# Optical Solutions of the Date–Jimbo–Kashiwara–Miwa Equation via the Extended Direct Algebraic Method

Authors: Ghazala Akram; Naila Sajid; Muhammad Abbas; Y. S. Hamed; Khadijah M. Abualnaja
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, the solutions of2+1-dimensional nonlinear Date–Jimbo–Kashiwara–Miwa (DJKM) equation are characterized, which can be used in mathematical physics to model water waves with low surface tension and long wavelengths. The integration scheme, namely, the extended direct algebraic method, is used to extract complex trigonometric, rational and hyperbolic functions. The complex-valued solutions represent traveling waves in different structures, such as bell-, V-, and W-shaped multiwaves. The results obtained in this article are novel and more general than those contained in the literature (Wang et al., 2014, Yuan et al., 2017, Pu and Hu 2019, Singh and Gupta 2018). Furthermore, the mechanical features and dynamical characteristics of the obtained solutions are demonstrated by three-dimensional graphics.


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## Body

## 1. Introduction

Nonlinear evolution equations (NLEEs) can represent various nonlinear problems that occur in a wide range of scientific fields such as nonlinear optics, mathematical physics, superconductivity, biophysics, optical fiber, modern optics, solid state physics, fluid mechanics, fluid dynamics, plasma physics, chemical physics, and chemical kinetics. In the literature, various effective approaches have been proposed to calculate the exact solutions for NLEEs [1–3], such as the Hereman–Nuseir method [4], inverse scattering transformation [5], Painlevé technique [6], Bäcklund transformation [7], extended modified auxiliary equation mapping method [8], Darboux transformation [9], Exp-function method [10], binary-bell-polynomial scheme [11], modified Khater method [12], ansatz method [13], sine-Gordon expansion method [14], trial equation method [15, 16], extended direct algebraic method [17], and auxiliary equation method [18].In this study, the nonlinear DJKM equation [19] is investigated to construct various solitary wave solutions. In the integrable systems of KP hierarchy, the Jimbo–Miwa equation is the second equation used to explain such interesting 2+1-dimensional waves in physics. The DJKM equation can be used in mathematical physics to model water waves with low surface tension and long wavelengths with weakly nonlinear restoring forces and frequency dispersion. Firstly, Hu and Li [19] applied bilinear Bäcklund transformations and nonlinear superposition formula for nonlinear DJKM equations, and after a gap of more than two decades, Wang et al. [20] used the bell polynomials to study the integrable properties of nonlinear DJKM equations such as Lax system, Bäcklund transformations, and infinite conservation laws along with multishock wave. Yuan et al. [21] presented Grammian- and Wronskian-type solutions by the Hirota method, and other types of solution are also obtained like auxiliary variables, the bilinear Bäcklund transformation, and N-soliton. Pu and Hu [22] employed the sine-Gordon expansion method in finding the traveling wave solutions of nonlinear DJKM equations and obtained hyperbolic, trigonometric, and complex solutions. Singh and Gupta [23] used the direct method and nonlinear self-adjointness to find the Painlevé analysis, symmetric properties, and conservation laws of the nonlinear DJKM equation. Sajid and Akram [24] utilized exp−Φξ-expansion method and derived some exact traveling wave solutions including trigonometric, hyperbolic, and rational functions and W-shaped soliton of the DJKM equation. The proposed research analyzes some more new exact solutions such as bell-, V-, and W-shaped multiwave types of the nonlinear DJKM equation which are not yet found in the literature. To our utmost understanding, the DJKM equations were not analyzed using the extended direct algebraic method. Therefore, the benefits of this article included evaluating a wide range of advanced and contextual solutions to the considered wave equations by the use of the extended direct algebraic method. Furthermore, this beneficial and powerful approach can be used to investigate other NLEEs which frequently emerge in different scientific real-world applications.The novelty of this paper lies in the following: (i) complex-valued solutions and solitons are in different shapes and (ii) 3-dimensional figures are first presented by the extended direct algebraic method. The limitations of this work include that the solution methods for the construction of exact solutions to the equation involve various parameters. Such parameters show up in the final precise solution expressions and create hurdles in some physical situations. These are resolved with a careful selection of appropriate parametric values which is possible through graphical interpretation and testing of the solution expressions.The structure of this paper is organized as follows: in Section2, detailed explanation of the extended direct algebraic method has been presented. Section 3 illustrates the method to solve the 2+1-dimensional DJKM equation. In Section 3.1, the physical explanation of the solutions by mechanical features and dynamical characteristics is demonstrated. Finally, conclusion is given in Section 4.


### 1.1. Governing Model

Considering the governing model,2+1-dimensional nonlinear DJKM equation, as(1)Φxxxxy+4ΦxxyΦx+2ΦxxxΦy+6ΦxyΦxx+Φyyy−2Φxxt=0,where Φ=Φx,y,t is the real-valued function. The DJKM equation belongs to the well-known KP hierarchy [25, 26] which can be obtained from the first two bilinear equations using transformation u=2logτx. The KP hierarchy is an infinite set of nonlinear PDEs.


## 1.1. Governing Model

Considering the governing model,2+1-dimensional nonlinear DJKM equation, as(1)Φxxxxy+4ΦxxyΦx+2ΦxxxΦy+6ΦxyΦxx+Φyyy−2Φxxt=0,where Φ=Φx,y,t is the real-valued function. The DJKM equation belongs to the well-known KP hierarchy [25, 26] which can be obtained from the first two bilinear equations using transformation u=2logτx. The KP hierarchy is an infinite set of nonlinear PDEs.


## 2. Extended Direct Algebraic Method [27]

According to extended direct algebraic method, we have the following.Step 1.
Consider NLEE in three independent variablesx, y, and t of the form, as(2)PΦ,Φx,Φy,Φt,Φxx,Φxy,Φxt,Φyy,Φyt,Φtt,…=0,where Φ=Φx,y,t and P is the polynomial in Φ. Using the wave transformation(3)Φx,y,t=Uξ,ξ=ωx+μy−kt,where ω is the wave number. After applying the transformation, equation (2) can be converted into the nonlinear ODE, as(4)QU,ωU′,ωμU′,−ωkU′,ω2U″,ω2μU″,−ω2kU″,ω2μ2U″,−ω2μkU″,ω2k2U″,…=0,where prime denotes the derivatives w.r.t. ξ.Step 2.
Consider that the formal solution of equation (4) has a form, as follows:(5)Uξ=∑j=0NbjQjξ,bN≠0,where b0,b1,…,bN are constants and Qξ satisfies the auxiliary equation, as(6)Q′ξ=LnAα+βQξ+σQ2ξ,A≠0,1,where σ, α, and β are constants. The solutions of equation (6) are given in the following.Family 1.
Ifβ2−4ασ<0 and σ≠0, then the solutions are given as(7)Q1ξ=−β2σ+−β2−4ασ2σtanA−β2−4ασ2ξ.Q2ξ=−β2σ−−β2−4ασ2σcotA−β2−4ασ2ξ,Q3ξ=−β2σ+−β2−4ασ2σtanA−β2−4ασξ±pqsecA−β2−4ασξ,Q4ξ−β2σ−−β2−4ασ2σcotA−β2−4ασξ∓pqcscA−β2−4ασξ,Q5ξ=−β2σ+−β2−4ασ4σtanA−β2−4ασ4ξ−cotA−β2−4ασ4ξ.Family 2.
Ifβ2−4ασ>0 and σ≠0, then the solutions are given as(8)Q6ξ=−β2σ−β2−4ασ2σtanhAβ2−4ασ2ξ,Q7ξ=−β2σ−β2−4ασ2σcothAβ2−4ασ2ξ,Q8ξ=−β2σ−β2−4ασ2σtanhAβ2−4ασξ∓ipqsechAβ2−4ασξ,Q9ξ=−β2σ−β2−4ασ2σcothAβ2−4ασξ∓pqcschAβ2−4ασξ,Q10ξ=−β2σ−β2−4ασ4σtanhAβ2−4ασ4ξ+cothAβ2−4ασ4ξ.Family 3.
Ifβ=0, and ασ>0, then the solutions are given as(9)Q11ξ=ασtanAασξ,Q12ξ=−ασcotAασξ,Q13ξ=ασtanA2ασξ±pqsecA2ασξ,Q14ξ=−ασcotA2ασξ∓pqcscA2ασξ,Q15ξ=12ασtanAασ2ξ−cotAασ2ξ.Family 4.
Ifβ=0 and ασ<0, then the solutions are given as(10)Q16ξ=−−ασtanhA−ασξ,Q17ξ=−−ασcothA−ασξ,Q18ξ=−−ασtanhA2−ασξ∓ipqsechA2−ασξ,Q19ξ=−−ασcothA2−ασξ∓pqcschA2−ασξ,Q20ξ−12−ασtanhA−ασ2ξ+cothA−ασ2ξ.Family 5.
Ifβ=0 and σ=α, then the solutions are given as(11)Q21ξ=tanAαξ,Q22ξ=−cotAαξ,Q23ξ=tanA2αξ±pqsecA2αξ,Q24ξ=−cotA2αξ±pqcscA2αξ,Q25ξ=12tanAα2ξ−cotAα2ξ.Family 6.
: Ifσ=−α and β=0, then the solutions are given as(12)Q26ξ=−tanhAαξ,Q27ξ=−cothAαξ,Q28ξ=−tanhA2αξ±ipqsechA2αξ,Q29ξ=−cothA2αξ±pqcschA2αξ,Q30ξ=−12tanhAα2ξ+cothAα2ξ.Family 7.
Ifβ2=4ασ, then the solution is given as(13)Q31ξ=−2αβξLnA+2β2ξLnA.Family 8.
Ifβ=l, σ=0, and α=mlm≠0, then the solution is given as(14)Q32ξ=Alξ−m.Family 9.
Ifβ=0=σ, then the solution is given as(15)Q33ξ=αξLnA.Family 10.
Ifβ=α=0, then the solution is given as(16)Q34ξ=−1αξLnA.Family 11.
Ifβ≠0 and α=0, then the solutions are given as(17)Q35ξ=−pβσcoshAβξ−sinhAβξ+p,Q36ξ=−βsinhAβξ+coshAβξσsinhAβξ+coshAβξ+q.Family 12.
Ifβ=l, α=0, and σ=mlm≠0, then the solution is given as(18)Q37ξ=pAlξq−mpAlξ.Remark 1.
The generalized triangular functions and hyperbolic functions [28] are defined as follows:(19)sinhAξ=pAξ−qA−ξ2,coshAξ=pAξ+qA−ξ2,tanhAξ=pAξ−qA−ξpAξ+qA−ξ,cothAξ=pAξ+qA−ξpAξ−qA−ξ,sechAξ=2pAξ+qA−ξ,cschAξ=2pAξ−qA−ξ,sinAξ=pAiξ−qA−iξ2i,cosAξ=pAiξ+qA−iξ2,tanAξ=−ipAiξ−qA−iξpAξ+qA−ξ,cotAξ=ipAiξ+qA−iξpAiξ−qA−iξ,secAξ=2pAiξ+qA−iξ,cscAξ=2ipAiξ−qA−iξ,where p,q>0 and ξ is an independent variable.Step 3.
Using homogeneous balancing principle in equation (4), the value of N can be determined. Substituting equation (6) along with equation (5) into equation (4), collecting the coefficients of each power Qjξj=0,1,2,…, and then setting each coefficient to zero give a system of equations.Step 4.
Unknowns can be found by calculating the system of equations. Putting the unknowns in equation (6), the required solutions of equation (2) are obtained.


## 3. Application to the DJKM Equation

The extended direct algebraic scheme is presented to obtain the optical solitons and other solutions to equation (1). After utilizing the transformation Φx,y,t=Vξ, where ξ=ωx+μy−kt, to equation (1), we obtain nonlinear ODE as follows:(20)μω2V4+6μωV′V″+μ3+2kV″=0.SettingU=V′, we obtain(21)μω2U‴+6μωUU′+μ3+2kU′=0.BalancingU‴ with UU′ in equation (21) gives N=2. Thus, the solution can be written as(22)Uξ=b0+b1Qξ+b2Q2ξ,where b0, b1, and b2 are constants to be determined. Substituting equations (22) into (21), collecting all terms with the same power of Qξi (i = 0, 1, 2, 3, 4, 5), and equating the coefficients of each polynomial to zero will yield a set of algebraic equations for b0, b1, b2, and ω as follows:(23)2kb1LnAα+2μω2b1LnA3σα2+6μωb0b1LnAα+μω2b1LnA3β2α+6μω2b2LnA3βα2+μ3b1LnAα=0,16μω2b2LnA3σα2+4kb2LnAα+μ3b1LnAβ+2kb1LnAβ+8μω2b1LnA3βσα+μω2b1LnA3β3+2μ3b2LnAα+6μωb0b1LnAβ+12μωb0b2LnAα+6μωb12LnAα+14μω2b2LnA3αβ2=0,μ3b1LnAσ+6μωb0b1LnAσ+18μωb1b2LnAα+8μω2b1LnA3σ2α+52μω2b2LnA3αβσ+4kb2LnAβ+2μ3b2LnAβ+2kb1LnAσ+7μω2b1LnA3β2σ+8μω2b2LnA3β3+6μωb12LnAβ+12μωb0b2LnAβ=0,38μω2b2LnA3β2σ+18μωb1b2LnAβ+4kb2LnAσ+2μ3b2LnAσ+12μω2b1LnA3βσ2+40μω2b2LnA3ασ2+12μωb22LnAα+6μωb12LnAσ+12μωb0b2LnAσ=0,18μωb1b2LnAσ+54μω2b2LnA3βσ2+6μω2b1LnA3σ3+12μωb22LnAβ=0,24μω2b2LnA3σ3+12μωb22LnAσ=0.Solving system (23) for b0, b1, b2, and ω gives(24)b0=−16μω2LnA2β2+8μω2LnA2σα+2k+μ3/μω,b1=−2ωLnA2βσ,b2=−2ωLnA2σ2,ω=ω.Five families of traveling wave solutions of the DJKM equation can be obtained, as shown in the following.Family 13.
Whenβ2−4ασ<0 and σ≠0, the dark, combined dark-bright, singular, combined dark-singular, and combined singular solutions are obtained, as follows:(25)Φ1=−16ξ−5μωLnA2β2+2k+8μωLnA2σα+μ3/μω+LnA2tanhA12γξγ+LnA2ξ−β2+2ασ,Φ2=−16ξ−5μωLnA2β2+2k+8μωLnA2σα+μ3/μω+LnA2cothA12γξγ+2LnA2−β2+2ασarctanhAcothA1/2γξγ,Φ3=16ξ−μωLnA2β2−2k+4μωLnA2σα−μ3/μω+12LnA2tanhAγξγ1+pq−iLnA2pqγcoshAγξ,Φ4=16ξ−μωLnA2β2−2k+4μωLnA2σα−μ3/μω+12LnA2γcothAγξ1+pq−LnA2pqγsinhAγξ.Φ5=16ξ−μωLnA2β2−2k+4μωLnA2σα−μ3/μω+12LnA2tanhA1/4γξγ+12LnA2cothA1/4γξγ.Family 14.
Whenβ2−4ασ>0 and σ≠0, the singular, dark, and combined dark-singular solutions are obtained, as follows:(26)Φ6=−16ξ−5μωLnA2β2+2k+8μωLnA2σα+μ3/μω+γLnA2tanhA12γξ−LnA2−β2+2ασLntanhA1/2γξ−1−LntanhA1/2γξ+1γ,Φ7=−16ξ−5μωLnA2β2+2k+8μωLnA2σα+μ3/μω+γLnA2cothA12γξ−LnA2−β2+2ασLncothA1/2γξ−1−LncothA1/2γξ+1γ,Φ8=16ξ−μωLnA2β2−2k+4μωLnA2σα−μ3/μω−LnA2ipqγcoshAγξ−LnA2βipqarctanAsinhAγξ−2arctanAeγξ+12LnA2tanhAγξγ1+pq+LnA2ipqγsinhAγξ2coshAγξ,Φ9=16ξ−μωLnA2β2−2k+4μωLnA2σα−μ3/μω+LnA2pqγsinhAγξ−LnA2βpqLntanhA12γξ+2arctanhAeγξ+12LnA2cothAγξ1+pqγ−LnA2pqγcoshAγξ2sinhAγξ,Φ10=16ξ−μωLnA2β2−2k+4μωLnA2σα−μ3/μω+12LnA2tanhA1/4γξγ+12LnA2cothA1/4γξγ.Family 15.
Whenασ>0 and β=0, the periodic-singular solutions are obtained as(27)Φ11=−16ξ2k+8μωLnA2σα+μ3/μω−2LnA2ασtanAασξ−ασξ,Φ12=−16ξ2k+8μωLnA2σα+μ3/μω+LnA2ασ2cotAασξ−π+2ασξ,Φ13=16ξ−2k+4μωLnA2σα−μ3/μω−2LnA2pqασcosA2ασξ−LnA2ασtanA2ασξ1+pq,Φ14=16ξ−2k+4μωLnA2σα−μ3/μω−2LnA2pqασsinA2ασξ+LnA2ασcotA2ασξ1+pq,Φ15=16ξ−2k+4μωLnA2σα−μ3/μω−LnA2ασtanA12ασξ−cotA12ασξ.Family 16.
Whenασ<0 and β=0, the singular, dark, combined dark-bright, combined dark-singular, and combined singular solutions are obtained, as follows:(28)Φ16=−16ξ2k+8μωLnA2σα+μ3/μω+LnA2−ασLntanhA−ασξ−1tanhA−ασξ+1+2LnA2−ασtanhA−ασξ,Φ17=−16ξ2k+8μωLnA2σα+μ3/μω+LnA2−ασLncothA−ασξ−1cothA−ασξ+1+2LnA2−ασcothA−ασξ,Φ18=−16ξ2k−4μωLnA2σα+μ3/μω+LnA2−ασtanhA2−ασξ1+pq+2iLnA2−pqσαcoshA2−ασξ,Φ19=−16ξ2k−4μωLnA2σα+μ3/μω+LnA2−ασcothA2−ασξ1+pq+2LnA2−pqασsinhA2−ασξ,Φ20=−16ξ2k−4μωLnA2σα+μ3/μω+LnA2−ασtanhA12−ασξ+LnA2−ασcothA12−ασξ.Family 17.
Whenβ=0 and σ=α, the periodic-singular solutions are obtained as follows:(29)Φ21=−16ξ2k+8LnA2α2μω+μ3/μω−2LnA2αtanAαξ−αξ,Φ22=−16ξ2k+8LnA2α2μω+μ3/μω+2LnA2αcotAαξ−12π+αξ,Φ23=−16ξ2k−4LnA2α2μω+μ3/μω−LnA2αtanA2αξ1+pq−2LnA2αpqcosA2αξ,Φ24=−16ξ2k−4LnA2α2μω+μ3/μω+LnA2αcotA2αξ1+pq−2LnA2αpqsinA2αξ,Φ25=−16ξ2k−4LnA2α2μω+μ3/μω−LnA2αtanA12αξ−cotA12αξ.Family 18.
Whenβ=0 and σ=−α, the singular dark, combined dark-bright, combined dark-singular, and combined singular solutions are obtained, as follows:(30)Φ26=−16ξ2k−8LnA2α2μω+μ3/μω+LnA2α2tanhAαξ+LntanhAαξ−1tanhAαξ+1,Φ27=−16ξ2k−8LnA2α2μω+μ3/μω+LnA2α2cothAαξ+LncothAαξ−1cothAαξ+1,Φ28=−16ξ2k+4LnA2α2μω+μ3/μω+LnA2αtanhA2αξ1+pq−2LnA2αipqcoshA2αξ,Φ29=−16ξ2k+4LnA2α2μω+μ3/μω+LnA2αcothA2αξ1+pq−2LnA2αpqsinhA2αξ,Φ30=−16ξ2k+4μωLnA2α2+μ3/μω+LnA2αtanhA12αξ+cothA12αξ.Family 19.
Whenβ2=4ασ, the rational solution is obtained, as(31)Φ31=−16x+μy−kt2k+μ3/μω+2ωx+μy−kt.Family 20.
Whenβ=l, σ=0, and α=mlm≠0, the rational solution is obtained, as follows:(32)Φ32=−16x+μy−ktLnA2l2μω+2k+μ3/μω.Family 21.
Whenβ=σ=0, the rational solution is obtained as(33)Φ33=−162k+μ3x+μy−kt/μω.Family 22.
Whenβ=α=0, the rational solution is obtained as(34)Φ34=−162ξ2k+μ3ξ2−12μω/μωξ.Family 23.
Whenα=0 and β≠0, the singular and dark-singular combo solitons solutions are obtained, as follows:(35)Φ35=−16ξμωLnA2β2+2k+μ3/μω−4LnA2pβp−1tanhA1/2βξp−tanhA1/2βξ+1+p,Φ36=−16ξμωLnA2β2+2k+μ3/μω−4LnA2βp−1+p−tanhA1/2βξ+tanhA1/2βξp−p−1+2LnA2βLnsinhAβξ+coshAβξ+p+2lnA2βLntanhA12βξ−1−2LnA2βLn−tanhA12βξ+tanhA1/2βξp−p−1.Family 24.
Whenβ=l, α=0, and σ=mlm≠0, the rational solution is obtained, as follows:(36)Φ37=−16ξLnA2l2μω+2k+μ3/μω+2LnAlLnq−mpAlξ−q+mpAlξ+2LnAlq−q+mpAlξ,where γ=β2−4ασ and ξ=ωx+μy−kt.


### 3.1. Physical Description of Solutions

A solitary wave is a restricted gravity wave that maintains its finite amplitude and propagates with consistent speed and constant shape. Solitons are the solitary wave with an elastic dispersive property. Solitons are the consequence of a delicate balance between nonlinear and dispersive impact in the medium. If the solution is in the form of tangent, secant, cotangent, and cosecant hyperbolic, then the solution is called dark, bright, singular, singular-soliton solutions, respectively. The solution of hyperbolic tangent plus hyperbolic secant form is called combined dark-bright soliton solution. The solution of hyperbolic cotangent plus hyperbolic cosecant form is called combined singular soliton solution, and the solution of hyperbolic tangent plus hyperbolic cotangent form is called dark-singular combo soliton solution.Figure1 demonstrates the solutions of Φ1,Φ2,Φ3,Φ4, and Φ5 for the particular parameters α=1.5, β=2, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.75, and A=2.5. The complex plane represents multiwaves having positive or negative jumps time to time. The modulus of this solution represents periodic long waves with positive amplitudes.Figure 1
3D profile of Family13 with α=1.5, β=2, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.75, and A=2.5. (a) Φ1. (b) Φ2. (c) Φ3. (d) Φ4. (e) Φ5.
(a)(b)(c)(d)(e)The solutions ofΦ6,Φ7,Φ8,Φ9, and Φ10 are depicted in Figure 2 for some particular choice of the parameters such as α=1.5, β=2, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.5, and A=2.5. In Figure 3, the solutions of Φ11,Φ12,Φ13,Φ14, and Φ15 are plotted in the finite domain for the parameters α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.75, and A=2.5. Figure 4 demonstrates the solutions of Φ16,Φ17,Φ18,Φ19, and Φ20 for the particular parameters α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=−0.75, andA=2.5. Figure 5 demonstrates the solutions of Φ21,Φ22,Φ23,Φ24, and Φ25 for the particular parameters α=1.5=σ, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, and A=2.5. Figure 6 demonstrates the solutions of Φ26,Φ27,Φ28,Φ29, and Φ30 for the particular parameters α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=−1.5, and A=2.5. Figure 7 demonstrates the 3D graphics of the solutions of Φ31,Φ32, and Φ33 under the particular values α=1.5, k=y=1, ω=0.5, μ=0.25, and A=2.5 for Φ31 with β=1 and σ=0.5, Φ32 with β=1, σ=0, l=1, and m=0.5, and Φ33 with β=0=σ. Figure 8 demonstrates the 3D graphics of the solutions Φ34,Φ35,Φ36, and Φ37 under the particular values α=0, k=y=1, ω=0.5, σ=0.5, μ=0.25, and A=2.5 for Φ34 with β=0, Φ35 and Φ36 with β=0.5, and Φ37 with β=0.5=l, σ=1, and m=2. Families 13, 14, 15 and 16 represent the singular, dark, combined dark-bright, combined dark-singular, combined singular, and solitary wave solutions. Families 15 and 17 represent the exact periodic traveling wave solutions, whereas the Families 19, 20, 21, 22, 23, and 24 show the rational solutions.Figure 2
3D profile ofFamily 14 with α=1.5, β=2, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.5, and A=2.5. (a) Φ6. (b) Φ7. (c) Φ8. (d) Φ9. (e) Φ10.
(a)(b)(c)(d)(e)Figure 3
3D profile ofFamily 15 with α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.75, and A=2.5. (a) Φ11. (b) Φ12. (c) Φ13. (d) Φ14. (e) Φ15.
(a)(b)(c)(d)(e)Figure 4
3D profile ofFamily 16 with α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=−0.75, and A=2.5. (a) Φ16. (b) Φ17. (c) Φ18. (d) Φ19. (e) Φ20.
(a)(b)(c)(d)(e)Figure 5
3D profile ofFamily 17 with α=1.5=σ, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, and A=2.5. (a) Φ21. (b) Φ22. (c) Φ23. (d) Φ24. (e) Φ25.
(a)(b)(c)(d)(e)Figure 6
3D profile ofFamily 18 with α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=−1.5, and A=2.5. (a) Φ26. (b) Φ27. (c) Φ28. (d) Φ29. (e) Φ30.
(a)(b)(c)(d)(e)Figure 7
3D graphics ofFamilies 19, 20, and 21 under the values of α=1.5, k=y=1, ω=0.5, μ=0.25, and A=2.5 for (a) Φ31 with β=1 and σ=0.5, (b) Φ32 with β=1σ=0, l=1, and m=0.5, and (c) Φ33 with β=0=σ.
(a)(b)(c)Figure 8
3D graphics ofFamilies 22, 23, and 24 under the values α=0, k=y=1, ω=0.5, σ=0.5, μ=0.25, and A=2.5 for (a) Φ34 with β=0, (b) Φ35, (c) Φ36 with β=0.5, and (d) Φ37 with β=0.5=l, σ=1, and m=2.
(a)(b)(c)(d)


## 3.1. Physical Description of Solutions

A solitary wave is a restricted gravity wave that maintains its finite amplitude and propagates with consistent speed and constant shape. Solitons are the solitary wave with an elastic dispersive property. Solitons are the consequence of a delicate balance between nonlinear and dispersive impact in the medium. If the solution is in the form of tangent, secant, cotangent, and cosecant hyperbolic, then the solution is called dark, bright, singular, singular-soliton solutions, respectively. The solution of hyperbolic tangent plus hyperbolic secant form is called combined dark-bright soliton solution. The solution of hyperbolic cotangent plus hyperbolic cosecant form is called combined singular soliton solution, and the solution of hyperbolic tangent plus hyperbolic cotangent form is called dark-singular combo soliton solution.Figure1 demonstrates the solutions of Φ1,Φ2,Φ3,Φ4, and Φ5 for the particular parameters α=1.5, β=2, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.75, and A=2.5. The complex plane represents multiwaves having positive or negative jumps time to time. The modulus of this solution represents periodic long waves with positive amplitudes.Figure 1
3D profile of Family13 with α=1.5, β=2, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.75, and A=2.5. (a) Φ1. (b) Φ2. (c) Φ3. (d) Φ4. (e) Φ5.
(a)(b)(c)(d)(e)The solutions ofΦ6,Φ7,Φ8,Φ9, and Φ10 are depicted in Figure 2 for some particular choice of the parameters such as α=1.5, β=2, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.5, and A=2.5. In Figure 3, the solutions of Φ11,Φ12,Φ13,Φ14, and Φ15 are plotted in the finite domain for the parameters α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.75, and A=2.5. Figure 4 demonstrates the solutions of Φ16,Φ17,Φ18,Φ19, and Φ20 for the particular parameters α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=−0.75, andA=2.5. Figure 5 demonstrates the solutions of Φ21,Φ22,Φ23,Φ24, and Φ25 for the particular parameters α=1.5=σ, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, and A=2.5. Figure 6 demonstrates the solutions of Φ26,Φ27,Φ28,Φ29, and Φ30 for the particular parameters α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=−1.5, and A=2.5. Figure 7 demonstrates the 3D graphics of the solutions of Φ31,Φ32, and Φ33 under the particular values α=1.5, k=y=1, ω=0.5, μ=0.25, and A=2.5 for Φ31 with β=1 and σ=0.5, Φ32 with β=1, σ=0, l=1, and m=0.5, and Φ33 with β=0=σ. Figure 8 demonstrates the 3D graphics of the solutions Φ34,Φ35,Φ36, and Φ37 under the particular values α=0, k=y=1, ω=0.5, σ=0.5, μ=0.25, and A=2.5 for Φ34 with β=0, Φ35 and Φ36 with β=0.5, and Φ37 with β=0.5=l, σ=1, and m=2. Families 13, 14, 15 and 16 represent the singular, dark, combined dark-bright, combined dark-singular, combined singular, and solitary wave solutions. Families 15 and 17 represent the exact periodic traveling wave solutions, whereas the Families 19, 20, 21, 22, 23, and 24 show the rational solutions.Figure 2
3D profile ofFamily 14 with α=1.5, β=2, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.5, and A=2.5. (a) Φ6. (b) Φ7. (c) Φ8. (d) Φ9. (e) Φ10.
(a)(b)(c)(d)(e)Figure 3
3D profile ofFamily 15 with α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=0.75, and A=2.5. (a) Φ11. (b) Φ12. (c) Φ13. (d) Φ14. (e) Φ15.
(a)(b)(c)(d)(e)Figure 4
3D profile ofFamily 16 with α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=−0.75, and A=2.5. (a) Φ16. (b) Φ17. (c) Φ18. (d) Φ19. (e) Φ20.
(a)(b)(c)(d)(e)Figure 5
3D profile ofFamily 17 with α=1.5=σ, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, and A=2.5. (a) Φ21. (b) Φ22. (c) Φ23. (d) Φ24. (e) Φ25.
(a)(b)(c)(d)(e)Figure 6
3D profile ofFamily 18 with α=1.5, β=0, k=y=1, ω=0.5, p=q=0.9, μ=0.25, σ=−1.5, and A=2.5. (a) Φ26. (b) Φ27. (c) Φ28. (d) Φ29. (e) Φ30.
(a)(b)(c)(d)(e)Figure 7
3D graphics ofFamilies 19, 20, and 21 under the values of α=1.5, k=y=1, ω=0.5, μ=0.25, and A=2.5 for (a) Φ31 with β=1 and σ=0.5, (b) Φ32 with β=1σ=0, l=1, and m=0.5, and (c) Φ33 with β=0=σ.
(a)(b)(c)Figure 8
3D graphics ofFamilies 22, 23, and 24 under the values α=0, k=y=1, ω=0.5, σ=0.5, μ=0.25, and A=2.5 for (a) Φ34 with β=0, (b) Φ35, (c) Φ36 with β=0.5, and (d) Φ37 with β=0.5=l, σ=1, and m=2.
(a)(b)(c)(d)


## 4. Conclusion

To investigate the2+1-dimensional DJKM equation for exact solutions, the extended direct algebraic method is applied. By the extended direct algebraic method, many new exact solitary wave solutions are constructed including the singular, dark, combined dark-bright, periodic-singular, combined dark-singular, combined singular, and rational kinds. Such observations show that the suggested approaches are highly helpful and efficient in solving the NEEs. The complex-valued solutions represent traveling waves in different structures. Even though some are of the well-known forms such as bell-, V-, and W-shaped multiwaves, the shape of some others are completely different from them which were not found in the previous literature. The results of this investigation can be useful in illustrating the physical meaning of the studied model by 3D graphics. The performance of the method is reliable and a computerized mathematical approach to conduct other NLEEs in the field of mathematical physics and applied sciences.



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*Source: 5591016-2021-07-26.xml*