing something, explaining an event, and commanding and in many other cases contain behavior physical of fractional equations with wide range of nonlinear physical phenomena in the vast areas of scientific disciplines which are explained by nonlinear evolution equations. Owing to the importance of studying this class of equations, several distinct techniques have been proposed such as the fractional Davey–Stewartson equation with power law nonlinearity [9], the Calogero–Bogoyavlenskii–Schiff equation in (2 + 1) dimensions with time-fractional conformable derivative [10], the fractional mitigating Internet bottleneck with quadratic cubic nonlinearity [11], the density-dependent conformable fractional diffusion-reaction equation [12], the space-time fractional Klein–Gordon equation with symmetry analysis [13], and the space-time fractional advection-diffusion equation with convergence analysis [14].The further analytical methods for the (2 + 1)-dimensional Jaulent–Miodek (JM) evolution equation have been well scrutinized by a lot of researchers containing the results such as the Hirota’s bilinear method [15], the exp-function method [16], the symmetry reductions method [17], the homotopy perturbation method [18], the optimal hidden symmetries [19], and the related topics of the Jaulent–Miodek equation with various topics, e.g., the (G′/G)-expansion method [20], the bifurcation and exact traveling wave solutions [21], the integrating factors method in an unbounded domain [22], the Adomian’s decomposition method [23], the finite-band solution method [24], N-fold Darboux transformation method [25], the Hermite wavelets method [26], and the modified Riemann–Liouville derivative and exterior derivatives [27]. In the following, we take the nonlinear time-fractional coupled Jaulent–Miodek (FCJM) equation, firstly introduced by Jaulent and Miodek [27–29], that is,(1)∂αu∂tα+∂3u∂x3+32v∂3v∂x3+92∂v∂x∂2v∂x2−6uv∂v∂x−6u∂u∂x−32v2∂u∂x=0,0≤α<1,∂αv∂tα+∂3v∂x3−6u∂v∂x−6v∂u∂x−152v2∂v∂x=0,0≤α<1.Gupta and Ray [30] used the wavelet method based on the Hermite wavelet expansion and found the numerical solution to a coupled system of nonlinear time-fractional Jaulent–Miodek equations. The Lie symmetry analysis has been performed on a coupled system of nonlinear time-fractional Jaulent–Miodek equations associated with energy-dependent Schrödinger potential to find the exact solution using Erdelyi–Kober fractional derivatives [31]. Moreover, the authors of [32–34] obtained new exact solutions for some of PDEs using the Hirota bilinear method.The foremost objective in this article is discovering some periodic, soliton, singular, and singular-kink solutions by using two methods, namely,tanθ/2-expansion method and modified exp−θξ-expansion method [35], for the time-fractional coupled JM equations. In [36], Boulkhemair et al. used geometrical variations of a state-constrained functional on star-shaped domains. Nachaoui et al. investigated parallel numerical computation of an analytical method for solving an inverse problem in [37]. Use the modified Riemann–Liouville derivative of order α [38] as follows:(2)Dtαut=1Γ1−α∫0tt−ϑ−αuϑ−u0dϑ,if0<α≤1,untα−n,ifn≤α<n+1,n≥1,with the below relations [39–41](3)DJαfxgx=∑j=0+∞αjfjxgR−Lα−jx−f0g0xαΓ1−α,DJαfgx=∑j=0+∞αjxj−αj!Γj−α+1∑m=1jfmg∑∏k=1j1Pk!gkk!Pk+fgx−fg0xαΓ1−α,Dtαtγ=Γγ+1Γ1+α−γtγ−α,γ>0,where Γ denotes the gamma function.In numerous scientific and engineering problems, to find exact analytical solutions of nonlinear partial differential equations (NLPDEs) for understanding many physical phenomena has great significance. So, establishing explicit traveling wave solutions of many physical systems has attracted considerable attention in the last few decades. Fortunately, in the literature, most of these methods are based on the hypothesis that the exact solution can be expressed as finite expansion of a function and use the following methodology in general to determine it. The method is efficiently used for several kinds of differential equations with respect to subsidiary conditions, types, etc. It includes initial value problems, boundary value problem, partial and ordinary differential equations, linear and nonlinear equations, and also nonlinear stochastic system as well as deterministic differential equations. In contrast to previously defined methods, this method extends more solutions containing periodic, hyperbolic, rational, kink, singular, and singular-kink solutions. In the continuation, we utilize concepts of local fractional calculus and its properties in Section2. By utilizing of the symbolic calculation and employing the used TEM and MEEM, we want to solve the CFJM system in Sections 3 and 4. As a consequence, we plot some periodic, soliton, singular, and singular-kink solutions in Section 5. At the end, some concluding remarks about the obtained results are given.


## 2. Description of Local Fractional Calculus and Its Properties

Definition 1.
(see [42–44]). Assume that φx is defined throughout some interval containing x0 and all points near x0, then φx is said to be local fractional continuous at x=x0, symbolized by limx⟶x0φx=φx0; then to each positive number ε and also to a positive constant Ω corresponding some positive ε, we have(4)φx−φx0<Ωεβ,0≤β<1,whenever x−x0<ρ, ρ>0 and ε,ρ∈ℝ. Therefore, the function φx is called local fractional continuous on the a,b,(5)φx∈Cβa,b,0≤β<1,in which β is the fractal dimension.Definition 2.
(see [42–44]). A function φx:ℝ⟶−ℝ,X↦φX is called to be nondifferentiable function of exponent β,0<β≤1 which satisfies Hölder function of exponent β, then for x,y∈X, one gets(6)φx−φy<Cx−yβ.Definition 3.
(see [42–44]). A function φx:ℝ⟶−ℝ,X↦φX is said to be local fractional continuous of order β,0<β≤1, or in short, β-local fractional continuous, when we have(7)φx−φx0=Ox−x0β,for any x0∈a,b and 0<β≤1.Definition 4.
(see [42–44]). A function φx is said to be in the space Cβa,b if and only if it can be expressed as(8)φx−φx0=Ox−x0β,for any x0∈a,b and 0<β≤1.Definition 5.
(see [42–44]). Get φx∈Cβa,b. Hence, the local fractional derivative of φx having order β,0<β≤1 at x=x0 is defined as(9)φβx=dβφxdxβ|x=x0=limx⟶x0Δβφx−φx0x−x0β,where Δβφx−φx0≊Γβ+1Δφx−φx0.Remark 1.
(see [42, 4, 45, 46]). More important properties of the local fractional derivative are as follows:(10)idβxkβdxβ=Γ1+kβΓ1+k−1βxk−1β,iidβEβκxβdxβ=κEβκxβ.Remark 2.
(see [42–47]). Moreover, the chain rule of the local fractional derivative is in the following:(11)dβφxdxβ=Λ1θxθβx,φx=Λθx,when Λ1θx and θβx exist.


## 3. Summarization of thetanθξ/2-Expansion Method

In this section, we will describe the main steps of thetanθξ/2-expansion method for finding traveling wave solutions of the time-fractional coupled Jaulent–Miodek equation in order to furnish its exact solutions:Step 1. Given a nonlinear physical model governed by partial differential equation,(12)Pu,ux,uxx,uxxx,…,Dtαu,Dt2αu,…=0,where fractional derivative is explained in local sense andP is a polynomial in ux,t and its different partial derivatives can be converted into an ordinary differential equation (ODE),(13)ℳu,ru′,r2u″,r3u‴,…,ku′,…=0,by utilizing the suitable fractional complex change(14)ux,t=uξ,ξ=rx+kΓα+1tα,wherer and k are the unknown values which would be computed subsequently. Also, the chain rule [44] is utilized as(15)Dtαu=σtuξDtαξ,Dxαu=σxuξDxαξ,whereσt and σx are the fractal indexes [43, 48], without loss of generality, consider σt=σx=κ, in which κ is a constant.Step 2. Suppose that the solution of ODE (13) can be expressed by a polynomial in tankθξ/2 as follows:(16)uξ=S1θ=∑k=0mAktankθ2,vξ=S2θ=∑k=0nBktankθ2,whereAk0≤k≤m and Bk1≤k≤n are constants to be found such that Am≠0,Bn≠0 and θ=θξ satisfies the following ordinary differential equation:(17)θ′=k1sinθ+k2cosθ+k3.We will take the below particular solutions of equation (17):Family 1: ifΔ=k12+k22−k32<0 and k2−k3≠0, afterwards θξ=2tan−1k1/k2−k3−−Δ/k2−k3tan−Δ/2ξ¯Family 2: ifΔ=k12+k22−k32>0 and k2−k3≠0, afterwards θξ=2tan−1k1/k2−k3+Δ/k2−k3tanhΔ/2ξ¯Family 3: ifΔ=k12+k22−k32>0, k2≠0, and k3=0, afterwards θξ=2tan−1k1/k2+k22+k12/k2tanhk22+k12/2ξ¯Family 4: ifΔ=k12+k22−k32<0, k3≠0, and k2=0, afterwards θξ=2tan−1−k1/k3+k32+k12/k3tank32+k12/2ξ¯Family 5: ifΔ=k12+k22−k32>0, k2−k3≠0, and k1=0, afterwards θξ=2tan−1k2+k3/k2−k3tanhk22−k32/2ξ¯.Family 6: ifk1=0 and k3=0, afterwards θξ=tan−1e2k2ξ¯−1/e2k2ξ¯+1,2ek2ξ¯/e2k2ξ¯+1Family 7: ifk2=0 and k3=0, afterwards θξ=tan−12ek1ξ¯/e2k1ξ¯+1,e2k1ξ¯−1/e2k1ξ¯+1Family 8: ifk12+k22=k32, afterwards θξ=2tan−1k1ξ¯+2/k2−k3ξ¯Family 9: Ifk1=k2=k3=kk1, afterwards θξ=2tan−1ekk1ξ¯−1Family 10: ifk1=k3=kk1 and k2=−kk1, afterwards θξ=−2tan−1ekk1ξ¯/−1+ekk1ξ¯Family 11: ifk3=k1, afterwards θξ=−2tan−1k1+k2ek2ξ¯−1/k1−k2ek2ξ¯−1Family 12: ifk1=k3, afterwards θξ=2tan−1k2+k3ek2ξ¯+1/k2−k3ek2ξ¯−1Family 13: ifk3=−k1, afterwards θξ=2tan−1ek2ξ¯+k2−k1/ek2ξ¯−k2−k1Family 14: ifk2=−k3, afterwards θξ=2tan−1k1ek1ξ¯/1−k3ek1ξ¯Family 15: ifk2=0 and k1=k3, afterwards θξ=−2tan−1k3ξ¯+2/k3ξ¯Family 16: ifk1=0 and k2=k3, afterwards θξ=2tan−1k3ξ¯Family 17: ifk1=0 and k2=−k3, afterwards θξ=−2tan−11/k3ξ¯Family 18: ifk1=0 and k2=0, afterwards θξ=k3ξ+CFamily 19: ifk2=k3, afterwards θξ=2tan−1ek1ξ¯−k3/k1 where A0, Ak,Bkk=1,2,…,m, k1,k2, and k3 are free constants to be determined later. To determine the positive integers m and n, it, usually, can be accomplished by the nonlinear terms of highest-order in equation (13) with the highest-order linear terms. Substituting equations (16) and (17) into equation (13) yields an equation of powers of tanθ/2k.Step 3. WithN determined, we then collect all coefficients of powers of tanθ/2k,cotθ/2kk=0,1,2,… in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters A0,Aii=1,2,…,m,Bii=1,2,…,n,k1,k2, and k3.


### 3.1. Periodic, Soliton, Singular, and Kink-Singular Solutions of the FCJM Equation by TEM

Take the fractional CJM equation as(18)∂αu∂tα+∂3u∂x3+32v∂3v∂x3+92∂v∂x∂2v∂x2−6uv∂v∂x−6u∂u∂x−32v2∂u∂x=0,∂αv∂tα+∂3v∂x3−6u∂v∂x−6v∂u∂x−152v2∂v∂x=0.By using the transformation (14), equation (18) will be reduced to the following ODE:(19)ku′+r3u‴+32r3vv‴+92r3v′v″−6ruvv′−6ruu′−32rv2u′=0,kv′+r3v‴−6ruv′−6rvu′−152rv2v′=0.Balancingu‴ with v2u′ and v‴ with v2v′ in equation (19) gives m=2 and n=1. We substitute the values of m=2 and n=1 in (16). To find the periodic waves of the fractional CJM equations, we would like to start from an ansatz as follows:(20)uξ=A0+A1tanθξ2+A2tanθξ22,vξ=B0+B1tanθξ2,where A0,A1,A2,B0, and B1 are free parameters. Inserting (20) into equation (19) and then collecting the coefficients in the front of the various functions including tanθξ/2 and cotθξ/2 give the following:(21)18−b+c18b2r3B12−36bcr3B12+18c2r3B12+24b2r3A2−48bcr3A2+24c2r3A2−36rA2B12−48rA22=0,14a18b2r3B12−36bcr3B12+18c2r3B12+24b2r3A2−48bcr3A2+24c2r3A2−36rA2B12−48rA22+18−b+c45acr3B12−45abr3B12+9b2r3B0B1−18bcr3B0B1+9c2r3B0B1−60abr3A2+60acr3A2+6b2r3A1−12bcr3A1+6c2r3A1−30rA1B12−48rA2B0B1−72rA1A2=0,18b+c18b2r3B12−36bcr3B12+18c2r3B12+24b2r3A2−48bcr3A2+24c2r3A2−36rA2B12−48rA22+14a−45abr3B12+45acr3B12+9b2r3B0B1−18bcr3B0B1+9c2r3B0B1−60abr3A2+60acr3A2+6b2r3A1−12bcr3A1+6c2r3A1−30rA1B12−48rA2B0B1−72rA1A2+18c−b24a2r3+12c2r3−12b2r3B12+18ar3B0B1b−c+32a2r3A2−12abr3A1+12acr3A1−16b2r3A2+16c2r3A2−24rA0B12−36rA1B0B1−12rA2B02−48rA0A2−24rA12+8kA2=0,18b+c−45abr3B1245acr3B12+9b2r3B0B1−18bcr3B0B1+9c2r3B0B1−60abr3A2+60acr3A2+6b2r3A1−12bcr3A1+6c2r3A1−30rA1B12−48rA2B0B1−72rA1A2+14a24a2r3B12−18abr3B0B1+18acr3B0B1−12b2r3B12+12c2r3B12+32a2r3A2−12abr3A1+12acr3A1−16b2r3A2+16c2r3A2−24rA0B12−36rA1B0B1−12rA2B02−48rA0A2−24rA12+8kA2+18−b+c6a2r3B0B1+9abr3B12+9acr3B12−3b2r3B0B1+3c2r3B0B1+4a2r3A1+12abr3A2+12acr3A2−2b2r3A1+2c2r3A1−24rA0B0B1−6rA1B02−24rA0A1+4kA1=0,18b+c24a2r3B12−18abr3B0B1+18acr3B0B1−12b2r3B12+12c2r3B12+32a2r3A2−12abr3A1+12acr3A1−16b2r3A2+16c2r3A2−24rA0B12−36rA1B0B1−12rA2B02−48rA0A2−24rA12+8kA2+14a6a2r3B0B1+9abr3B12+9acr3B12−3b2r3B0B1+3c2r3B0B1+4a2r3A1+12abr3A2+12acr3A2−2b2r3A1+2c2r3A1−24rA0B0B1−6rA1B02−24rA0A1+4kA118b+c24a2r3B12−18abr3B0B1+18acr3B0B1−12b2r3B12+12c2r3B12+32a2r3A2−12abr3A1+12acr3A1−16b2r3A2+16c2r3A2−24rA0B12−36rA1B0B1−12rA2B02−48rA0A2−24rA12+8kA2+14a6a2r3B0B1+9abr3B12+9acr3B12−3b2r3B0B1+3c2r3B0B1+4a2r3A1+12abr3A2+12acr3A2−2b2r3A1+2c2r3A1−24rA0B0B1−6rA1B02−24rA0A1+4kA1=0,18b+c6a2r3B0B1+9abr3B12+9acr3B12−3b2r3B0B1+3c2r3B0B1+4a2r3A1+12abr3A2+12acr3A2−2b2r3A1+2c2r3A1−24rA0B0B1−6rA1B02−24rA0A1+4kA1=0,14−b+c3b2r3B1−6bcr3B1+3c2r3B1−15rB13−36rA2B1=0,12a3b2r3B1−6bcr3B1+3c2r3B1−15rB13−36rA2B1+14−b+c−6abr3B1+6acr3B1−30rB0B12−24rA1B1−24rA2B0=0,14b+c3b2r3B1−6bcr3B1+3c2r3B1−15rB13−36rA2B1+12a−6abr3B1+6acr3B1−30rB0B12−24rA1B1−24rA2B0+14−b+c2a2r3B1−b2r3B1+c2r3B1−15rB02B1−12rA0B1−12rA1B0+2kB1=0,14b+c−6abr3B1+6acr3B1−30rB0B12−24rA1B1−24rA2B0+12a2a2r3B1−b2r3B1+c2r3B1−15rB02B1−12rA0B1−12rA1B0+2kB1=0,14b+c2a2r3B1−b2r3B1+c2r3B1−15rB02B1−12rA0B1−12rA1B0+2kB1=0.By solving the nonlinear algebraic system of equations can conclude the following results:Set I:(22)k=−14r34k12+k22−k32,r=r,A0=−18k22−k32r2,A1=−14r2k1k2−k3,A2=316r2k2−k32,B0=0,B1=12rk2−k3−1.Putting (22) into (20), we obtain a periodic solution of equation (18) by utilizing Family 1 as follows:(23)u1=−18k22−k32r2−14r2k1k2−k3k1k2−k3−−Δk2−k3tan−Δ2ξ¯+316r2k2−k32k1k2−k3−−Δk2−k3tan−Δ2ξ¯2,(24)v1=12rk2−k3−1k1k2−k3−−Δk2−k3tan−Δ2ξ¯,whereΔ=k12+k22−k32<0 and ξ¯=rx−1/4r34k12+k22−k32tα/Γα+1+C.Appending (22) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(25)u2=−18k22−k32r2−14r2k1k2−k3k1k2−k3+Δk2−k3tanhΔ2ξ¯+316r2k2−k32k1k2−k3+Δk2−k3tanhΔ2ξ¯2,(26)v2=12rk2−k3−1k1k2−k3+Δk2−k3tanhΔ2ξ¯,whereΔ=k12+k22−k32>0 and ξ¯=rx−1/4r34k12+k22−k32tα/Γα+1+C.Substituting (22) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows:(27)u3=−18k22−k32r2−14r2k1k2−k3k1ξ¯+2k2−k3ξ¯+316r2k2−k32k1ξ¯+2k2−k3ξ¯2,(28)v3=12rk2−k3−1316r2k2−k32k1ξ¯+2k2−k3ξ¯,wherek12=k32−k22 and ξ¯=rx−3/3r3k12tα/Γα+1+C.Inserting (22) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 12 as follows:(29)u4=−18k22−k32r2−14r2k3k2−k3k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1+316r2k2−k32k2+k3ek2ξ¯+1k2−k3ek2ξ¯−12,(30)v4=12rk2−k3−1316r2k2−k32k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1,whereξ¯=rx−1/4r33k32+k22tα/Γα+1+C.Set II:(31)k=24k12B12+6k1k2B0B1−6k1k3B0B1+3b2B02+k22B12−6k2k3B02+3k32B02−k32B12−1B1k2−k33,r=2B1−1k2−k3,A0=−14k2B02−2k2B12−k3B02−2k3B12k2−k3,A1=122k1B1−k2B0+k3B0B1k2−k3,A2=−34B12,B0=B0,B1=B1.Substituting (31) into (20), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows:(32)u5=−14k2B02−2k2B12−k3B02−2k3B12k2−k3+122k1B1−k2B0+k3B0B1k2−k3k1k2−k3−−Δk2−k3tan−Δ2ξ¯−34B12k1k2−k3−−Δk2−k3tan−Δ2ξ¯2,v5=B0+B1k1k2−k3−−Δk2−k3tan−Δ2ξ¯,whereΔ=k12+k22−k32<0 and ξ¯=2B1−1/k2−k3x+24k12B12+6k1k2B0B1−6k1k3B0B1+3k22B02+k22B12−6k2k3B02+3k32B02−k32B12−1B1/k2−k33tα/Γα+1+C.Appending (31) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(33)u6=−14k2B02−2k2B12−k3B02−2k3B12k2−k3+122k1B1−k2B0+k3B0B1k2−k3k1k2−k3+Δk2−k3tanhΔ2ξ¯−34B12k1k2−k3+Δk2−k3tanhΔ2ξ¯2,v6=B0+B1k1k2−k3+Δk2−k3tanhΔ2ξ¯,whereΔ=k12+k22−k32>0 and ξ¯=2B1−1/k2−k3x+24k12B12+6k1k2B0B1−6k1k3B0B1+3k22B02+k22B12−6k2k3B02+3k32B02−k32B12−1B1/k2−k33tα/Γα+1+C.Substituting (31) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows:(34)u7=−14k2B02−2k2B12−k3B02−2k3B12k2−k3+122k1B1−k2B0+k3B0B1k2−k3k1ξ¯+2k2−k3ξ¯−34B12k1ξ¯+2k2−k3ξ¯2,v7=B0+B1k1ξ¯+2k2−k3ξ¯,wherek12+k22−k32=0 and ξ¯=2B1−1/k2−k3x+24k12B12+6k1k2B0B1−6k1k3B0B1+3k22B02+k22B12−6k2k3B02+3k32B02−k32B12−1B1/k2−k33tα/Γα+1+C.Inserting (31) into (20), we receive a hyperbolic solution of equation (18) by utilizing Family 12 as follows:(35)u8=−14k2B02−2k2B12−k3B02−2k3B12k2−k3+122k3B1−k2B0+k3B0B1k2−k3k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1−34B12k2+k3ek2ξ¯+1k2−k3ek2ξ¯−12,v8=B0+B1k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1,wherek1=k3 and ξ¯=2B1−1/k2−k3x+23k32B12+6k3k2B0B1−6k32B0B1+3k22B02+k22B12−6k2k3B02+3k32B02−1B1/k2−k33tα/Γα+1+C.Set III:(36)k=−1476k12+k22−k32r3,r=r,A0=18r218k12−k22+k32,A1=−k2−k3ar2,A2=316r2k2−k32,B0=−3k1r−1,B1=12rk2−k3−1.Inserting (36) into (20), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows:(37)u9=18r218k12−k22+k32−k2−k3k1r2k1k2−k3−−Δk2−k3tan−Δ2ξ¯+316r2k2−k32k1k2−k3−−Δk2−k3tan−Δ2ξ¯2,v9=−3k1r−1+12rk2−k3−1k1k2−k3−−Δk2−k3tan−Δ2ξ¯,whereΔ=k12+k22−k32<0 and ξ¯=rx−1/476k12+k22−k32r3tα/Γα+1+C.Appending (36) into (20), we obtain a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(38)u10=18r218k12−k22+k32−k2−k3k1r2k1k2−k3+Δk2−k3tanhΔ2ξ¯+316r2k2−k32k1k2−k3+Δk2−k3tanhΔ2ξ¯2,v10=−3k1r−1+12rk2−k3−1k1k2−k3+Δk2−k3tanhΔ2ξ¯,whereΔ=k12+k22−k32<0 and ξ¯=rx−1/476k12+k22−k32r3tα/Γα+1+C.Inserting (36) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows:(39)u11=18r218k12−k22+k32−k2−k3k1r2k1ξ¯+2k2−k3ξ¯+316r2k2−k32k1ξ¯+2k2−k3ξ¯2,v11=−3k1r−1+12rk2−k3−1k1ξ¯+2k2−k3ξ¯,wherek12+k22−k32=0 and ξ¯=rx−75/4k12r3tα/Γα+1+C.Appending (36) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 12 as follows:(40)u12=18r219k12−k22−k2−k3k3r2k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1+316r2k2−k32k2+k3ek2ξ¯+1k2−k3ek2ξ¯−12,v12=−3k3r−1+12rk2−k3−1k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1,wherek1=k3 and ξ¯=rx−1/475k32+k22r3tα/Γα+1+C.Set IV:(41)k=−k12+k22−k32r3,r=r,A0=14r2k12−k22+k32,A1=−k2−k3k1r2,A2=12r2k2−k32,B0=−k1r−1,B1=rk2−k3−1.Substituting (41) into (20), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows:(42)u13=14r2k12−k22+k32−k2−k3k1r2k1k2−k3−−Δk2−k3tan−Δ2ξ¯+12r2k2−k32k1k2−k3−−Δk2−k3tan−Δ2ξ¯2,v13=−k1r−1+rk2−k3−1k1k2−k3−−Δk2−k3tan−Δ2ξ¯,where Δ=k12+k22−k32<0 and ξ¯=rx−k12+k22−k32r3tα/Γα+1+C.Inserting (41) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(43)u14=14r2k12−k22+k32−k2−k3k1r2k1k2−k3+Δk2−k3tanhΔ2ξ¯+12r2k2−k32k1k2−k3+Δk2−k3tanhΔ2ξ¯2,v14=−k1r−1+rk2−k3−1k1k2−k3+Δk2−k3tanhΔ2ξ¯,where Δ=k12+k22−k32<0 and ξ¯=rx−k12+k22−k32r3tα/Γα+1+C.Plugging (41) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows:(44)u15=14r2k12−k22+k32−k2−k3k1r2k1ξ¯+2k2−k3ξ¯+12r2k2−k32k1ξ¯+2k2−k3ξ¯2,v15=−k1r−1+rk2−k3−1k1ξ¯+2k2−k3ξ¯,where Δ=k12+k22−k32⟶0 and ξ¯=rx−Δ⟶0r3r3tα/Γα+1+C.Inserting (41) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 12 as follows:(45)u16=14r22k32−k22−k2−k3k3r2k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1+12r2k2−k32k2+k3ek2ξ¯+1k2−k3ek2ξ¯−12,v16=−k3r−1+rk2−k3−1k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1,where k1=k3 and ξ¯=rx−k22r3tα/Γα+1+C.


## 3.1. Periodic, Soliton, Singular, and Kink-Singular Solutions of the FCJM Equation by TEM

Take the fractional CJM equation as(18)∂αu∂tα+∂3u∂x3+32v∂3v∂x3+92∂v∂x∂2v∂x2−6uv∂v∂x−6u∂u∂x−32v2∂u∂x=0,∂αv∂tα+∂3v∂x3−6u∂v∂x−6v∂u∂x−152v2∂v∂x=0.By using the transformation (14), equation (18) will be reduced to the following ODE:(19)ku′+r3u‴+32r3vv‴+92r3v′v″−6ruvv′−6ruu′−32rv2u′=0,kv′+r3v‴−6ruv′−6rvu′−152rv2v′=0.Balancingu‴ with v2u′ and v‴ with v2v′ in equation (19) gives m=2 and n=1. We substitute the values of m=2 and n=1 in (16). To find the periodic waves of the fractional CJM equations, we would like to start from an ansatz as follows:(20)uξ=A0+A1tanθξ2+A2tanθξ22,vξ=B0+B1tanθξ2,where A0,A1,A2,B0, and B1 are free parameters. Inserting (20) into equation (19) and then collecting the coefficients in the front of the various functions including tanθξ/2 and cotθξ/2 give the following:(21)18−b+c18b2r3B12−36bcr3B12+18c2r3B12+24b2r3A2−48bcr3A2+24c2r3A2−36rA2B12−48rA22=0,14a18b2r3B12−36bcr3B12+18c2r3B12+24b2r3A2−48bcr3A2+24c2r3A2−36rA2B12−48rA22+18−b+c45acr3B12−45abr3B12+9b2r3B0B1−18bcr3B0B1+9c2r3B0B1−60abr3A2+60acr3A2+6b2r3A1−12bcr3A1+6c2r3A1−30rA1B12−48rA2B0B1−72rA1A2=0,18b+c18b2r3B12−36bcr3B12+18c2r3B12+24b2r3A2−48bcr3A2+24c2r3A2−36rA2B12−48rA22+14a−45abr3B12+45acr3B12+9b2r3B0B1−18bcr3B0B1+9c2r3B0B1−60abr3A2+60acr3A2+6b2r3A1−12bcr3A1+6c2r3A1−30rA1B12−48rA2B0B1−72rA1A2+18c−b24a2r3+12c2r3−12b2r3B12+18ar3B0B1b−c+32a2r3A2−12abr3A1+12acr3A1−16b2r3A2+16c2r3A2−24rA0B12−36rA1B0B1−12rA2B02−48rA0A2−24rA12+8kA2=0,18b+c−45abr3B1245acr3B12+9b2r3B0B1−18bcr3B0B1+9c2r3B0B1−60abr3A2+60acr3A2+6b2r3A1−12bcr3A1+6c2r3A1−30rA1B12−48rA2B0B1−72rA1A2+14a24a2r3B12−18abr3B0B1+18acr3B0B1−12b2r3B12+12c2r3B12+32a2r3A2−12abr3A1+12acr3A1−16b2r3A2+16c2r3A2−24rA0B12−36rA1B0B1−12rA2B02−48rA0A2−24rA12+8kA2+18−b+c6a2r3B0B1+9abr3B12+9acr3B12−3b2r3B0B1+3c2r3B0B1+4a2r3A1+12abr3A2+12acr3A2−2b2r3A1+2c2r3A1−24rA0B0B1−6rA1B02−24rA0A1+4kA1=0,18b+c24a2r3B12−18abr3B0B1+18acr3B0B1−12b2r3B12+12c2r3B12+32a2r3A2−12abr3A1+12acr3A1−16b2r3A2+16c2r3A2−24rA0B12−36rA1B0B1−12rA2B02−48rA0A2−24rA12+8kA2+14a6a2r3B0B1+9abr3B12+9acr3B12−3b2r3B0B1+3c2r3B0B1+4a2r3A1+12abr3A2+12acr3A2−2b2r3A1+2c2r3A1−24rA0B0B1−6rA1B02−24rA0A1+4kA118b+c24a2r3B12−18abr3B0B1+18acr3B0B1−12b2r3B12+12c2r3B12+32a2r3A2−12abr3A1+12acr3A1−16b2r3A2+16c2r3A2−24rA0B12−36rA1B0B1−12rA2B02−48rA0A2−24rA12+8kA2+14a6a2r3B0B1+9abr3B12+9acr3B12−3b2r3B0B1+3c2r3B0B1+4a2r3A1+12abr3A2+12acr3A2−2b2r3A1+2c2r3A1−24rA0B0B1−6rA1B02−24rA0A1+4kA1=0,18b+c6a2r3B0B1+9abr3B12+9acr3B12−3b2r3B0B1+3c2r3B0B1+4a2r3A1+12abr3A2+12acr3A2−2b2r3A1+2c2r3A1−24rA0B0B1−6rA1B02−24rA0A1+4kA1=0,14−b+c3b2r3B1−6bcr3B1+3c2r3B1−15rB13−36rA2B1=0,12a3b2r3B1−6bcr3B1+3c2r3B1−15rB13−36rA2B1+14−b+c−6abr3B1+6acr3B1−30rB0B12−24rA1B1−24rA2B0=0,14b+c3b2r3B1−6bcr3B1+3c2r3B1−15rB13−36rA2B1+12a−6abr3B1+6acr3B1−30rB0B12−24rA1B1−24rA2B0+14−b+c2a2r3B1−b2r3B1+c2r3B1−15rB02B1−12rA0B1−12rA1B0+2kB1=0,14b+c−6abr3B1+6acr3B1−30rB0B12−24rA1B1−24rA2B0+12a2a2r3B1−b2r3B1+c2r3B1−15rB02B1−12rA0B1−12rA1B0+2kB1=0,14b+c2a2r3B1−b2r3B1+c2r3B1−15rB02B1−12rA0B1−12rA1B0+2kB1=0.By solving the nonlinear algebraic system of equations can conclude the following results:Set I:(22)k=−14r34k12+k22−k32,r=r,A0=−18k22−k32r2,A1=−14r2k1k2−k3,A2=316r2k2−k32,B0=0,B1=12rk2−k3−1.Putting (22) into (20), we obtain a periodic solution of equation (18) by utilizing Family 1 as follows:(23)u1=−18k22−k32r2−14r2k1k2−k3k1k2−k3−−Δk2−k3tan−Δ2ξ¯+316r2k2−k32k1k2−k3−−Δk2−k3tan−Δ2ξ¯2,(24)v1=12rk2−k3−1k1k2−k3−−Δk2−k3tan−Δ2ξ¯,whereΔ=k12+k22−k32<0 and ξ¯=rx−1/4r34k12+k22−k32tα/Γα+1+C.Appending (22) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(25)u2=−18k22−k32r2−14r2k1k2−k3k1k2−k3+Δk2−k3tanhΔ2ξ¯+316r2k2−k32k1k2−k3+Δk2−k3tanhΔ2ξ¯2,(26)v2=12rk2−k3−1k1k2−k3+Δk2−k3tanhΔ2ξ¯,whereΔ=k12+k22−k32>0 and ξ¯=rx−1/4r34k12+k22−k32tα/Γα+1+C.Substituting (22) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows:(27)u3=−18k22−k32r2−14r2k1k2−k3k1ξ¯+2k2−k3ξ¯+316r2k2−k32k1ξ¯+2k2−k3ξ¯2,(28)v3=12rk2−k3−1316r2k2−k32k1ξ¯+2k2−k3ξ¯,wherek12=k32−k22 and ξ¯=rx−3/3r3k12tα/Γα+1+C.Inserting (22) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 12 as follows:(29)u4=−18k22−k32r2−14r2k3k2−k3k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1+316r2k2−k32k2+k3ek2ξ¯+1k2−k3ek2ξ¯−12,(30)v4=12rk2−k3−1316r2k2−k32k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1,whereξ¯=rx−1/4r33k32+k22tα/Γα+1+C.Set II:(31)k=24k12B12+6k1k2B0B1−6k1k3B0B1+3b2B02+k22B12−6k2k3B02+3k32B02−k32B12−1B1k2−k33,r=2B1−1k2−k3,A0=−14k2B02−2k2B12−k3B02−2k3B12k2−k3,A1=122k1B1−k2B0+k3B0B1k2−k3,A2=−34B12,B0=B0,B1=B1.Substituting (31) into (20), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows:(32)u5=−14k2B02−2k2B12−k3B02−2k3B12k2−k3+122k1B1−k2B0+k3B0B1k2−k3k1k2−k3−−Δk2−k3tan−Δ2ξ¯−34B12k1k2−k3−−Δk2−k3tan−Δ2ξ¯2,v5=B0+B1k1k2−k3−−Δk2−k3tan−Δ2ξ¯,whereΔ=k12+k22−k32<0 and ξ¯=2B1−1/k2−k3x+24k12B12+6k1k2B0B1−6k1k3B0B1+3k22B02+k22B12−6k2k3B02+3k32B02−k32B12−1B1/k2−k33tα/Γα+1+C.Appending (31) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(33)u6=−14k2B02−2k2B12−k3B02−2k3B12k2−k3+122k1B1−k2B0+k3B0B1k2−k3k1k2−k3+Δk2−k3tanhΔ2ξ¯−34B12k1k2−k3+Δk2−k3tanhΔ2ξ¯2,v6=B0+B1k1k2−k3+Δk2−k3tanhΔ2ξ¯,whereΔ=k12+k22−k32>0 and ξ¯=2B1−1/k2−k3x+24k12B12+6k1k2B0B1−6k1k3B0B1+3k22B02+k22B12−6k2k3B02+3k32B02−k32B12−1B1/k2−k33tα/Γα+1+C.Substituting (31) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows:(34)u7=−14k2B02−2k2B12−k3B02−2k3B12k2−k3+122k1B1−k2B0+k3B0B1k2−k3k1ξ¯+2k2−k3ξ¯−34B12k1ξ¯+2k2−k3ξ¯2,v7=B0+B1k1ξ¯+2k2−k3ξ¯,wherek12+k22−k32=0 and ξ¯=2B1−1/k2−k3x+24k12B12+6k1k2B0B1−6k1k3B0B1+3k22B02+k22B12−6k2k3B02+3k32B02−k32B12−1B1/k2−k33tα/Γα+1+C.Inserting (31) into (20), we receive a hyperbolic solution of equation (18) by utilizing Family 12 as follows:(35)u8=−14k2B02−2k2B12−k3B02−2k3B12k2−k3+122k3B1−k2B0+k3B0B1k2−k3k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1−34B12k2+k3ek2ξ¯+1k2−k3ek2ξ¯−12,v8=B0+B1k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1,wherek1=k3 and ξ¯=2B1−1/k2−k3x+23k32B12+6k3k2B0B1−6k32B0B1+3k22B02+k22B12−6k2k3B02+3k32B02−1B1/k2−k33tα/Γα+1+C.Set III:(36)k=−1476k12+k22−k32r3,r=r,A0=18r218k12−k22+k32,A1=−k2−k3ar2,A2=316r2k2−k32,B0=−3k1r−1,B1=12rk2−k3−1.Inserting (36) into (20), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows:(37)u9=18r218k12−k22+k32−k2−k3k1r2k1k2−k3−−Δk2−k3tan−Δ2ξ¯+316r2k2−k32k1k2−k3−−Δk2−k3tan−Δ2ξ¯2,v9=−3k1r−1+12rk2−k3−1k1k2−k3−−Δk2−k3tan−Δ2ξ¯,whereΔ=k12+k22−k32<0 and ξ¯=rx−1/476k12+k22−k32r3tα/Γα+1+C.Appending (36) into (20), we obtain a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(38)u10=18r218k12−k22+k32−k2−k3k1r2k1k2−k3+Δk2−k3tanhΔ2ξ¯+316r2k2−k32k1k2−k3+Δk2−k3tanhΔ2ξ¯2,v10=−3k1r−1+12rk2−k3−1k1k2−k3+Δk2−k3tanhΔ2ξ¯,whereΔ=k12+k22−k32<0 and ξ¯=rx−1/476k12+k22−k32r3tα/Γα+1+C.Inserting (36) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows:(39)u11=18r218k12−k22+k32−k2−k3k1r2k1ξ¯+2k2−k3ξ¯+316r2k2−k32k1ξ¯+2k2−k3ξ¯2,v11=−3k1r−1+12rk2−k3−1k1ξ¯+2k2−k3ξ¯,wherek12+k22−k32=0 and ξ¯=rx−75/4k12r3tα/Γα+1+C.Appending (36) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 12 as follows:(40)u12=18r219k12−k22−k2−k3k3r2k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1+316r2k2−k32k2+k3ek2ξ¯+1k2−k3ek2ξ¯−12,v12=−3k3r−1+12rk2−k3−1k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1,wherek1=k3 and ξ¯=rx−1/475k32+k22r3tα/Γα+1+C.Set IV:(41)k=−k12+k22−k32r3,r=r,A0=14r2k12−k22+k32,A1=−k2−k3k1r2,A2=12r2k2−k32,B0=−k1r−1,B1=rk2−k3−1.Substituting (41) into (20), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows:(42)u13=14r2k12−k22+k32−k2−k3k1r2k1k2−k3−−Δk2−k3tan−Δ2ξ¯+12r2k2−k32k1k2−k3−−Δk2−k3tan−Δ2ξ¯2,v13=−k1r−1+rk2−k3−1k1k2−k3−−Δk2−k3tan−Δ2ξ¯,where Δ=k12+k22−k32<0 and ξ¯=rx−k12+k22−k32r3tα/Γα+1+C.Inserting (41) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(43)u14=14r2k12−k22+k32−k2−k3k1r2k1k2−k3+Δk2−k3tanhΔ2ξ¯+12r2k2−k32k1k2−k3+Δk2−k3tanhΔ2ξ¯2,v14=−k1r−1+rk2−k3−1k1k2−k3+Δk2−k3tanhΔ2ξ¯,where Δ=k12+k22−k32<0 and ξ¯=rx−k12+k22−k32r3tα/Γα+1+C.Plugging (41) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows:(44)u15=14r2k12−k22+k32−k2−k3k1r2k1ξ¯+2k2−k3ξ¯+12r2k2−k32k1ξ¯+2k2−k3ξ¯2,v15=−k1r−1+rk2−k3−1k1ξ¯+2k2−k3ξ¯,where Δ=k12+k22−k32⟶0 and ξ¯=rx−Δ⟶0r3r3tα/Γα+1+C.Inserting (41) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 12 as follows:(45)u16=14r22k32−k22−k2−k3k3r2k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1+12r2k2−k32k2+k3ek2ξ¯+1k2−k3ek2ξ¯−12,v16=−k3r−1+rk2−k3−1k2+k3ek2ξ¯+1k2−k3ek2ξ¯−1,where k1=k3 and ξ¯=rx−k22r3tα/Γα+1+C.


## 4. Summarization of the Modifiedexp−θξ-Expansion Method

In this section, we will describe the main steps of the modifiedexp−θξ-expansion method for finding traveling wave solutions of NLPDEs to the time-fractional coupled Jaulent–Miodek equation in order to furnish its exact solutions:Step 1. Given a nonlinear physical model governed by partial differential equation of fractional order is expressed as(46)pu,ux,uxx,uxxx,…,Dtαu,Dt2αu,…=0,the above equation can be converted into an ordinary differential equation (ODE),(47)ℳu,ru′,r2u″,r3u‴,…,ku′,…=0,by utilizing the suitable fractional complex transform(48)ux,t=uξ,ξ=rx+kΓα+1tα,wherer and k are free parameters which would be computed subsequently. Also, the chain rule [44] is used as(49)Dtαu=σtuξDtαξ,Dxαu=σxuξDxαξ,whereσt and σx are the fractal indexes [43, 48]; without loss of generality, take σt=σx=κ, where κ is a constant.Step 2. Suppose that the solution of ODE (47) can be expressed by a polynomial in exp−θξ as follows:(50)uξ=S1θ=∑k=0N1Akexp−θξk∑k=0M1Bkexp−θξk,vξ=S2θ=∑k=0N2Ckexp−θξk∑k=0M2Dkexp−θξk,whereAk0≤k≤m and Bk1≤k≤n are constants to be found such that AN1≠0,CN2≠0, and θ=θξ satisfy the below ordinary differential equation:(51)θ′=μexpθ+exp−θ+λ,the solutions of equation (51) is given byFamily 1: ifμ≠0 and λ2−4μ>0, afterwards θξ=ln−λ/2μ−λ2−4μ/2μtanhλ2−4μ/2ξ¯Family 2: ifμ≠0 and λ2−4μ<0, afterwards θξ=ln−λ/2μ+4μ−λ2/2μtanh4μ−λ2/2ξ¯Family 3: ifμ=0,λ≠0 and λ2−4μ>0, afterwards θξ=−lnλ/expλξ¯−1Family 4: ifμ≠0,λ≠0 and λ2−4μ=0, afterwards θξ=ln−2λξ¯+4/λ2ξ¯Family 5: ifμ=0,λ=0 and λ2−4μ=0, afterwards θξ=lnξ¯ where Akk=0,1,…,N1,Bkk=0,1,…,M1,Ckk=0,1,…,N2,Dkk=1,2,…,M2, λ, and μ are also the free constants.Step 3. To determine the positive integerN1,M1,N2,M2, it, usually, can be accomplished by the nonlinear terms of highest-order in equation (47) with the highest-order linear terms. Substituting equations (50) and (51) into equation (13) yields an equation of powers of exp−kθ.Step 4. WithN1,M1,N2,M2 determined, we then collect all coefficients of powers of exp−kθ in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters Akk=0,1,…,N1,Bkk=0,1,…,M1,Ckk=0,1,…,N2,Dkk=1,2,…,M2, λ, and μ.


### 4.1. Periodic, Soliton, Singular, and Kink-Singular Solutions of the FCJM Equation by MEM

Balancingu‴ with v2u′ and v‴ with v2v′ in equation (19) gives the following cases:(52)Case I:N1=2,M1=0,N2=1,M1=0,Case II:N1=3,M1=1,N2=2,M1=1.Putting the values of (52) and B0=D0=1 to search for different wave solutions of the fractional CJM equations, we would like to start from ansatz as the following form:(53)uξ=A0+A1exp−θξ+A2exp−2θξ,vξ=C0+C1exp−θξ,where A0,A1,A2,C0, and C1 are unknown values. Inserting (53) into equation (56) and then collecting the coefficients in the front of the various functions containing exp−θξ are given as follows:(54)12r3C1−15rC13−36rA2C1=0,24λr3C1−15λrC13−36λrA2C1−30rC0C12−24rA1C1−24rA2C0=0,14λ2r3C1−30λrC0C12+16μr3C1−15μrC13−24λrA1C1−24λrA2C0−36μrA2C1−15rC02C1−12rA0C1−12rA1C0+2kC1=0,2λ3r3C1+16λμr3C1−15λrC02C1−30μrC0C12−12λrA0C1−12λrA1C0−24μrA1C1−24μrA2C0+2kλC1=0,2λ2μr3C1+4μ2r3C1−15μrC02C1−12μrA0C1−12μrA1C0+2kμC1=0,36r3C12+48r3A2−18rA2C12−24rA22=0,81λr3C12+108λr3A2−18λrA2C12+18r3C0C1−24λrA22+12r3A1−15rA1C12−24rA2C0C1−36rA1A2=0,57λ2r3C12+76λ2r3A2+36λr3C0C1+60μr3C12+24λr3A1−15λrA1C12−24λrA2C0C1+80μr3A2−18μrA2C12−36λrA1A2−24μrA22−12rA0C12−18rA1C0C1−6rA2C02−24rA0A2−12rA12+4kA2=0,12λ3r3C12+16λ3r3A2+21λ2r3C0C1+78λμr3C12+14λ2r3A1+104λμr3A2+24μr3C0C1−12λrA0C12−18λrA1C0C1−6λrA2C02+16μr3A1−15μrA1C12−24μrA2C0C1−24λrA0A2−12λrA12−36μrA1A2−12rA0C0C1−3rA1C02+4kλA2−12rA0A1+2kA1=0,3λ3r3C0C1+21λ2μr3C12+2λ3r3A1+28λ2μr3A2+24λμr3C0C1+24μ2r3C12+16λμr3A1+32μ2r3A2−12λrA0C0C1−3λrA1C02−12μrA0C12−18μrA1C0C1−6μrA2C02−12λrA0A1−24μrA0A2−12μrA12+2kλA1+4kμA2=0,3λ2μr3C0C1+9λμ2r3C12+2λ2μr3A1+1λμ2r3A2+6μ2r3C0C1+4μ2r3A1−12μrA0C0C1−3μrA1C02−12μrA0A1+2kμA1=0.Solving the nonlinear algebraic system of equations can be concluded as the following solutions:Set I:(55)k=3rC02,λ=0,μ=0,r=r,A0=−14C02,A1=−12C0r−1,A2=34r2,C0=C0,C1=r−1.Plugging (55) into (53), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows:(56)u1=−14C02−12C0r−11ξ¯+34r21ξ¯2,v1=C0+r−11ξ¯,ξ¯=rx+3rC02tαΓα+1+C.Set II:(57)k=−λ2r3,λ=λ,μ=0,r=r,A0=0,A1=12λr2,A2=34r2,C0=0,C1=r−1.Appending (57) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 3 as follows:(58)u2=12λr2expλξ¯−1λ+34r2expλξ¯−1λ2,v2=r−1expλξ¯−1λ,ξ¯=rx−λ2r3tαΓα+1+C.Set III:(59)k=−494λ2r3,λ=λ,μ=0,r=r,A0=916λ2r2,A1=−14λr2,A2=34r2,C0=−32λr−1,C1=r−1.Appending (59) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 3 as follows:(60)u3=916λ2r2−14λr2expλξ¯−1λ+34r2expλξ¯−1λ2,v3=−32λr−1+r−1expλξ¯−1λ,ξ¯=rx−494λ2r3tαΓα+1+C.Set IV:(61)k=rC02,λ=−C0−1r,μ=0,r=r,A0=−14C02,A1=−2C0r−1,A2=2r2,C0=C0,C1=2r−1.Appending (61) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 3 as follows:(62)u4=−14C02−2C0r−1expλξ¯−1λ+2r2expλξ¯−1λ2,v4=C0+2r−1expλξ¯−1λ,ξ¯=rx+rC02tαΓα+1+C.Set V:(63)k=2116rC02,λ=−34C0−1r,μ=0,r=r,A0=−14C02,A1=−78C0r−1,A2=34r2,C0=C0,C1=r−1.Appending (63) into (20), we receive a hyperbolic solution of equation (18) by utilizing Family 3 as follows:(64)u5=−14C02−78C0r−1expλξ¯−1λ+34r2expλξ¯−1λ2,v5=C0+2r−1expλξ¯−1λ,ξ¯=rx+2116rC02tαΓα+1+C.Set VI:(65)k=−r3λ2−4μ,λ=λ,μ=μ,r=r,A0=14r2λ2+4μ,A1=2λr2,A2=2r2,C0=λr−1,C1=2r−1.Appending (65) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 1 as follows:(66)u6=14r2λ2+4μ+2λr2−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−1+2r2−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−2,v6=λr−1+2r−1−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−1,ξ¯=rx−r3λ2−4μtαΓα+1+C,whereλ2−4μ>0. Plugging (65) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(67)u7=14r2λ2+4μ+2λr2−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−1+2r2−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−2,v7=λr−1+2r−1−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−1,ξ¯=rx−r3λ2−4μtαΓα+1+C,whereλ2−4μ<0.Set VII:Appending (68) into (20), we get a hyperbolic solution of equation (18) by utilizing Family 1 as follows:(68)k=λ2C12−3λC0C1−μC12+3C02C1−1,λ=λ,μ=μ,r=C1−1,A0=−12μC12−14C02,A1=−12C1λC1+C0,A2=−34C12,C0=C0,C1=C1.(69)u8=−12μC12−14C02+−12C1λC1+C0−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−1−34C12−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−2,v8=C0+C1−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−1,ξ¯=C1−1x+λ2C12−3λC0C1−μC12+3C02C1−1tαΓα+1+C,where λ2−4μ>0. Appending (68) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(70)u9=−12μC12−14C02+−12C1λC1+C0−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−1−34C12−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−2,v9=C0+C1−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−1,ξ¯=C1−1x+λ2C12−3λC0C1−μC12+3C02C1−1tαΓα+1+C,where λ2−4μ>0.


## 4.1. Periodic, Soliton, Singular, and Kink-Singular Solutions of the FCJM Equation by MEM

Balancingu‴ with v2u′ and v‴ with v2v′ in equation (19) gives the following cases:(52)Case I:N1=2,M1=0,N2=1,M1=0,Case II:N1=3,M1=1,N2=2,M1=1.Putting the values of (52) and B0=D0=1 to search for different wave solutions of the fractional CJM equations, we would like to start from ansatz as the following form:(53)uξ=A0+A1exp−θξ+A2exp−2θξ,vξ=C0+C1exp−θξ,where A0,A1,A2,C0, and C1 are unknown values. Inserting (53) into equation (56) and then collecting the coefficients in the front of the various functions containing exp−θξ are given as follows:(54)12r3C1−15rC13−36rA2C1=0,24λr3C1−15λrC13−36λrA2C1−30rC0C12−24rA1C1−24rA2C0=0,14λ2r3C1−30λrC0C12+16μr3C1−15μrC13−24λrA1C1−24λrA2C0−36μrA2C1−15rC02C1−12rA0C1−12rA1C0+2kC1=0,2λ3r3C1+16λμr3C1−15λrC02C1−30μrC0C12−12λrA0C1−12λrA1C0−24μrA1C1−24μrA2C0+2kλC1=0,2λ2μr3C1+4μ2r3C1−15μrC02C1−12μrA0C1−12μrA1C0+2kμC1=0,36r3C12+48r3A2−18rA2C12−24rA22=0,81λr3C12+108λr3A2−18λrA2C12+18r3C0C1−24λrA22+12r3A1−15rA1C12−24rA2C0C1−36rA1A2=0,57λ2r3C12+76λ2r3A2+36λr3C0C1+60μr3C12+24λr3A1−15λrA1C12−24λrA2C0C1+80μr3A2−18μrA2C12−36λrA1A2−24μrA22−12rA0C12−18rA1C0C1−6rA2C02−24rA0A2−12rA12+4kA2=0,12λ3r3C12+16λ3r3A2+21λ2r3C0C1+78λμr3C12+14λ2r3A1+104λμr3A2+24μr3C0C1−12λrA0C12−18λrA1C0C1−6λrA2C02+16μr3A1−15μrA1C12−24μrA2C0C1−24λrA0A2−12λrA12−36μrA1A2−12rA0C0C1−3rA1C02+4kλA2−12rA0A1+2kA1=0,3λ3r3C0C1+21λ2μr3C12+2λ3r3A1+28λ2μr3A2+24λμr3C0C1+24μ2r3C12+16λμr3A1+32μ2r3A2−12λrA0C0C1−3λrA1C02−12μrA0C12−18μrA1C0C1−6μrA2C02−12λrA0A1−24μrA0A2−12μrA12+2kλA1+4kμA2=0,3λ2μr3C0C1+9λμ2r3C12+2λ2μr3A1+1λμ2r3A2+6μ2r3C0C1+4μ2r3A1−12μrA0C0C1−3μrA1C02−12μrA0A1+2kμA1=0.Solving the nonlinear algebraic system of equations can be concluded as the following solutions:Set I:(55)k=3rC02,λ=0,μ=0,r=r,A0=−14C02,A1=−12C0r−1,A2=34r2,C0=C0,C1=r−1.Plugging (55) into (53), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows:(56)u1=−14C02−12C0r−11ξ¯+34r21ξ¯2,v1=C0+r−11ξ¯,ξ¯=rx+3rC02tαΓα+1+C.Set II:(57)k=−λ2r3,λ=λ,μ=0,r=r,A0=0,A1=12λr2,A2=34r2,C0=0,C1=r−1.Appending (57) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 3 as follows:(58)u2=12λr2expλξ¯−1λ+34r2expλξ¯−1λ2,v2=r−1expλξ¯−1λ,ξ¯=rx−λ2r3tαΓα+1+C.Set III:(59)k=−494λ2r3,λ=λ,μ=0,r=r,A0=916λ2r2,A1=−14λr2,A2=34r2,C0=−32λr−1,C1=r−1.Appending (59) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 3 as follows:(60)u3=916λ2r2−14λr2expλξ¯−1λ+34r2expλξ¯−1λ2,v3=−32λr−1+r−1expλξ¯−1λ,ξ¯=rx−494λ2r3tαΓα+1+C.Set IV:(61)k=rC02,λ=−C0−1r,μ=0,r=r,A0=−14C02,A1=−2C0r−1,A2=2r2,C0=C0,C1=2r−1.Appending (61) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 3 as follows:(62)u4=−14C02−2C0r−1expλξ¯−1λ+2r2expλξ¯−1λ2,v4=C0+2r−1expλξ¯−1λ,ξ¯=rx+rC02tαΓα+1+C.Set V:(63)k=2116rC02,λ=−34C0−1r,μ=0,r=r,A0=−14C02,A1=−78C0r−1,A2=34r2,C0=C0,C1=r−1.Appending (63) into (20), we receive a hyperbolic solution of equation (18) by utilizing Family 3 as follows:(64)u5=−14C02−78C0r−1expλξ¯−1λ+34r2expλξ¯−1λ2,v5=C0+2r−1expλξ¯−1λ,ξ¯=rx+2116rC02tαΓα+1+C.Set VI:(65)k=−r3λ2−4μ,λ=λ,μ=μ,r=r,A0=14r2λ2+4μ,A1=2λr2,A2=2r2,C0=λr−1,C1=2r−1.Appending (65) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 1 as follows:(66)u6=14r2λ2+4μ+2λr2−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−1+2r2−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−2,v6=λr−1+2r−1−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−1,ξ¯=rx−r3λ2−4μtαΓα+1+C,whereλ2−4μ>0. Plugging (65) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(67)u7=14r2λ2+4μ+2λr2−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−1+2r2−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−2,v7=λr−1+2r−1−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−1,ξ¯=rx−r3λ2−4μtαΓα+1+C,whereλ2−4μ<0.Set VII:Appending (68) into (20), we get a hyperbolic solution of equation (18) by utilizing Family 1 as follows:(68)k=λ2C12−3λC0C1−μC12+3C02C1−1,λ=λ,μ=μ,r=C1−1,A0=−12μC12−14C02,A1=−12C1λC1+C0,A2=−34C12,C0=C0,C1=C1.(69)u8=−12μC12−14C02+−12C1λC1+C0−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−1−34C12−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−2,v8=C0+C1−λ2μ−λ2−4μ2μtanhλ2−4μ2ξ¯−1,ξ¯=C1−1x+λ2C12−3λC0C1−μC12+3C02C1−1tαΓα+1+C,where λ2−4μ>0. Appending (68) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 2 as follows:(70)u9=−12μC12−14C02+−12C1λC1+C0−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−1−34C12−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−2,v9=C0+C1−λ2μ+4μ−λ22μtan4μ−λ22ξ¯−1,ξ¯=C1−1x+λ2C12−3λC0C1−μC12+3C02C1−1tαΓα+1+C,where λ2−4μ>0.


## 5. Interpretation Solutions with Physical Concepts

This section shows some soliton solutions of fractional form of the coupled Jaulent–Miodek equation.Remark 3.
The three-dimensional dynamic plots of the wave and corresponding density designs, contour designs, and two-dimensional designs were successfully plotted in Figures1–7 by utilizing the Maple.
In order to analyze the dynamics characteristics briefly, we would like to discuss the evolution characteristic. By taking appropriate parameters of these parameters in the following, we set(71)Eq.23:r=2,k1=1,k2=2,k3=3,α=1,Eq.25:r=2,k1=1,k2=2,k3=1,α=1,Eq.27:r=2,k1=0.3,k2=0.4,k3=0.5,α=1,Eq.29:r=2,k2=1,k3=2,α=1,Eq.24:r=2,k1=1,k2=2,k3=3,α=1,Eq.26:r=2,k1=1,k2=2,k3=1,α=1,Eq.30:r=2,k1=1,k2=2,k3=1,α=1.
We can see that the trigonometric function is as periodic solution, hyperbolic function is as soliton wave solution, exponential function is as kink solution, rational exponential function is as singular-kink solution, and rational function is as singular solution.Figure 1
Evolution of the periodic solution (23) with data r=,k1=1,k2=2,k3=3,α=1.Figure 2
Evolution of the periodic solution (25) with data r=,k1=1,k2=2,k3=1,α=1.Figure 3
Evolution of the periodic solution (27) with data r=2,k1=0.3,k2=0.4,k3=0.5,α=1.Figure 4
Evolution of the periodic solution (29) with data r=2,k2=1,k3=2,α=1.Figure 5
Evolution of the periodic solution (24) with data r=,k1=1,k2=2,k3=3,α=1.Figure 6
Evolution of the periodic solution (26) with data r=,k1=1,k2=2,k3=1,α=1.Figure 7
Evolution of the periodic solution (30) with data r=,k1=1,k2=2,k3=1,α=1.


## 6. Conclusion

We have investigated in detail the time-fractional coupled Jaulent–Miodek equation. By utilizing thetanθ/2-expansion technique and modified exp−θξ-expansion technique, we find bright, dark soliton, analytically periodic, singular, and kink-soliton solutions with the constraints under which these solutions are obtained. The tanθ/2-expansion technique and modified exp−θξ-expansion technique are helpful to obtain solutions in the form of hyperbolic and trigonometric forms which are exact and helpful in understanding the fractional forms of it. Finally, a transformation is used to draw soliton solutions of system (1) by the use of Maple software. So, this gives the efficient applications of aforementioned techniques for fractional PDEs.By utilizing Maple, the evolution phenomenon of these waves is seen in Figures1–7, respectively. Moreover, we can discover the physical structure and properties of the new periodic wave, soliton, singular-kink, and bell soliton solutions for the time FCJM equations. These new solutions can be utilized to perceive their physical morphology and density curves by computer computations, which will be extensively used in optics, DWDM systems, metasurfaces, and metamaterials, and so forth.



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