# N-Lump to the (2+1)-Dimensional Variable-Coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada Equation

Authors: Junjie Li; Jalil Manafian; Aditya Wardhana; Ali J. Othman; Ismail Husein; Mohaimen Al-Thamir; Mostafa Abotaleb
Journal: Complexity
 (2022)

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## Abstract

In this research, the (2 + 1)-dimensional (D) variable-coefficient (VC) Caudrey–Dodd–Gibbon–Kotera–Sawada model used in soliton hypothesis and implemented by operating the Hirota bilinear scheme is studied. A few modern exact analytical outcomes containing interaction between a lump-two kink soliton, interaction between two-lump, the interaction between two-lump soliton, lump-periodic, and lump-three kink outcomes for the (2 + 1)-D VC Caudrey–Dodd–Gibbon–Kotera–Sawada equation by Maple Symbolic packages are obtained. By employing Hirota’s bilinear technique, the extended soliton solutions according to bilinear frame equation are received. For this model, the contemplated model can be got by multi-D binary Bell polynomials (bBPs). In addition, the analytical analysis of the high-order soliton outcomes to present the discipline of outcomes. The effect of the free parameters on the behavior of acquired figures of a few obtained solutions for the nonlinear rational exact cases was also discussed. The above technique could also be employed to get exact solutions for other nonlinear models in physics, applied mathematics, and engineering.


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## Body

## 1. Introduction

Nonlinear mathematical models (NLMs) are broadly implemented to express a lot of significant phenomena and nonlinear dynamic applications in applied mathematics, mathematical physics, engineering, electromagnetic, communications, acoustics, genetic algorithms, electrochemistry, transport systems, material science, finance, image processing, stochastic dynamical systems, plasma physics, nonlinear control theory, and so many. In accordance to determine the exact answers of NLMs, countless influential and well-organized schemes have been presented and industrialized, such as the modifiedtanφ/2 -expansion technique [1], the homotopy perturbation technique [2], Lie symmetry analysis technique [3], the csch-function technique [4], the Bäcklund transformation technique [5], the sine-Gordon expansion technique [6], the differential transform method [7], an improved differential transform method [8], the multiple Exp-function method [9, 10], the reduced chi-square and root mean square error [11], Hirota’s bilinear method [12, 13], a partial least squares structural equation modeling [14], solving absolute value problems [15], the inverse scattering transformation method [16], an optical flow approach [17], the multidimensional binary Bell polynomials [18], perturbed conformable Boussinesq-like equation [19], and so on. A generalized (2 + 1)-D BKP equation with more second-order nonlinear terms was investigated, and divergent kinds of lumpkink outputs and lump soliton outputs were established by utilizing a combination of Hirota’s bilinear scheme and Maple symbolic estimations [20]. The resonant multisoliton outputs of a combined pKP-BKP equation got by making out the linear superposition principle and 1-breather, 2-breather, 3-breather, and hybrid outputs between breather and soliton solutions were reported [21] by Feng and Bilige. Li et al. [22] studied the generalized Burgers equation with variable coefficients by contemplating the Hirota bilinear scheme, and they retrieve plenty of different exact analytical outputs, including 2-lump-type wave outputs, interaction between 2-lump and one kink wave outputs, and interaction between two lump and two kink wave outputs. The lump waves and collision of lump with periodic waves by way of the Hirota technique for the (2 + 1)-D Bogoyavlenskii’s breaking soliton model was contemplated in [23]. Also, by means of the bilinear Bell polynomials scheme, Wang and Chen [24] investigated the higher-order Sawada–Kotera-type equation and the higher-order Lax-type equation in fluids. The bilinear technique was operated to put together the multiple lump outputs of the (3 + 1)-D potential Yu–Toda–Sasa–Fukuyama equation, and also, the solutions including 1-lump solutions, 3-lump outputs, and 6-lump outputs were explicitly shown by Cheng et al. [25]. A validated computational fluid dynamic model was executed to analyze the effects of pin angle on the thermomechanical action during the friction stir welding process [26, 27]. In the study by Manafian [28], the new solitary wave solutions for the (3 + 1)-dimensional extended Jimbo–Miwa equations were inquired into. Besides, Sadat et al. [29] got lump-type solutions and their interaction solutions with one or two-stripe solutions through the Hirota bilinear scheme and the Cole–Hopf transformation for a generalized (3 + 1) shallow water-like equation. The lump and lump strip outputs of the (3 + 1)-D generalized Kadomtsev–Petviashvili equation were found by way of the generalized bilinear operator with the parameter p=3 [30]. In the study by Wu et al. [31], by using an extended Bäcklund transformation, the symmetry breaking soliton, breather, and lump outputs of a nonlocal KP were obtained. A (3 + 1)-D variable-coefficient generalized shallow water wave model in oceanography and atmospheric science was explored, and lump solutions were demonstrated according to Hirota’s bilinear frame, and some characteristics of lump outputs, containing the bright lump wave and the bright-dark lump wave, were obtained by Liu and co-workers [32].Consider the (2 + 1)-D variable coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada (VC CDGKS) model [33, 34] as follows:(1)ut+ϕ1tuxxxxx+ϕ2tuxuxx+ϕ3tuuxxx+ϕ4tu2ux+ϕ5tuxxy,(2)ϕ6t∂x−1uyy+ϕ7tux∂x−1uy+ϕ7tuuy+ϕ9tu=0,where u=ux,y,t is an unfamiliar function and ϕjtj=1:9 are all optional values. Based upon the created simplifications in [33], we can find the following relations:(3)ϕ2t=ϕ3t=15ϕ1tc0exp∫0tϕ9tdt,ϕ4t=45ϕ1tc02exp2∫0tϕ9tdt,ϕ5t=5c1ϕ1t,(4)ϕ6t=−5c12ϕ1t,ϕ7t=ϕ8t=15c1ϕ1tc0exp∫0tϕ9tdt,where c0 and c1 are the arbitrary values.Hirota bilinear differential equations have attracted special interest during the past one decade owing to their ability to model many phenomena in different research areas and engineering applications. Some physical applications in sciences and engineering can be solved by way of Hirota operators, which are extremely useful for a wide range of nonlinear physical problems. The Hirota bilinear scheme is used to represent these equations, and solving nonlinear partial differential equations is essential [35–37]. Nonlinear equations are used to describe the world’s most important processes. Finding the exact solution to nonlinear partial differential equations is still a major problem in physics and applied mathematics, necessitating the use of various techniques to obtain at innovative approximate or exact solutions. One of most important analytical techniques that has been used to solve nonlinear differential equations is Hirota transformation scheme. Lately, many new approaches to nonlinear partial differential equations have been proposed, and a few of these methods are as follows: two-D lumps in nonlinear dispersive systems [38], rogue waves in the Davey–Stewartson I equation [39], the nonlocal Boussinesq-KdV type systems [40], asymmetric solitons in three fifth-order systems via velocity resonance [41], fusion and fission phenomena for the (2 + 1)-dimensional fifth-order KdV system [42], and superposition between lump waves and other waves for Kadomtsev–Petviashvili I equation [43].Many researchers used various methods to study the nonlinear models, such as effectiveness of understanding the design model in science teaching [44], guided discovery learning and problem-based learning models [45], study a panel data analysis, pooled mean group, and cross-sectionally augmented autoregressive distributed lag estimators [46]. Through these methods, some exact solutions of the nonlinear models of equations were obtained. In order to really understand these physical phenomena, it is of immense importance to solve nonlinear partial differential equations (NLPDEs), which govern these aforementioned phenomena. However, there is no general systematic theory that can be applied to NLPDEs, so that their analytic solutions can be obtained. Nevertheless, in recent times, scientists have developed effective techniques to obtain viable analytical solutions to NLPDEs. In our current effort, we employ the bBPs scheme for constructing exact outcomes of aforementioned equation. We can therefore easily convert NLPDE equation into nonlinear ODE via appropriate conversion (Hirota bilinear scheme), with the purpose of everybody acquainted with differential calculus lacking any trouble.In order to make up for this regret, the explanations of multidimensional binary Bell polynomials described in Section2 are perfected to find the exact solutions. Also, on the basis of bilinear frame equation, the (2 + 1)-D VC CDGKS equation is established in Section 3. In Sections 4 and 5, we conclude the multilump outputs, interaction between two-lump, interaction between two-lump soliton, lump-periodic, and lump-three kink outputs along with depicting 3D, density and 2D designs for the VC CDGKS equation. The conclusion is given in Section 6.


## 2. Multidimensional Binary Bell Polynomials

According to [18, 47], consider θ=θx1,x2,…,xn be a C∞ function with multivariables, and the polynomials,(5)Υn1x1,…,njxjθ≡Υn1,…,njθs1x1,…,sjxj=e−θ∂x1n1…∂xjnjeθ,n!r!n−r!,are named the multi-D Bell polynomials (BPs).(6)θs1x1,…,sjxj=∂x1s1…∂xjsjθ,θ0xi≡θ,s1=0,…,n1;…;sj=0,…,nj.We get(7)Υ1ξ=θx,Υ2ξ=θ2x+θx2,Υ3ξ=θ3x+3θxθ2x+θx3,…,θ=θx,t,Υx,tθ=θx,t+θxθt,Υ2x,tθ=θ2x,t+θ2xθt+2θx,tθx+θx2θt,…,.The multi-D binary BPs can be written as(8)∑n1x1,…,njxjτ,σ=Υn1,…,njθ|θs1x1,…,sjxj=τs1x1,…,sjxj,s1+s2+⋯+sj,is odd,σs1x1,…,sjxj,s1+s2+⋯+sj,is even.We have the following conditions:(9)∑xτ=τx,∑2xτ,σ=σ2x+τx2,∑x,tτ,σ=σx,t+τxτt,…,.Proposition 1.
Letτ=lnK1/K2,σ=lnK1/K2; then, the relations between binary Bell polynomials and Hirota D-operator can be stated as follows:(10)∑n1x1,…,njxjτ,σ|τ=lnK1/K2,σ=lnK1/K2=K1K2−1Dx1n1…DxjnjK1K2,with Hirota operator(11)∏i=1jDxinig.η=∏i=1j∂∂xi−∂∂xi′niK1x1,…,xjK2x1′,…,xj′|x1=x1′,…,xj=xj′.Proposition 2.
TakeΞγ=∑iδiPs1x1,…,sjxj=0 and τ=lnK1/K2,σ=lnK1/K2, and we have(12)∑iδ1iΥn1x1,…,njxjτ,σ=0,∑iδ1iΥs1x1,…,sjxjτ,σ=0,which need to satisfy(13)ℜγ′,γ=ℜγ′−ℜγ=ℜσ+τ−ℜσ−τ=0.
The generalized Bell polynomialsΥn1x1,…,njxjθ is(14)K1K2−1Dx1n1…DxjnjK1K2=∑n1x1,…,njxjτ,σ|τ=lnK1K2,σ=lnK1K2=∑n1x1,…,njxjτ,τ+γ|τ=lnK1/K2,γ=lnK1K2,=∑k1n1…∑kjnj∏i=1jnikiPk1x1,…,kjxjγΥn1−k1x1,…,nj−kjxjτ.The Cole–Hopf transformation is(15)Υk1x1,…,kjxjτ=lnφ=φn1x1,…,njxjφ,(16)K1K2−1Dx1n1…DxjnjΘΔ|Δ=expγ/2,K1/K2=φ,(17)=φ−1∑k1n1…∑kjnj∏d=1jndkdPk1x1,…,kdxdγφn1−k1x1,…,nd−kdxd,with(18)Υtτ=φtφ,Υ2xτ,σ=γ2x+φ2xφ,Υ2x,yτ,σ=γ2xφyφ+2γx,yφxφ+φ2x,yφ.


## 3. Bilinear Form Equation to the (2 + 1)-D VC CDGKS Equation

To find the linearizing representation, we consider(19)u=Ωγxx+u0,γ=γx,y,t,Ω=Ωt.Inserting equation (28) into equation (1), one obtains(20)ℜγ=ddtΩtgx,y,t+Ωt∂∂tgx,y,t+ϕ1tΩt∂5∂x5gx,y,t+ϕ2tΩt2∂∂xgx,y,t∂2∂x2gx,y,t,(21)+ϕ3tΩt2gx,y,t∂3∂x3gx,y,t+ϕ4tΩt3gx,y,t2∂∂xgx,y,t+ϕ5tΩt∂3∂x2∂ygx,y,t+ϕ6tΩt∫∂2∂y2gx,y,tdx+ϕ7tΩt2∂∂xgx,y,t∫∂∂ygx,y,tdx+ϕ8tΩt2gx,y,t∂∂ygx,y,t+ϕ9tΩtgx,y,t=0,with(22)Ω=2c0e∫0tϕ9tdt.The novel equationℜγ is(23)ℜγ=Px,t+ϕ1tP6x+6P3x2+5c1ϕ1tP2x−5c12ϕ1tP2y=0.Implementing a change of dependent variable,(24)γ=lng⇔u=2c0e∫0tϕ9tdtlngxx.Theorem 1.
With the following relations(25)γ=lng⇔u=2c0e∫0tϕ9tdtlngxx,into equation (1), the (3 + 1)-D VC nonlinear wave equation can be linearized as the following bilinear model:(26)ℜg=2ggxt−gxgt+2ϕ1tggxxxxxx−6gxgxxxxx+15gxxgxxxx−10gxxx2+10c1ϕ1tggxxxy−gygxxx−3gxxygx+3gxxgxy−10c12ϕ1tggyy−gy2,(27)=12DxDt+ϕ1tDx6+5c1ϕ1tDx3Dy−5c12ϕ1tDy2g⋅g=0,where g=gx,y,t and γ=γx,y,t.


## 4. Multisoliton of the (2 + 1)-Dimensional VC CDGKS Equation

TheN-soliton function for studying the unction solution of (1) based on the Hirota operator can be written as follows:(28)f=fN=∑r=0,1exp∑i<jNrirjSij+∑i=1Nriλi,where(29)λi=kix+piy+ϕi5c12pi2−5c1ki2pi−ki4t+λi0,(30)expSij=c12pi−pj2+c1ki−kj2kipi+kipj−kjpi−2kjpj+ki2−kikj+kj2ki−kj2c12pi−pj2+c1kj+ki2kip1+kipj+kjpi+2kjpj+ki2+kikj+kj2kj+ki2.Letr1=0,1,r2=0,1,…,rN=0,1. In the following, equation (29) can be written as(31)f1=1+eλ1,(32)f2=1+eλ1+eλ2+ℵ12eλ1+λ2,f3=1+eλ1+eλ2+eλ3+ℵ12eλ1+λ2+ℵ13eλ1+λ3+ℵ23eλ2+λ3+ℵ12ℵ13ℵ23eλ1+λ2+λ3,where ℵij=expSij,i<j. To detect M-lump solutions of equation (5), by catching expλi0=−1 in equation (29), then the following fN reads:(33)fN=∑r=0,1∏i=1N−1σiexpriζi∏i<jNexprirjSij,where(34)ζi=kix+piy+ϕ15c12p12−5c1k12p1−k14t.Via employing a limit ofki⟶0, one gets(35)fN=∑r=0,1∏i=1N−1ri1+rikiχi∏i<jN1+rirjkikjℑij+OkN+1.By factorizationfN by ∏i=1Nki and by using u=2c0e−∫ϕ9tdtlnfxx, we get(36)fN=∏i=1Nχi+12∑i,jNℑij∏l≠i,jNχl+12!22∑i,j,s,rNℑijℑsr∏l≠i,j,s,rNχl+⋯+1M!2M∑i,j,…,m,nNℑijℑkl…ℑmn∏p≠i,j,k,l,…,m,nNξp,where(37)χi=x+piy+ϕ15c12p12−5c1k12p1−k14t,ℑij=−6p1+p2c1p1−p22.With consideringpM+i=pi∗,qM+i=qi∗,i=1:M for N=2M, the statement ℑij>0, M-lump outputs are obtained. By keeping doing something, 1-lump and multiple lump outputs of equation (5) will be furnished in the following situations.


### 4.1. 1-Lump Outputs

Supposingexpλi0=−1,i=1,2 [47, 48], 2-soliton will be as follows:(38)f2=1−eμ1−eμ2+eμ1+μ2+S12,where(39)μl=klx+ply+ϕ15c12pl2−5c1k12pl−k14t,l=1,2.Through employing limit,kl⟶0 for l=1,2, with k1/k2=O1 and p1/p2=O1, one gets(40)expSij=1−6k1k2pi+pjc1pi−pj2+Ok3,and(41)f2=k1k2χ1χ2−6p1+p2c1p1−p22+Ok.By eliminating the factork1k2 of f2, we admit the following:(42)f2=χ1χ2−6p1+p2c1p1−p22=ξ1ξ2+ℑ12,ξi=x+piy+ϕ15c12pi2t,i=1,2.By utilizingp2=p1∗ in equation (42), one becomes(43)f2=ξ1ξ1∗−6p1+p1∗c1p1−p1∗2,withℑ12>0.Appending equation (43) into u=2c0e−∫ϕ9tdtlnf2xx and inserting p1=a+bi becomes(44)f2=1c1b2(25a4b2t2c15ϕ12+50a2b4t2c15ϕ12+25b6t2c15ϕ12+10a3b2tyc13ϕ1+10ab4tyc13ϕ1+10a2b2txc13ϕ1−10b4txc13ϕ1+a2b2y2c1+b4y2c1+2ab2xyc1+b2x2c1+3a),(45)u=2c0e−∫ϕ9tdt2c1b2Ω1−Ω22Ω12,Ω1=25a4b2t2c15ϕ12+50a2b4t2c15ϕ12+25b6t2c1512+10a3b2tyc13ϕ1+10ab4tyc13ϕ1+10a2b2txc13ϕ1−10b4txc13ϕ1+a2b2y2c1+b4y2c1+2ab2xyc1+b2x2c1+3a,Ω2=10a2b2tc13ϕ1−10b4tc13ϕ1+2ab2yc1+2b2xc1.Three critical points were reached includingx1,y1=5a2tc12ϕ1+5b2tc12ϕ1,−10atc12ϕ1 and(46)x2,3,y2,3=5a2btc13ϕ1+5b3tc13ϕ1+3ac1c1b,−10atc12ϕ1.At the pointx1,y1, the second-order derivative and Hessian matrix [49] are obtained as(47)Θ1=∂2∂x2ux,y|x1,y1=−83c0e−∫ϕ9tdtc12b4a2,Δ1=det∂2∂x2ux,y∂2∂x∂yux,y∂2∂x∂yux,y∂2∂y2ux,yx1,y1=64c02e−∫ϕ9tdt2c14b1027a4.In pointx1,y1=5a2tc12ϕ1+5b2tc12ϕ1,−10atc12ϕ1, we have the following.Ifc0>0, then the solution x1,y1 is the lump output, as shown in Figures 1–3; while, if c0<0, then the local minimum point of function u does arises. The point x1,y1 is a maximum amount point at which umax. By utilizing different a and b amounts, the lump solution ux,y has one maximum value including umax=4/3c0e−∫ϕ9tdtb2c1/a.Figure 1
Plot of equation (44) by taking a=1.5,b=1.1,c0=1,c1=2,ϕ1t=t2,ϕ9t=sin6t+3,t=0. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 2
Plot of equation (44) by taking a=0.5,b=−1.1,c0=1,c1=2,ϕ1t=t2,ϕ9t=sin6t+3,t=0. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 3
Plot of equation (44) by taking a=1,b=−1,c0=1,c1=2,ϕ1t=t2,ϕ9t=cost,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)


### 4.2. 2 and 3-Lump Soliton

SupposingN=4,M=2 in equation (36), f4 can be represented as(48)f4=∂1∂2∂3∂4+℘12∂3∂4+ℑ13∂2∂4+ℑ14∂2∂3+ℑ23∂1∂4+ℑ24∂1∂3+ℑ34∂1∂2+ℑ12ℑ34+ℑ13ℑ24+ℑ14ℑ23,where(49)∂m=x+pmy+ϕ15c12pm2t,m=1,2,3,4,ℑmn=−6pm+pnc1pm−pn2,m<n,j=2,3,4.Withu=2c0e−∫ϕ9tdtlnf4xx, a nonsingular solution by considering p3=p1∗,p4=p2∗, and ℜpi>0i=1,2 will be reached. In Figures 4–6, the solution equation (48) is designed with obtained amounts p1=pR1+pI1i,p2=pR2+pI2i, pR1,pI1,pR2,pI2,q1=qR1+qI1i,q2=qR2+qI2i, qR1,qI1,qR2, and qI2. Also, equation (36) can be reduced to f6 in the following form:(50)f6=∂1∂2∂3∂4∂5∂6+ℑ12∂3∂4∂5∂6+ℑ13∂2∂4∂5∂6+ℑ14∂2∂3∂5∂6+ℑ15∂2∂3∂4∂6+ℑ16∂2∂3∂4∂5+ℑ23∂1∂4∂5∂6+ℑ24∂1∂3∂5∂6+ℑ25∂1∂3∂4∂6+ℑ26∂1∂3∂4∂5+ℑ34∂1∂2∂5∂6+ℑ35∂1∂2∂4∂6+ℑ36∂1∂2∂4∂5+ℑ45∂1∂2∂3∂6+ℑ46∂1∂2∂3∂5+ℑ56∂1∂2∂3∂4+ℑ12ℑ34∂5∂6+ℑ12ℑ35∂4∂6+ℑ12ℑ36∂4∂5+ℑ12ℑ45∂3∂6+ℑ12ℑ46∂3∂5+ℑ12ℑ56∂3∂4+ℑ13ℑ24∂5∂6+ℑ13ℑ25∂4∂6+ℑ13ℑ26∂4∂5+ℑ13ℑ45∂2∂6+ℑ13ℑ46∂2∂5+ℑ13ℑ56∂2∂4+ℑ14ℑ23∂5∂6+ℑ14ℑ25∂3∂6+ℑ14ℑ26∂3∂5+ℑ14ℑ35∂2∂6+ℑ14ℑ36∂2∂5+ℑ14ℑ56∂2∂3+ℑ15ℑ23∂4∂6+ℑ15ℑ24∂3∂6+ℑ15ℑ26∂3∂4+ℑ15ℑ34∂2∂6+ℑ15ℑ36∂2∂4+ℑ15ℑ46∂2∂3+ℑ16ℑ23∂4∂5+ℑ16ℑ24∂3∂5+ℑ16ℑ25∂3∂4+ℑ16ℑ34∂2∂5+ℑ16ℑ35∂2∂4+ℑ16ℑ45∂2∂3+ℑ23ℑ45∂1∂6+ℑ23ℑ46∂1∂5+ℑ23ℑ56∂1∂4+ℑ24ℑ35∂1∂6+ℑ24ℑ36∂1∂5+ℑ24ℑ56∂1∂3+ℑ25ℑ34∂1∂4+ℑ25ℑ36∂1∂4+ℑ25ℑ46∂1∂3+ℑ26ℑ34∂1∂5+ℑ26ℑ35∂1∂4+ℑ26ℑ45∂1∂3+ℑ34ℑ56∂1∂2+ℑ35ℑ46∂1∂2+ℑ36ℑ45∂1∂2+ℑ12ℑ34ℑ56+ℑ12ℑ35ℑ46+ℑ12ℑ36ℑ45+ℑ13ℑ24ℑ56+ℑ13ℑ25ℑ46+ℑ13ℑ26ℑ45+℘14ℑ23ℑ56+ℑ14ℑ25ℑ36+ℑ14ℑ26ℑ35+ℑ15ℑ23ℑ46+ℑ15ℑ24ℑ36+ℑ15ℑ26ℑ34+ℑ16ℑ23ℑ45+ℑ16ℑ24ℑ35+ℑ16ℑ25ℑ34,where(51)∂i=x+piy+ϕ15c12p12t,i=1,…,6,ℑij=−6pi+pjc1pi−pj2,i<j,j=2,…,6.Figure 4
Plot of equation (48) by taking a=1,b=1.5,c=1,d=2,c0=1,c1=2,ϕ1t=t2,ϕ9t=cost,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 5
Plot of equation (48) by taking a=1,b=−2.5,c=1,d=2,c0=1,c1=2,ϕ1t=t2,ϕ9t=cost,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 6
Plot of equation (48) by taking a=c=1,b=−2.5,d=2,ϕ1t=t3,ϕ9t=t+t2,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)u=2c0e−∫ϕ9tdtlnf6xx along with f6 and considering p4=p1∗,p5=p2∗,p6=p3∗, and ℜpm>0m=1,2,3, the three-lump solutions of equation (5) are concluded. While, in Figures 7–9, consider p1=pR1+pI1i,p2=pR2+pI2i,p3=pR3+pI3i.Figure 7
Plot of equation (50) by taking a=c=c0=1,b=1.5,d=e=c1=2,f=4.1,ϕ1t=t2,ϕ9t=cos6t+8,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 8
Plot of equation (50) by taking a=c=d=e=c0=1,b=1.5,f=4,c1=2,ϕ1t=t2,ϕ9t=cos6t+8,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 9
Plot of equation (50) by taking a=c=d=e=c0=1,b=1.5,e=f=c1=2,ϕ1t=t2,ϕ9t=cos6t+8,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)


## 4.1. 1-Lump Outputs

Supposingexpλi0=−1,i=1,2 [47, 48], 2-soliton will be as follows:(38)f2=1−eμ1−eμ2+eμ1+μ2+S12,where(39)μl=klx+ply+ϕ15c12pl2−5c1k12pl−k14t,l=1,2.Through employing limit,kl⟶0 for l=1,2, with k1/k2=O1 and p1/p2=O1, one gets(40)expSij=1−6k1k2pi+pjc1pi−pj2+Ok3,and(41)f2=k1k2χ1χ2−6p1+p2c1p1−p22+Ok.By eliminating the factork1k2 of f2, we admit the following:(42)f2=χ1χ2−6p1+p2c1p1−p22=ξ1ξ2+ℑ12,ξi=x+piy+ϕ15c12pi2t,i=1,2.By utilizingp2=p1∗ in equation (42), one becomes(43)f2=ξ1ξ1∗−6p1+p1∗c1p1−p1∗2,withℑ12>0.Appending equation (43) into u=2c0e−∫ϕ9tdtlnf2xx and inserting p1=a+bi becomes(44)f2=1c1b2(25a4b2t2c15ϕ12+50a2b4t2c15ϕ12+25b6t2c15ϕ12+10a3b2tyc13ϕ1+10ab4tyc13ϕ1+10a2b2txc13ϕ1−10b4txc13ϕ1+a2b2y2c1+b4y2c1+2ab2xyc1+b2x2c1+3a),(45)u=2c0e−∫ϕ9tdt2c1b2Ω1−Ω22Ω12,Ω1=25a4b2t2c15ϕ12+50a2b4t2c15ϕ12+25b6t2c1512+10a3b2tyc13ϕ1+10ab4tyc13ϕ1+10a2b2txc13ϕ1−10b4txc13ϕ1+a2b2y2c1+b4y2c1+2ab2xyc1+b2x2c1+3a,Ω2=10a2b2tc13ϕ1−10b4tc13ϕ1+2ab2yc1+2b2xc1.Three critical points were reached includingx1,y1=5a2tc12ϕ1+5b2tc12ϕ1,−10atc12ϕ1 and(46)x2,3,y2,3=5a2btc13ϕ1+5b3tc13ϕ1+3ac1c1b,−10atc12ϕ1.At the pointx1,y1, the second-order derivative and Hessian matrix [49] are obtained as(47)Θ1=∂2∂x2ux,y|x1,y1=−83c0e−∫ϕ9tdtc12b4a2,Δ1=det∂2∂x2ux,y∂2∂x∂yux,y∂2∂x∂yux,y∂2∂y2ux,yx1,y1=64c02e−∫ϕ9tdt2c14b1027a4.In pointx1,y1=5a2tc12ϕ1+5b2tc12ϕ1,−10atc12ϕ1, we have the following.Ifc0>0, then the solution x1,y1 is the lump output, as shown in Figures 1–3; while, if c0<0, then the local minimum point of function u does arises. The point x1,y1 is a maximum amount point at which umax. By utilizing different a and b amounts, the lump solution ux,y has one maximum value including umax=4/3c0e−∫ϕ9tdtb2c1/a.Figure 1
Plot of equation (44) by taking a=1.5,b=1.1,c0=1,c1=2,ϕ1t=t2,ϕ9t=sin6t+3,t=0. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 2
Plot of equation (44) by taking a=0.5,b=−1.1,c0=1,c1=2,ϕ1t=t2,ϕ9t=sin6t+3,t=0. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 3
Plot of equation (44) by taking a=1,b=−1,c0=1,c1=2,ϕ1t=t2,ϕ9t=cost,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)


## 4.2. 2 and 3-Lump Soliton

SupposingN=4,M=2 in equation (36), f4 can be represented as(48)f4=∂1∂2∂3∂4+℘12∂3∂4+ℑ13∂2∂4+ℑ14∂2∂3+ℑ23∂1∂4+ℑ24∂1∂3+ℑ34∂1∂2+ℑ12ℑ34+ℑ13ℑ24+ℑ14ℑ23,where(49)∂m=x+pmy+ϕ15c12pm2t,m=1,2,3,4,ℑmn=−6pm+pnc1pm−pn2,m<n,j=2,3,4.Withu=2c0e−∫ϕ9tdtlnf4xx, a nonsingular solution by considering p3=p1∗,p4=p2∗, and ℜpi>0i=1,2 will be reached. In Figures 4–6, the solution equation (48) is designed with obtained amounts p1=pR1+pI1i,p2=pR2+pI2i, pR1,pI1,pR2,pI2,q1=qR1+qI1i,q2=qR2+qI2i, qR1,qI1,qR2, and qI2. Also, equation (36) can be reduced to f6 in the following form:(50)f6=∂1∂2∂3∂4∂5∂6+ℑ12∂3∂4∂5∂6+ℑ13∂2∂4∂5∂6+ℑ14∂2∂3∂5∂6+ℑ15∂2∂3∂4∂6+ℑ16∂2∂3∂4∂5+ℑ23∂1∂4∂5∂6+ℑ24∂1∂3∂5∂6+ℑ25∂1∂3∂4∂6+ℑ26∂1∂3∂4∂5+ℑ34∂1∂2∂5∂6+ℑ35∂1∂2∂4∂6+ℑ36∂1∂2∂4∂5+ℑ45∂1∂2∂3∂6+ℑ46∂1∂2∂3∂5+ℑ56∂1∂2∂3∂4+ℑ12ℑ34∂5∂6+ℑ12ℑ35∂4∂6+ℑ12ℑ36∂4∂5+ℑ12ℑ45∂3∂6+ℑ12ℑ46∂3∂5+ℑ12ℑ56∂3∂4+ℑ13ℑ24∂5∂6+ℑ13ℑ25∂4∂6+ℑ13ℑ26∂4∂5+ℑ13ℑ45∂2∂6+ℑ13ℑ46∂2∂5+ℑ13ℑ56∂2∂4+ℑ14ℑ23∂5∂6+ℑ14ℑ25∂3∂6+ℑ14ℑ26∂3∂5+ℑ14ℑ35∂2∂6+ℑ14ℑ36∂2∂5+ℑ14ℑ56∂2∂3+ℑ15ℑ23∂4∂6+ℑ15ℑ24∂3∂6+ℑ15ℑ26∂3∂4+ℑ15ℑ34∂2∂6+ℑ15ℑ36∂2∂4+ℑ15ℑ46∂2∂3+ℑ16ℑ23∂4∂5+ℑ16ℑ24∂3∂5+ℑ16ℑ25∂3∂4+ℑ16ℑ34∂2∂5+ℑ16ℑ35∂2∂4+ℑ16ℑ45∂2∂3+ℑ23ℑ45∂1∂6+ℑ23ℑ46∂1∂5+ℑ23ℑ56∂1∂4+ℑ24ℑ35∂1∂6+ℑ24ℑ36∂1∂5+ℑ24ℑ56∂1∂3+ℑ25ℑ34∂1∂4+ℑ25ℑ36∂1∂4+ℑ25ℑ46∂1∂3+ℑ26ℑ34∂1∂5+ℑ26ℑ35∂1∂4+ℑ26ℑ45∂1∂3+ℑ34ℑ56∂1∂2+ℑ35ℑ46∂1∂2+ℑ36ℑ45∂1∂2+ℑ12ℑ34ℑ56+ℑ12ℑ35ℑ46+ℑ12ℑ36ℑ45+ℑ13ℑ24ℑ56+ℑ13ℑ25ℑ46+ℑ13ℑ26ℑ45+℘14ℑ23ℑ56+ℑ14ℑ25ℑ36+ℑ14ℑ26ℑ35+ℑ15ℑ23ℑ46+ℑ15ℑ24ℑ36+ℑ15ℑ26ℑ34+ℑ16ℑ23ℑ45+ℑ16ℑ24ℑ35+ℑ16ℑ25ℑ34,where(51)∂i=x+piy+ϕ15c12p12t,i=1,…,6,ℑij=−6pi+pjc1pi−pj2,i<j,j=2,…,6.Figure 4
Plot of equation (48) by taking a=1,b=1.5,c=1,d=2,c0=1,c1=2,ϕ1t=t2,ϕ9t=cost,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 5
Plot of equation (48) by taking a=1,b=−2.5,c=1,d=2,c0=1,c1=2,ϕ1t=t2,ϕ9t=cost,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 6
Plot of equation (48) by taking a=c=1,b=−2.5,d=2,ϕ1t=t3,ϕ9t=t+t2,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)u=2c0e−∫ϕ9tdtlnf6xx along with f6 and considering p4=p1∗,p5=p2∗,p6=p3∗, and ℜpm>0m=1,2,3, the three-lump solutions of equation (5) are concluded. While, in Figures 7–9, consider p1=pR1+pI1i,p2=pR2+pI2i,p3=pR3+pI3i.Figure 7
Plot of equation (50) by taking a=c=c0=1,b=1.5,d=e=c1=2,f=4.1,ϕ1t=t2,ϕ9t=cos6t+8,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 8
Plot of equation (50) by taking a=c=d=e=c0=1,b=1.5,f=4,c1=2,ϕ1t=t2,ϕ9t=cos6t+8,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 9
Plot of equation (50) by taking a=c=d=e=c0=1,b=1.5,e=f=c1=2,ϕ1t=t2,ϕ9t=cos6t+8,t=1. (a) f1. (b) f2. (c) f3.
(a)(b)(c)


## 5. N-Soliton Outputs of the (2 + 1)-D VC CDGKS Equation

To the VC CDGKS equation, consider the following function:(52)f=1+∑ℏ1=1N+∑ℏ1<ℏ2NSℏ1ℏ2eΛℏ1+Λℏ2+∑ℏ1<ℏ2<ℏ3NSℏ1ℏ2Sℏ1ℏ3Sℏ2ℏ3eΛℏ1+Λℏ2+Λℏ3+⋯+∏ℏ1<ℏ2NSℏ1ℏ2e∑ℏ1=1NΛℏ1,with(53)Λℏ1=aℏ1x+bℏ1y+ωℏ1t+ιℏ1k,ωℏ1=ϕ15c12pℏ12−5c1kℏ12pℏ1−kℏ14,(54)Sℏ1ℏ2=c12p1−p22+c1k1−k22k1p1+k1p2−k2p1−2k2p2+k12−k1k2+k22k1−k22c12p1−p22+c1k2+k12k1p1+k1p2+k2p1+2k2p2+k12+k1k2+k22k2+k12,where the constants aℏ1,bℏ1,cℏ1, and Sℏ1ℏ2 are the values of the Nth soliton. In the following subsections, various kinds of interaction breather and lump are contemplated.


### 5.1. 1-Breather and 1-Lump

By takingN=2 in equation (52), we get(55)f=1+eΛ1+eΛ2+S12eΛ1+Λ2.We have the following cases.Option I: by consideringa1=a2=ρ1,b1=b2∗=ϱ1+iτ1,ι01=ι02=0, where i2=−1, equation (55) will be(56)f=1+S12e2ξ1+2eξ1cosτ1y−δ1t.Here,(57)S12=c12τ12−3ρ16−3c1ρ13ϱ1+c12τ12,ξ1=θ1t+ρ1x+ϱ1y,(58)θ1=ϕ1ρ16−5c1ρ13ϱ1+c1τ12−c1ϱ12ρ1,δ1=−ϕ1−5ρ13+10c1ϱ1c1τ1ρ1.By consideringτ1=1/5,ρ1=1/3 and value of ϱ1=−1/3, three cases of breathers can be obtained, as shown in Figure 10, which are x,y -periodic breathers.Choice II: via consideringa1=l1ε,a2=l2ε,b1=ν1a1,b2=ν2a2,ι01=ι02∗=iπ in equation (55) and letting limit ε⟶0, fx,y,t can be written as(59)fx,y,t=η12+Υ1Υ2l1l2ε2+Oε3.Here,(60)η12=1,Υi=x+νiy+5ϕ1c12mi2t,i=1.2.Putting equation (59) into equation (9) and considering ν1=ν2=γ1+iσ1, the two-soliton transforms into the following, equation (5), as(61)u=2c0e−∫ϕ9tdtν1+ν2η12+Υ1Υ2−Υ1+Υ2ν2Υ1+ν1Υ2η12+Υ1Υ22,With the following conditions:(62)∂u∂x=0,∂u∂y=0.Hence, we obtain(63)x1,y1=5tc12γ12ϕ1+5tc12ϕ1σ12,−10tc12γ1ϕ1,x2,3,y2,3=5tc12γ12ϕ1+5tc12ϕ1σ12±3,−10tc12γ1ϕ1.At the pointx1,y1, the second-order derivative and Hessian matrix is(64)CaseA:Γ1=∂2∂x2ux,y,t|x1,y1=−24c0e−∫ϕ9tdt,Δ1=det∂2∂x2ux,y,t∂2∂x∂yux,y,t∂2∂x∂yux,y,t∂2∂y2ux,y,tx1,y1=192c02e−∫ϕ9tdt2σ12,(65)CaseB:Γ2=∂2∂x2ux,y,t|x2,3,y2,3=34c0e−∫ϕ9tdt,Δ2=det∂2∂x2ux,y,t∂2∂x∂yux,y,t∂2∂x∂yux,y,t∂2∂y2ux,y,tx2,3,y2,3=34c02e−∫ϕ9tdt2σ12.In case A, ifc0<0 and Δ1>0, then the point x1,y1 is the extreme value. Also, the point x1,y1 is a maximum amount with measure umax=4c0e−∫ϕ9tdt and is shown in Figure 11. In case B, if c0>0 and Δ1>0, then the point x2,3,y2,3 is extreme amount. Therefore, the point x2,3,y2,3 is a maximum amount with measure umax=−1/2c0e−∫ϕ9tdt. The related plots are designed with amounts of parameters, such as γ1=1,σ1=1/2 for (f1, f4), γ1=−1,σ1=1/2 for (f2, f5), and γ1=1/3,σ1=1/5 for (f3, f6).Set III: through consideringa1=a2∗=δ1+iτ1,c1=c2∗=ϱ°1+iρ°1,ν01=ν02=0 and values ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10 in (55), many of localized waves in (2 + 1)-D obtained are shown in Figure 12. Figures 12(a) and 12(c) show that the output u are t-periodic breathers, while Figure 12(b) shows that the output u are x-periodic breathers. Also, Figure 12 shows the periodic breather outputs. The related plots are plotted with amounts of parameters, such as δ1=0,τ1=1,α1=1,β1=1 for (f1, f4), δ1=1/3,τ1=1,α1=0,β1=1 for (f2, f5), and δ1=1/3,τ1=1,α1=1,β1=1 for (f3, f6).Figure 10
Plots of equation (55) by taking ϱ1=2/3,τ1=2/5,ρ1=−2/3,ϕ1t=2,ϕ9t=cos6t+3,c0=1,c1=2,t=−2. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)Figure 11
Plots of equation (61) by taking ϱ1=2/3,τ1=2/5,ρ1=−2/3,ϕ1t=2,ϕ9t=cos6t+3,c0=1,c1=2,t=2. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)Figure 12
Plots of equation (61) by taking λ1=2,ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10,t=−2. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)


### 5.2. Interaction 1-Bell-Shaped Soliton with 1-Lump or 1-Breather

By consideringN=3, the following function is taken:(66)f=1+eΛ1+eΛ2+eΛ3+Σ12eΛ1+Λ2+S13eΛ1+Λ3+S23eΛ2+Λ3+S12S13S23eΛ1+Λ2+Λ3.Some special cases are as follows.Set I: via supposinga1=a2=ρ1,a3=ρ2,b1=b2∗=ϱ1+iτ1,b3=ϱ2,ι01=ι02=ι03=0, equation (66) can be written as follows:(67)fx,y,t=1+S12e2Ξ1+2eΞ1cosΞ2+eΞ31+2eΞ1S13cosΞ2+ϕS13+S12S132e2Ξ1.In whichθ1,Υ1, and Σ12 are stated in equation (57) and have the following relations:(68)Ξ1=tθ1+xρ1+yϱ1,Ξ2=δ1t−τ1y,Ξ3=ω3t+ρ2x+ϱ2y,S13=S13eiϕS13,(69)θ1=ϕ1−ρ16−5c1ρ13ϱ1−5c12τ12+5c12ϱ12ρ1,δ1=ϕ15ic1ρ13τ1−10ic12τ1ϱ1ρ1,(70)ω3=ϕ1−ρ26−5c1ρ23ϱ2+5c12ϱ22ρ2,S23=S13∗=S13e−iϕS13.By consideringτ1=3,ρ1=1,ρ2=1,ϕ1t=t2,ϕ9t=sint+3 and various values of ϱ1 and ϱ2, the dynamical behaviors of related solutions obtained are shown in Figure 13. Also, Figure 13 shows that the solution contains kinds of behavior properties of solutions with diverse values of ϱ1 and ϱ2. The related plots are plotted with amounts of parameters like ϱ1=−1,ϱ2=1,τ1=3,ρ1=1,ρ2=1 for (f1, f4), ϱ1=−2,ϱ2=1,τ1=3,ρ1=1,ρ2=1 for (f2, f5), and ϱ1=−2,ϱ2=2,τ1=3,ρ1=1,ρ2=1 for (f3, f6).Set II: by takinga1=l1ε,a2=l2ε,bi=νiai,i=1,2,3,ι01=ι02∗=iπ,ι03=0 in equation (55) and ε⟶0, fx,y,t one becomes(71)fx,y,t=η12+Υ1Υ2l1l2ε2+Υ1Υ2+η23Υ1+η13Υ2+η12+η13η23eΞ3l1l2ε2+Oε3.Here,(72)η12=η13=η23=1,Υi=x+νiy+εit,i=1,2,ε1=λ4σ12−2iγ1λ4σ1−γ12λ4,(73)ε2=2iγ1λ4σ1−γ12λ4+λ4σ12,η3=κ1x+γ2y−κ1κ14λ3+γ2κ12λ5+γ22λ4+κ12λ1t.Substituting equation (73) into equation (67), also consider ν1=ν2=ϱ1+iτ1,ν3=ϱ2. The solution u obtained by equation (68) will transform to the behavior output between one-lump and one-breather soliton, whose dynamical property at different times is found in Figure 14. Also, from Figure 14, the related graphs are plotted with amounts of various times like t=−40 for (f1), t=−20 for (f2), t=0 for (f3), t=20 for (f4), and t=40 for (f5). Also, from Figure 15, the related graphs are plotted with amounts of various times like t=−40 for (f1), t=−20 for (f2), t=0 for (f3), t=20 for (f4), and t=40 for (f5). Moreover, from Figure 16, the related graphs are plotted with amounts of various times like t=−40 for (f1, f6), t=−20 for (f2, f7), t=0 for (f3, f8), t=20 for (f4, f9), and t=40 for (f5, f10).Set III: considera1=a2∗=δ1+iτ1,a3=δ2,b1=b2∗=ϱ1+iρ1,b3=ϱ2,ι01=ι02=ι03=0, in equation (66). Figure 17(a) shows the cross-behavior between one-breather and one-breather-shaped soliton with values δ1=1/3,σ1=−1,δ2=1,α1=1,β1=1 in x,y-plane. Figure 17(b) shows the cross-behavior between one-breather and one-breather-shaped soliton with values δ1=0,σ1=−1,δ2=−4,α1=1,β1=1 in x,y-plane. Therefore, Figure 17(c) shows the cross-behavior between one-breather and one-breather-shaped soliton with values δ1=−1/3,σ1=−1,δ2=1,α1=1,β1=1 in x,y-plane.Figure 13
Plots of equation (67) by using c0=1,c1=1/10,t=−2. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 14
Plots of equation (71) by using γ1=1,σ1=1,κ1=1/3,σ2=1,γ2=1/5,ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5.
(a)(b)(c)(d)(e)Figure 15
Plots of equation (71) by using γ1=1,σ1=1,κ1=1/3,σ2=1,γ2=1/5,ϕ1t=t+1/100,ϕ9t=sint+3,c0=1,c1=1/10. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5.
(a)(b)(c)(d)(e)Figure 16
Plots of equation (71) by using γ1=1,σ1=1,κ1=1/3,σ2=1,γ2=1/5,ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6. (g) f7. (h) f8. (i) f9. (j) f10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 17
Plots of equation (71) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10,t=−2. (a) f1. (b) f2. (c) f3.
(a)(b)(c)


### 5.3. The Abundant Solutions WhenN=4

WithN=4, the following function is considered:(74)fx,y,t=1+eΛ1+eΛ2+eΛ3+eΛ4+S12eΛ1+Λ2+S13eΛ1+Λ3+S14eΛ1+Λ4+S23eΛ2+Λ3+S24eΛ2Λ4+S34eΛ3+Λ4+S12S13S23eΛ1+Λ2+Λ3+S12S14S24eΛ1+Λ2+Λ4+S13S14S34eΛ1+Λ3+Λ4+S23S24S34eΛ2+Λ3+Λ4+S12S13S14S23S24S34eΛ1+Λ2+Λ3+Λ4.The cases are as follows.Option I: via supposinga1=a2=ρ1,a3=a4=ρ2,b1=b2∗=ϱ1+iτ1,b3=b4∗=ϱ2+iτ2,ι01=ι02=ι03=ι04=0, four-soliton solution can be written as follows:(75)fx,y,t=1+S12e2Ξ1+S34e2Ξ2+2eΞ1cos∅1+2eΞ2cos∅2+2S13eΞ1+Ξ2cos∅1+∅2+ϕS13+2S23eΞ1+Ξ2cos∅1−∅2−ϕS23+2S12e2Ξ1+Ξ2cos∅2+ϕS13+ϕS23+2S34eΞ1+2Ξ2cos∅1+ϕS13−ϕS23+S12S34S132S232e2Ξ1+2Ξ2.Here,θ1,Ω1, and S12 are taken in equation (57), and the following relations will be reached:(76)Ξi=tAi+xδi+yαi,∅i=tBi+yβi,Ai=ϕ1−δi6−5αic1δi3+5αi2c12−5βi2c12δi,i=1,2,(77)Bi=ϕ1−5βic1δi3+10αiβic12δi,i=1,2,S12=−β12c123δ16+3α1c1δ13−β12c12,(78)S34=−c12β223δ26+3α2c1δ13−c12β22,S14=S23∗,S24=S13∗,S13=ℑ1+iℑ2ℵ1+iℵ2=S13eiϕS13,S23=ℑ3+iℑ4ℵ3+iℵ4=S23eiϕS23,ℑ1=δ12δ22δ12−δ1δ2+δ22−δ2+δ12+c1δ1δ2−δ2+δ12α1δ1δ2−α1δ22+α2δ12−2α2δ1δ2+c12α1δ2−α2δ12−β1δ2−β2δ12,ℑ2=c1δ1δ2−δ2+δ12β1δ1δ2−β1δ22+β2δ12−2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℵ1=δ12δ22δ12+δ1δ2+δ22δ2+δ12+c1δ1δ2δ2+δ12α1δ1δ2+α1δ22+α2δ12+2α2δ1δ2+c12α1δ2−α2δ12−β1δ2−β2δ12,ℵ2=c1δ1δ2δ2+δ12β1δ1δ2+β1δ22+β2δ12+2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℑ3=δ12δ22δ12−δ1δ2+δ22−δ2+δ12+c1δ1δ2−δ2+δ12α1δ1δ2−α1δ22+α2δ12−2α2δ1δ2+c12α1δ2−α2δ1−β1δ2−β2δ12,ℑ4=c1δ1δ2−δ2+δ12β1δ1δ2−β1δ22+β2δ12−2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℵ3=δ12δ22δ12+δ1δ2+δ22δ2+δ12+c1δ1δ2δ2+δ12α1δ1δ2+α1δ22+α2δ12+2α2δ1δ2+c12α1δ2−α2δ12−β1δ2−β2δ12,ℵ4=c1δ1δ2δ2+δ12β1δ1δ2+β1δ22+β2δ12+2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1.Take(79)δ1=13,δ2=−23,α1=15,α2=12,β1=25,β2=12,t=−10,δ1=13,δ2=−23,α1=15,α2=12,β1=0,β2=13,t=−5,δ1=13,δ2=23,α1=0,α2=−38,β1=27,β2=18,t=0,δ1=13,δ2=23,α1=0,α2=−38,β1=27,β2=18,t=5,And(80)δ1=13,δ2=23,α1=0,α2=−38,β1=27,β2=18,t=10.For Figure18(a)t=−10, Figure 18(b)t=−5, Figure 18(c)t=0, Figure 18(d)t=5, and (f5) t=10. Also, Figure 18 shows two y-periodic breather.Option II: by supposinga1=l1ε,a2=l2ε,bi=νiai,i=1,…,4,ι01=ι02∗=iπ,ι03=ι04=0 in equation (74), and ε⟶0, fx,y,t is given as(81)f=η12+Υ1Υ2l1l2ε2+Υ1Υ2+η23Υ1+η13Υ2+η12+η13η23eΞ3l1l2ε2+Υ1Υ2+η24Υ1+η14Υ2+η12+η14η24eΞ4l1l2ε2+η34Υ1Υ2+η23+η24Υ1+η13+η14Υ2+η12+η13+η14η23+η24eΞ3+Ξ4l1l2ε2+Oε3.Here,(82)η12=η13=η14=η23=η24=1,η34=c12σ22c12σ22−3c1γ2κ12−3κ14,Υl=x+νly+ωlt,l=1,2,ω1=−ϕ1−10ic12γ1σ1−5c12γ12+5c12σ12,ω2=ϕ1−5c12σ12−10ic12γ1σ1+5c12γ12,Ξ3=κ1x+κ1ν3y+ϕ1κ1−5c12σ22+10ic12γ2σ2−5ic1κ12σ2+5c12γ22−5c1γ2κ12−κ14t,Ξ4=κ1x+κ1ν4y−ϕ1κ110ic12γ2σ2−5ic1κ12σ2−5c12γ22+5c12σ22+5c1γ2κ12+κ14t.Set II-A: pluggingν1=ν2∗=ϱ1+iσ1,a3=a4=ρ1,ν3=ν4∗=ϱ2+iσ2, the solution u is given by equation (81), and choosing values γ1=1,σ1=1,γ2=0,σ2=3/5,κ1=2/3, the dynamical properties at different times are shown in Figure 19.Option II-B: consideringν1=ν2∗=ϱ1+iσ1,a3=a4=ρ1,ν3=ϱ2,ν4=ϱ3, the solution u given by equation (81) shows the dynamical property solitons with related plots. Putting γ1=1,σ1=1,γ2=1,γ3=−2,κ1=2/3, the dynamical behaviors at different times are shown in Figure 20.Option III: takingai=liε,bi=aiνi,i=1,…,4,ι01=ι02∗=ι03=ι04∗=iπ in equation (81) and utilizing limit as ε⟶0 transforms(83)fx,y,t=Υ1Υ2Υ3Υ4+η34Υ1Υ2+η24Υ1Υ3+η23Υ1Υ4+η14Υ2Υ3+η13Υ2Υ4+η12Υ3Υ4+η12η34+η13η24+η14η23l1l2l3l4ε4+Oε5.Here,(84)η12=η13=η14=η23=η24=η34=1,Υi=x+νiy+5ϕ1c12νi2t,i=1,2,3,4.By consideringν1=ν2∗=ϱ1+iσ1,ν3=ν4∗=ϱ2+iσ2 and by getting γ1=0,γ2=0,σ1=1,σ2=2, the structural behaviors obtained are shown in Figure 21. Moreover, substituting γ1=1/3,γ2=−2/3,σ1=1,σ2=2, the structural behaviors obtained are shown in Figure 22 with related plots.Figure 18
Plots of equation (75) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5.
(a)(b)(c)(d)(e)Figure 19
Plots of equation (81) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−150. (b) t=0. (c) t=150. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 20
Plots of equation (81) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a), (d) t=−100. (b), (e) t=0. (c), (f) t=100. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)Figure 21
Plots of equation (83) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a), (d) t=−50. (b), (e) t=0. (c), (f) t=50. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)Figure 22
Plots of equation (83) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10, (a), (d) t=−50. (b), (e) t=0. (c), (f) t=50. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)


### 5.4. The Dynamical Behavior Solutions WhenN=5

With gettingN=5, the following function will be arisen:(85)fx,y,t=1+eΛ1+eΛ2+eΛ3+eΛ4+eΛ5+S12eΛ1+Λ2+S13eΛ1+Λ3+S14eΛ1+Λ4+S15eΛ1+Λ5+S23eΛ2+Λ3+S24eΛ2+Λ4+S25eΛ2+Λ5+S34eΛ3+Λ4+S35eΛ3+Λ5+S45eΛ4+Λ5+S12S13S23eΛ1+Λ2+Λ3+S12S14S24eΛ1+Λ2+Λ4+S12S15S25eΛ1+Λ2+Λ5+S13S14S34eΛ1+Λ3+Λ4+S13S15S35eΛ1+Λ3+Λ5+S14S15S45eΛ1+Λ4+Λ5+S23S24S34eΛ2+Λ3+Λ4+S23S25S35eΛ2+Λ3+Λ5+S24S25S45eΛ2+Λ4+Λ5+S34S35S45eΛ3+Λ4+Λ5+S12S13S14S23S24S34eΛ1+Λ2+Λ3+Λ4+S12S13S15S23S25S35eΛ1+Λ2+Λ3+Λ5+S12S14S15S24S25S45eΛ1+Λ2+Λ4+Λ5+S13S14S15S34S35S45eΛ1+Λ3+Λ4+Λ5+S23S24S25S34S35S45eΛ2+Λ3+Λ4+Λ5+S12S13S14S15S23S24S25S34S35S45eΛ1+Λ2+Λ3+Λ4+Λ5.The following cases will be considered.Option I: by supposinga1=a2=ρ1,a3=a4=ρ2,b1=b2∗=ϱ1+iτ1,b3=b4∗=ϱ2+iτ2,a5=ρ3,b5=ϱ3,ι01=ι02=ι03=ι04=ι05=0, therewith, 5-soliton becomes(86)f=1+S12e2Ξ1+S34e2Ξ2+2eΞ1cos∅1+2eΞ2cos∅2+eΞ3+eΞ1+Ξ2(2S13cos∅1+∅2+ϕS13+2S23cos∅1−∅2−ϕS23+2S12S13‖S23eΞ1cosη2+ϕS13+ϕS23+2S34S13‖S23eΞ1cosη1+ϕS13−ϕS23+S12S34S132S232eΞ1+Ξ2)+eΞ1+Ξ3S12S152eΞ1+2S15cos∅1+ϕS15+eΞ2+Ξ3S34S352eΞ2+2S35cos∅2+ϕS35+S15‖S35eΞ1+Ξ2+Ξ3(2S13cos∅1+∅2+ϕS13+ϕS15+ϕS35+2S23cos∅1−∅2−ϕS23+ϕS15−ϕS35+2S12S13S23S15eΞ1cosη2+ϕS13+ϕS23+ϕS35+2S34S13S23S35eΞ2cos∅1+ϕS13−ϕS23+ϕS15+S12S34S132S232S15‖S35eΞ1+Ξ2),In whichA1,A2,∅1,∅2,Ξ1,Ξ2, and Sij1≤i<j≤4 are given in equations (76)–(78), and the following relations can be received:(87)Ξi=tAi+xδi+yαi,Ξ3=xδ3+yα3+ϕ1−δ36−5α3c1δ33+5α32c12δ3t,∅i=tBi+yβi,i=1,2,Ai=ϕ1−δi6−5αic1δi3+5αi2c12−5βi2c12δi,Bi=ϕ1−5βic1δi3+10αiβic12δi,i=1,2,S25=S15∗,S45=S35∗,S15=ℑ1+iℑ2ℵ1+iℵ2=S15eiϕS15,S23=ℑ3+iℑ4ℵ3+iℵ4=S23eiϕS23,(88)ℑ1=δ12δ22δ12−δ1δ2+δ22−δ2+δ12+c1δ1δ2−δ2+δ12α1δ1δ2−α1δ22+α2δ12−2α2δ1δ2+c12α1\δ2−α2δ12−β1δ2−β2δ12,ℑ2=c1δ1δ2−δ2+δ12β1δ1δ2−β1δ22+β2δ12−2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℵ1=δ12δ22δ12+δ1δ2+δ12δ2+δ12+c1δ1δ2δ2+δ12α1δ1δ2+α1δ22+α2δ12+2α2δ1δ2+c12α1δ2−α2δ12−β1δ2−β2δ12,ℵ2=c1δ1δ2δ2+δ12β1δ1δ2+β1δ22+β2δ12+2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℑ3=−δ12δ22δ12−δ1δ2+δ22−δ2+δ12−c1δ1δ2−δ2+δ12α1δ1δ2−α1δ22+α2δ12−2α2δ1δ2−c12α1δ2−α2δ12−β1δ2+β2δ12,ℑ4=c1δ1δ2−δ2+δ12β1δ1δ2−β1δ22−β2δ12+2β2δ1δ2+2c1β1δ2+β2δ1α1δ2−α2δ1,ℵ3=−δ12δ22δ12+δ1δ2+δ22δ2+δ12−c1δ1δ2δ2+δ12α1δ1δ2+α1δ22+α2δ12+2α2δ1δ2−c12α1δ2−α2δ12−β1δ2+β2δ12,ℵ4=c1δ1δ2δ2+δ12β1δ1δ2+β1δ22−β2δ12−2β2δ1δ2+2c1β1δ2+β2δ1α1δ2−α2δ1.Option I-A: by consideringδ1=1/3,δ2=2/3,δ3=1,α1=1/5α2=0,α3=1/3,β1=2/5,β2=1/2, u given by equation (86) returns the structural properties of solutions as shown in Figure 23 with density plot.Option I-B: by puttingδ1=1/3,δ2=−2/3,δ3=1,α1=0,α2=0,α3=1/2,β1=2/5,β2=1/2 in equation (86), the structural property among two parallel y-periodic of solutions obtained is shown in Figure 24 with density plot.Option I-C: via substitutingδ1=1/3,δ2=2/3,δ3=1,α1=0,α2=−3/8,α3=1/2,β1=2/5,β2=1/8i in equation (86), we can get the properties y-periodic of solutions as shown in Figure 25 with density plot.Set II: consideringbi=aiνi,i=1,…,5,a1=l1ε,a2=l2ε,ι01=ι02∗=iπ,ι03=ι04=ι04=0 in equation (86) and getting limit as ε⟶0, the 5-soliton one becomes(89)f=η12+Υ1Υ2l1l2ε2+Υ1Υ2+η23Υ1+η13Υ2+η12+η13η23eΞ3l1l2ε2+Υ1Υ2+η24Υ1+η14Υ2+η12+η14η24eΞ4l1l2ε2+Υ1Υ2+η25Υ1+η15Υ2+η12+η15η25eΞ5l1l2ε2+η34Υ1Υ2+η23+η24Υ1+η13+η14Υ2+η12+η13+η14η23+η24eΞ3+Ξ4l1l2ε2+Oε3+η35Υ1Υ2+η23+η25Υ1+η13+η15Υ2+η12+η13+η15η23+η25eΞ3+Ξ5l1l2ε2+Oε3+η45Υ1Υ2+η24+η25Υ1+η14+η15Υ2+η12+η14+η15η24+η25eΞ4+Ξ5l1l2ε2+Oε3+η34η35η45Υ1Υ2+η23+η24+η25Υ1+η13+η14+η15Υ2+η12+η13+η14+η15η23+η24+η25eΞ3+Ξ4+Ξ5l1l2ε2+Oε3.Here,(90)η12=η13=η14=η15=η23=η24=η25=1,ηij=ai2aj2ai2−aiaj+aj2ai−aj2+aiajc1ai−ajai2bj+2aiajbi−2aiajbj−aj2bi+c12aibj−ajbi2ai2aj2ai2+aiaj+aj2ai+aj2+aiajc1ai+ajai2bj+2aiajbi+2aiajbj+aj2bi+c12aibj−ajbi2,3≤i<j≤5,Υi=x+νiy−ϕ1γi2−2iγiσi−σi2t,i=1,2,Ξj=κ1x+κ1γjy+ωjt,ω1=−ϕ1κ16+5κ14γ2+iσ2c1−5κ12γ2+iσ22c12κ1,ω2=−ϕ1κ16+5κ14γ2−iσ2c1−5κ12γ2−iσ22c12κ1,ω3=−ϕ1−5c12γ12κ22+5c1γ3κ24+κ26κ2.Option II-A: insertingν1=ν2∗=ϱ1+iσ1,a3=a4=ρ1,ν3=ν4∗=ϱ2+iσ2,a5=ρ2,ν5=ϱ3, the u received by equation (89) explains the structural property of special solutions. Again, considering γ1=1/3,σ1=2/3,κ1=2/3,γ2=0,σ2=3/5,κ2=1/2,γ3=−1/2, the formation of graphs at different times makes expressiveness and are shown in Figure 26 with density plot.Set II-B: substitutingν1=ν2∗=ϱ1+iσ1,a3=a4=ρ1,ν3=ϱ2,ν4=ϱ3,ν5=ϱ4,a5=ρ2, ux,y,t found by equation (89) transforms the structural property of solutions. Again, considering γ1=1,σ1=2/3,κ1=2/3,γ2=−1/2,γ3=−1/4,κ2=1/2,γ4=1, the formation of graphs at various times makes expressiveness and is shown in Figure 27.Set III: via insertingai=liε,ci=aiνi,i=1,…,4,b5=a5ν5,ι01=ι02∗=ι03=ι04∗=iπ,ι05=0 in equation (85) and providing limit as ε⟶0, the 5-soliton transforms(91)fx,y,t=Υ1Υ2Υ3Υ4+η34Υ1Υ2+η24Υ1Υ3+η23Υ1Υ4+η14Υ2Υ3+η13Υ2Υ4+η12Υ3Υ4+η12η34+η13η24+η14η23)l1l2l3l4ε4+Υ1Υ2Υ3Υ4+η45Υ1Υ2Υ3+η35Υ1Υ2Υ4+η25Υ1Υ3Υ4+η15Υ2Υ3Υ4+Υ1η35η45+η34Υ2+η25η45+η24Υ3+η25η35+η23Υ4+Υ2η15η45+η14Υ3+η15η35+η13Υ4+η15η25+η12Υ3Υ4+Υ1η25η35+η23η45+η24η35+η25η34+Υ2η15η35+η13η45+η14η35+η15η34Υ3η15η25+η12η45+η14η25+η15η24+Υ4η15η25+η12η35+η13η25+σ15η23+η13η24+η23η14+η34η15η25+η12+η35η14η25+η15η24+η45η15η25+η12η35+η13η25+η15η23expΞ5l1l2l3l4ε4+Oε5.Here,(92)η12=η13=η14=η23=η24=η34=1,Υi=x+νiy−ϕ1c1νi2t,i=1,2,3,4,Ξ5=κ1x+γ5y−ϕ1−5c12γ32κ12+5c1γ3κ14+κ16κ1t.Takingγ1=1/3,σ1=3/2,κ1=2/3,γ2=−1/3,σ2=2,γ3=1, the conformation of graphs at different times makes expressiveness and is shown in Figure 28.Figure 23
Plots of equation (86) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−15. (b) t=0. (c) t=15. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 24
Plots of equation (86) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−15. (b) t=0. (c) t=15. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 25
Plots of equation (86) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−30. (b) t=0. (c) t=30. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 26
Plots of equation (89) by considering ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−150. (b) t=0. (c) t=150. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 27
Plots of equation (89) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−30. (b) t=0. (c) t=10. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 28
Plots of equation (91) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (f1, f4) t=−30. (f2, f5) t=0. (f3, f6) t=30. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)It is remarkable to detect that the found N-lump solutions of the above equation are general, and for the individual values of the involved parameters, some exacting solutions available in the proceeding literature are new and do not develop in the former study.


## 5.1. 1-Breather and 1-Lump

By takingN=2 in equation (52), we get(55)f=1+eΛ1+eΛ2+S12eΛ1+Λ2.We have the following cases.Option I: by consideringa1=a2=ρ1,b1=b2∗=ϱ1+iτ1,ι01=ι02=0, where i2=−1, equation (55) will be(56)f=1+S12e2ξ1+2eξ1cosτ1y−δ1t.Here,(57)S12=c12τ12−3ρ16−3c1ρ13ϱ1+c12τ12,ξ1=θ1t+ρ1x+ϱ1y,(58)θ1=ϕ1ρ16−5c1ρ13ϱ1+c1τ12−c1ϱ12ρ1,δ1=−ϕ1−5ρ13+10c1ϱ1c1τ1ρ1.By consideringτ1=1/5,ρ1=1/3 and value of ϱ1=−1/3, three cases of breathers can be obtained, as shown in Figure 10, which are x,y -periodic breathers.Choice II: via consideringa1=l1ε,a2=l2ε,b1=ν1a1,b2=ν2a2,ι01=ι02∗=iπ in equation (55) and letting limit ε⟶0, fx,y,t can be written as(59)fx,y,t=η12+Υ1Υ2l1l2ε2+Oε3.Here,(60)η12=1,Υi=x+νiy+5ϕ1c12mi2t,i=1.2.Putting equation (59) into equation (9) and considering ν1=ν2=γ1+iσ1, the two-soliton transforms into the following, equation (5), as(61)u=2c0e−∫ϕ9tdtν1+ν2η12+Υ1Υ2−Υ1+Υ2ν2Υ1+ν1Υ2η12+Υ1Υ22,With the following conditions:(62)∂u∂x=0,∂u∂y=0.Hence, we obtain(63)x1,y1=5tc12γ12ϕ1+5tc12ϕ1σ12,−10tc12γ1ϕ1,x2,3,y2,3=5tc12γ12ϕ1+5tc12ϕ1σ12±3,−10tc12γ1ϕ1.At the pointx1,y1, the second-order derivative and Hessian matrix is(64)CaseA:Γ1=∂2∂x2ux,y,t|x1,y1=−24c0e−∫ϕ9tdt,Δ1=det∂2∂x2ux,y,t∂2∂x∂yux,y,t∂2∂x∂yux,y,t∂2∂y2ux,y,tx1,y1=192c02e−∫ϕ9tdt2σ12,(65)CaseB:Γ2=∂2∂x2ux,y,t|x2,3,y2,3=34c0e−∫ϕ9tdt,Δ2=det∂2∂x2ux,y,t∂2∂x∂yux,y,t∂2∂x∂yux,y,t∂2∂y2ux,y,tx2,3,y2,3=34c02e−∫ϕ9tdt2σ12.In case A, ifc0<0 and Δ1>0, then the point x1,y1 is the extreme value. Also, the point x1,y1 is a maximum amount with measure umax=4c0e−∫ϕ9tdt and is shown in Figure 11. In case B, if c0>0 and Δ1>0, then the point x2,3,y2,3 is extreme amount. Therefore, the point x2,3,y2,3 is a maximum amount with measure umax=−1/2c0e−∫ϕ9tdt. The related plots are designed with amounts of parameters, such as γ1=1,σ1=1/2 for (f1, f4), γ1=−1,σ1=1/2 for (f2, f5), and γ1=1/3,σ1=1/5 for (f3, f6).Set III: through consideringa1=a2∗=δ1+iτ1,c1=c2∗=ϱ°1+iρ°1,ν01=ν02=0 and values ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10 in (55), many of localized waves in (2 + 1)-D obtained are shown in Figure 12. Figures 12(a) and 12(c) show that the output u are t-periodic breathers, while Figure 12(b) shows that the output u are x-periodic breathers. Also, Figure 12 shows the periodic breather outputs. The related plots are plotted with amounts of parameters, such as δ1=0,τ1=1,α1=1,β1=1 for (f1, f4), δ1=1/3,τ1=1,α1=0,β1=1 for (f2, f5), and δ1=1/3,τ1=1,α1=1,β1=1 for (f3, f6).Figure 10
Plots of equation (55) by taking ϱ1=2/3,τ1=2/5,ρ1=−2/3,ϕ1t=2,ϕ9t=cos6t+3,c0=1,c1=2,t=−2. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)Figure 11
Plots of equation (61) by taking ϱ1=2/3,τ1=2/5,ρ1=−2/3,ϕ1t=2,ϕ9t=cos6t+3,c0=1,c1=2,t=2. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)Figure 12
Plots of equation (61) by taking λ1=2,ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10,t=−2. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)


## 5.2. Interaction 1-Bell-Shaped Soliton with 1-Lump or 1-Breather

By consideringN=3, the following function is taken:(66)f=1+eΛ1+eΛ2+eΛ3+Σ12eΛ1+Λ2+S13eΛ1+Λ3+S23eΛ2+Λ3+S12S13S23eΛ1+Λ2+Λ3.Some special cases are as follows.Set I: via supposinga1=a2=ρ1,a3=ρ2,b1=b2∗=ϱ1+iτ1,b3=ϱ2,ι01=ι02=ι03=0, equation (66) can be written as follows:(67)fx,y,t=1+S12e2Ξ1+2eΞ1cosΞ2+eΞ31+2eΞ1S13cosΞ2+ϕS13+S12S132e2Ξ1.In whichθ1,Υ1, and Σ12 are stated in equation (57) and have the following relations:(68)Ξ1=tθ1+xρ1+yϱ1,Ξ2=δ1t−τ1y,Ξ3=ω3t+ρ2x+ϱ2y,S13=S13eiϕS13,(69)θ1=ϕ1−ρ16−5c1ρ13ϱ1−5c12τ12+5c12ϱ12ρ1,δ1=ϕ15ic1ρ13τ1−10ic12τ1ϱ1ρ1,(70)ω3=ϕ1−ρ26−5c1ρ23ϱ2+5c12ϱ22ρ2,S23=S13∗=S13e−iϕS13.By consideringτ1=3,ρ1=1,ρ2=1,ϕ1t=t2,ϕ9t=sint+3 and various values of ϱ1 and ϱ2, the dynamical behaviors of related solutions obtained are shown in Figure 13. Also, Figure 13 shows that the solution contains kinds of behavior properties of solutions with diverse values of ϱ1 and ϱ2. The related plots are plotted with amounts of parameters like ϱ1=−1,ϱ2=1,τ1=3,ρ1=1,ρ2=1 for (f1, f4), ϱ1=−2,ϱ2=1,τ1=3,ρ1=1,ρ2=1 for (f2, f5), and ϱ1=−2,ϱ2=2,τ1=3,ρ1=1,ρ2=1 for (f3, f6).Set II: by takinga1=l1ε,a2=l2ε,bi=νiai,i=1,2,3,ι01=ι02∗=iπ,ι03=0 in equation (55) and ε⟶0, fx,y,t one becomes(71)fx,y,t=η12+Υ1Υ2l1l2ε2+Υ1Υ2+η23Υ1+η13Υ2+η12+η13η23eΞ3l1l2ε2+Oε3.Here,(72)η12=η13=η23=1,Υi=x+νiy+εit,i=1,2,ε1=λ4σ12−2iγ1λ4σ1−γ12λ4,(73)ε2=2iγ1λ4σ1−γ12λ4+λ4σ12,η3=κ1x+γ2y−κ1κ14λ3+γ2κ12λ5+γ22λ4+κ12λ1t.Substituting equation (73) into equation (67), also consider ν1=ν2=ϱ1+iτ1,ν3=ϱ2. The solution u obtained by equation (68) will transform to the behavior output between one-lump and one-breather soliton, whose dynamical property at different times is found in Figure 14. Also, from Figure 14, the related graphs are plotted with amounts of various times like t=−40 for (f1), t=−20 for (f2), t=0 for (f3), t=20 for (f4), and t=40 for (f5). Also, from Figure 15, the related graphs are plotted with amounts of various times like t=−40 for (f1), t=−20 for (f2), t=0 for (f3), t=20 for (f4), and t=40 for (f5). Moreover, from Figure 16, the related graphs are plotted with amounts of various times like t=−40 for (f1, f6), t=−20 for (f2, f7), t=0 for (f3, f8), t=20 for (f4, f9), and t=40 for (f5, f10).Set III: considera1=a2∗=δ1+iτ1,a3=δ2,b1=b2∗=ϱ1+iρ1,b3=ϱ2,ι01=ι02=ι03=0, in equation (66). Figure 17(a) shows the cross-behavior between one-breather and one-breather-shaped soliton with values δ1=1/3,σ1=−1,δ2=1,α1=1,β1=1 in x,y-plane. Figure 17(b) shows the cross-behavior between one-breather and one-breather-shaped soliton with values δ1=0,σ1=−1,δ2=−4,α1=1,β1=1 in x,y-plane. Therefore, Figure 17(c) shows the cross-behavior between one-breather and one-breather-shaped soliton with values δ1=−1/3,σ1=−1,δ2=1,α1=1,β1=1 in x,y-plane.Figure 13
Plots of equation (67) by using c0=1,c1=1/10,t=−2. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 14
Plots of equation (71) by using γ1=1,σ1=1,κ1=1/3,σ2=1,γ2=1/5,ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5.
(a)(b)(c)(d)(e)Figure 15
Plots of equation (71) by using γ1=1,σ1=1,κ1=1/3,σ2=1,γ2=1/5,ϕ1t=t+1/100,ϕ9t=sint+3,c0=1,c1=1/10. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5.
(a)(b)(c)(d)(e)Figure 16
Plots of equation (71) by using γ1=1,σ1=1,κ1=1/3,σ2=1,γ2=1/5,ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6. (g) f7. (h) f8. (i) f9. (j) f10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 17
Plots of equation (71) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10,t=−2. (a) f1. (b) f2. (c) f3.
(a)(b)(c)


## 5.3. The Abundant Solutions WhenN=4

WithN=4, the following function is considered:(74)fx,y,t=1+eΛ1+eΛ2+eΛ3+eΛ4+S12eΛ1+Λ2+S13eΛ1+Λ3+S14eΛ1+Λ4+S23eΛ2+Λ3+S24eΛ2Λ4+S34eΛ3+Λ4+S12S13S23eΛ1+Λ2+Λ3+S12S14S24eΛ1+Λ2+Λ4+S13S14S34eΛ1+Λ3+Λ4+S23S24S34eΛ2+Λ3+Λ4+S12S13S14S23S24S34eΛ1+Λ2+Λ3+Λ4.The cases are as follows.Option I: via supposinga1=a2=ρ1,a3=a4=ρ2,b1=b2∗=ϱ1+iτ1,b3=b4∗=ϱ2+iτ2,ι01=ι02=ι03=ι04=0, four-soliton solution can be written as follows:(75)fx,y,t=1+S12e2Ξ1+S34e2Ξ2+2eΞ1cos∅1+2eΞ2cos∅2+2S13eΞ1+Ξ2cos∅1+∅2+ϕS13+2S23eΞ1+Ξ2cos∅1−∅2−ϕS23+2S12e2Ξ1+Ξ2cos∅2+ϕS13+ϕS23+2S34eΞ1+2Ξ2cos∅1+ϕS13−ϕS23+S12S34S132S232e2Ξ1+2Ξ2.Here,θ1,Ω1, and S12 are taken in equation (57), and the following relations will be reached:(76)Ξi=tAi+xδi+yαi,∅i=tBi+yβi,Ai=ϕ1−δi6−5αic1δi3+5αi2c12−5βi2c12δi,i=1,2,(77)Bi=ϕ1−5βic1δi3+10αiβic12δi,i=1,2,S12=−β12c123δ16+3α1c1δ13−β12c12,(78)S34=−c12β223δ26+3α2c1δ13−c12β22,S14=S23∗,S24=S13∗,S13=ℑ1+iℑ2ℵ1+iℵ2=S13eiϕS13,S23=ℑ3+iℑ4ℵ3+iℵ4=S23eiϕS23,ℑ1=δ12δ22δ12−δ1δ2+δ22−δ2+δ12+c1δ1δ2−δ2+δ12α1δ1δ2−α1δ22+α2δ12−2α2δ1δ2+c12α1δ2−α2δ12−β1δ2−β2δ12,ℑ2=c1δ1δ2−δ2+δ12β1δ1δ2−β1δ22+β2δ12−2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℵ1=δ12δ22δ12+δ1δ2+δ22δ2+δ12+c1δ1δ2δ2+δ12α1δ1δ2+α1δ22+α2δ12+2α2δ1δ2+c12α1δ2−α2δ12−β1δ2−β2δ12,ℵ2=c1δ1δ2δ2+δ12β1δ1δ2+β1δ22+β2δ12+2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℑ3=δ12δ22δ12−δ1δ2+δ22−δ2+δ12+c1δ1δ2−δ2+δ12α1δ1δ2−α1δ22+α2δ12−2α2δ1δ2+c12α1δ2−α2δ1−β1δ2−β2δ12,ℑ4=c1δ1δ2−δ2+δ12β1δ1δ2−β1δ22+β2δ12−2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℵ3=δ12δ22δ12+δ1δ2+δ22δ2+δ12+c1δ1δ2δ2+δ12α1δ1δ2+α1δ22+α2δ12+2α2δ1δ2+c12α1δ2−α2δ12−β1δ2−β2δ12,ℵ4=c1δ1δ2δ2+δ12β1δ1δ2+β1δ22+β2δ12+2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1.Take(79)δ1=13,δ2=−23,α1=15,α2=12,β1=25,β2=12,t=−10,δ1=13,δ2=−23,α1=15,α2=12,β1=0,β2=13,t=−5,δ1=13,δ2=23,α1=0,α2=−38,β1=27,β2=18,t=0,δ1=13,δ2=23,α1=0,α2=−38,β1=27,β2=18,t=5,And(80)δ1=13,δ2=23,α1=0,α2=−38,β1=27,β2=18,t=10.For Figure18(a)t=−10, Figure 18(b)t=−5, Figure 18(c)t=0, Figure 18(d)t=5, and (f5) t=10. Also, Figure 18 shows two y-periodic breather.Option II: by supposinga1=l1ε,a2=l2ε,bi=νiai,i=1,…,4,ι01=ι02∗=iπ,ι03=ι04=0 in equation (74), and ε⟶0, fx,y,t is given as(81)f=η12+Υ1Υ2l1l2ε2+Υ1Υ2+η23Υ1+η13Υ2+η12+η13η23eΞ3l1l2ε2+Υ1Υ2+η24Υ1+η14Υ2+η12+η14η24eΞ4l1l2ε2+η34Υ1Υ2+η23+η24Υ1+η13+η14Υ2+η12+η13+η14η23+η24eΞ3+Ξ4l1l2ε2+Oε3.Here,(82)η12=η13=η14=η23=η24=1,η34=c12σ22c12σ22−3c1γ2κ12−3κ14,Υl=x+νly+ωlt,l=1,2,ω1=−ϕ1−10ic12γ1σ1−5c12γ12+5c12σ12,ω2=ϕ1−5c12σ12−10ic12γ1σ1+5c12γ12,Ξ3=κ1x+κ1ν3y+ϕ1κ1−5c12σ22+10ic12γ2σ2−5ic1κ12σ2+5c12γ22−5c1γ2κ12−κ14t,Ξ4=κ1x+κ1ν4y−ϕ1κ110ic12γ2σ2−5ic1κ12σ2−5c12γ22+5c12σ22+5c1γ2κ12+κ14t.Set II-A: pluggingν1=ν2∗=ϱ1+iσ1,a3=a4=ρ1,ν3=ν4∗=ϱ2+iσ2, the solution u is given by equation (81), and choosing values γ1=1,σ1=1,γ2=0,σ2=3/5,κ1=2/3, the dynamical properties at different times are shown in Figure 19.Option II-B: consideringν1=ν2∗=ϱ1+iσ1,a3=a4=ρ1,ν3=ϱ2,ν4=ϱ3, the solution u given by equation (81) shows the dynamical property solitons with related plots. Putting γ1=1,σ1=1,γ2=1,γ3=−2,κ1=2/3, the dynamical behaviors at different times are shown in Figure 20.Option III: takingai=liε,bi=aiνi,i=1,…,4,ι01=ι02∗=ι03=ι04∗=iπ in equation (81) and utilizing limit as ε⟶0 transforms(83)fx,y,t=Υ1Υ2Υ3Υ4+η34Υ1Υ2+η24Υ1Υ3+η23Υ1Υ4+η14Υ2Υ3+η13Υ2Υ4+η12Υ3Υ4+η12η34+η13η24+η14η23l1l2l3l4ε4+Oε5.Here,(84)η12=η13=η14=η23=η24=η34=1,Υi=x+νiy+5ϕ1c12νi2t,i=1,2,3,4.By consideringν1=ν2∗=ϱ1+iσ1,ν3=ν4∗=ϱ2+iσ2 and by getting γ1=0,γ2=0,σ1=1,σ2=2, the structural behaviors obtained are shown in Figure 21. Moreover, substituting γ1=1/3,γ2=−2/3,σ1=1,σ2=2, the structural behaviors obtained are shown in Figure 22 with related plots.Figure 18
Plots of equation (75) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5.
(a)(b)(c)(d)(e)Figure 19
Plots of equation (81) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−150. (b) t=0. (c) t=150. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 20
Plots of equation (81) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a), (d) t=−100. (b), (e) t=0. (c), (f) t=100. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)Figure 21
Plots of equation (83) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a), (d) t=−50. (b), (e) t=0. (c), (f) t=50. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)Figure 22
Plots of equation (83) by utilizing ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10, (a), (d) t=−50. (b), (e) t=0. (c), (f) t=50. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)


## 5.4. The Dynamical Behavior Solutions WhenN=5

With gettingN=5, the following function will be arisen:(85)fx,y,t=1+eΛ1+eΛ2+eΛ3+eΛ4+eΛ5+S12eΛ1+Λ2+S13eΛ1+Λ3+S14eΛ1+Λ4+S15eΛ1+Λ5+S23eΛ2+Λ3+S24eΛ2+Λ4+S25eΛ2+Λ5+S34eΛ3+Λ4+S35eΛ3+Λ5+S45eΛ4+Λ5+S12S13S23eΛ1+Λ2+Λ3+S12S14S24eΛ1+Λ2+Λ4+S12S15S25eΛ1+Λ2+Λ5+S13S14S34eΛ1+Λ3+Λ4+S13S15S35eΛ1+Λ3+Λ5+S14S15S45eΛ1+Λ4+Λ5+S23S24S34eΛ2+Λ3+Λ4+S23S25S35eΛ2+Λ3+Λ5+S24S25S45eΛ2+Λ4+Λ5+S34S35S45eΛ3+Λ4+Λ5+S12S13S14S23S24S34eΛ1+Λ2+Λ3+Λ4+S12S13S15S23S25S35eΛ1+Λ2+Λ3+Λ5+S12S14S15S24S25S45eΛ1+Λ2+Λ4+Λ5+S13S14S15S34S35S45eΛ1+Λ3+Λ4+Λ5+S23S24S25S34S35S45eΛ2+Λ3+Λ4+Λ5+S12S13S14S15S23S24S25S34S35S45eΛ1+Λ2+Λ3+Λ4+Λ5.The following cases will be considered.Option I: by supposinga1=a2=ρ1,a3=a4=ρ2,b1=b2∗=ϱ1+iτ1,b3=b4∗=ϱ2+iτ2,a5=ρ3,b5=ϱ3,ι01=ι02=ι03=ι04=ι05=0, therewith, 5-soliton becomes(86)f=1+S12e2Ξ1+S34e2Ξ2+2eΞ1cos∅1+2eΞ2cos∅2+eΞ3+eΞ1+Ξ2(2S13cos∅1+∅2+ϕS13+2S23cos∅1−∅2−ϕS23+2S12S13‖S23eΞ1cosη2+ϕS13+ϕS23+2S34S13‖S23eΞ1cosη1+ϕS13−ϕS23+S12S34S132S232eΞ1+Ξ2)+eΞ1+Ξ3S12S152eΞ1+2S15cos∅1+ϕS15+eΞ2+Ξ3S34S352eΞ2+2S35cos∅2+ϕS35+S15‖S35eΞ1+Ξ2+Ξ3(2S13cos∅1+∅2+ϕS13+ϕS15+ϕS35+2S23cos∅1−∅2−ϕS23+ϕS15−ϕS35+2S12S13S23S15eΞ1cosη2+ϕS13+ϕS23+ϕS35+2S34S13S23S35eΞ2cos∅1+ϕS13−ϕS23+ϕS15+S12S34S132S232S15‖S35eΞ1+Ξ2),In whichA1,A2,∅1,∅2,Ξ1,Ξ2, and Sij1≤i<j≤4 are given in equations (76)–(78), and the following relations can be received:(87)Ξi=tAi+xδi+yαi,Ξ3=xδ3+yα3+ϕ1−δ36−5α3c1δ33+5α32c12δ3t,∅i=tBi+yβi,i=1,2,Ai=ϕ1−δi6−5αic1δi3+5αi2c12−5βi2c12δi,Bi=ϕ1−5βic1δi3+10αiβic12δi,i=1,2,S25=S15∗,S45=S35∗,S15=ℑ1+iℑ2ℵ1+iℵ2=S15eiϕS15,S23=ℑ3+iℑ4ℵ3+iℵ4=S23eiϕS23,(88)ℑ1=δ12δ22δ12−δ1δ2+δ22−δ2+δ12+c1δ1δ2−δ2+δ12α1δ1δ2−α1δ22+α2δ12−2α2δ1δ2+c12α1\δ2−α2δ12−β1δ2−β2δ12,ℑ2=c1δ1δ2−δ2+δ12β1δ1δ2−β1δ22+β2δ12−2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℵ1=δ12δ22δ12+δ1δ2+δ12δ2+δ12+c1δ1δ2δ2+δ12α1δ1δ2+α1δ22+α2δ12+2α2δ1δ2+c12α1δ2−α2δ12−β1δ2−β2δ12,ℵ2=c1δ1δ2δ2+δ12β1δ1δ2+β1δ22+β2δ12+2β2δ1δ2+2c1β1δ2−β2δ1α1δ2−α2δ1,ℑ3=−δ12δ22δ12−δ1δ2+δ22−δ2+δ12−c1δ1δ2−δ2+δ12α1δ1δ2−α1δ22+α2δ12−2α2δ1δ2−c12α1δ2−α2δ12−β1δ2+β2δ12,ℑ4=c1δ1δ2−δ2+δ12β1δ1δ2−β1δ22−β2δ12+2β2δ1δ2+2c1β1δ2+β2δ1α1δ2−α2δ1,ℵ3=−δ12δ22δ12+δ1δ2+δ22δ2+δ12−c1δ1δ2δ2+δ12α1δ1δ2+α1δ22+α2δ12+2α2δ1δ2−c12α1δ2−α2δ12−β1δ2+β2δ12,ℵ4=c1δ1δ2δ2+δ12β1δ1δ2+β1δ22−β2δ12−2β2δ1δ2+2c1β1δ2+β2δ1α1δ2−α2δ1.Option I-A: by consideringδ1=1/3,δ2=2/3,δ3=1,α1=1/5α2=0,α3=1/3,β1=2/5,β2=1/2, u given by equation (86) returns the structural properties of solutions as shown in Figure 23 with density plot.Option I-B: by puttingδ1=1/3,δ2=−2/3,δ3=1,α1=0,α2=0,α3=1/2,β1=2/5,β2=1/2 in equation (86), the structural property among two parallel y-periodic of solutions obtained is shown in Figure 24 with density plot.Option I-C: via substitutingδ1=1/3,δ2=2/3,δ3=1,α1=0,α2=−3/8,α3=1/2,β1=2/5,β2=1/8i in equation (86), we can get the properties y-periodic of solutions as shown in Figure 25 with density plot.Set II: consideringbi=aiνi,i=1,…,5,a1=l1ε,a2=l2ε,ι01=ι02∗=iπ,ι03=ι04=ι04=0 in equation (86) and getting limit as ε⟶0, the 5-soliton one becomes(89)f=η12+Υ1Υ2l1l2ε2+Υ1Υ2+η23Υ1+η13Υ2+η12+η13η23eΞ3l1l2ε2+Υ1Υ2+η24Υ1+η14Υ2+η12+η14η24eΞ4l1l2ε2+Υ1Υ2+η25Υ1+η15Υ2+η12+η15η25eΞ5l1l2ε2+η34Υ1Υ2+η23+η24Υ1+η13+η14Υ2+η12+η13+η14η23+η24eΞ3+Ξ4l1l2ε2+Oε3+η35Υ1Υ2+η23+η25Υ1+η13+η15Υ2+η12+η13+η15η23+η25eΞ3+Ξ5l1l2ε2+Oε3+η45Υ1Υ2+η24+η25Υ1+η14+η15Υ2+η12+η14+η15η24+η25eΞ4+Ξ5l1l2ε2+Oε3+η34η35η45Υ1Υ2+η23+η24+η25Υ1+η13+η14+η15Υ2+η12+η13+η14+η15η23+η24+η25eΞ3+Ξ4+Ξ5l1l2ε2+Oε3.Here,(90)η12=η13=η14=η15=η23=η24=η25=1,ηij=ai2aj2ai2−aiaj+aj2ai−aj2+aiajc1ai−ajai2bj+2aiajbi−2aiajbj−aj2bi+c12aibj−ajbi2ai2aj2ai2+aiaj+aj2ai+aj2+aiajc1ai+ajai2bj+2aiajbi+2aiajbj+aj2bi+c12aibj−ajbi2,3≤i<j≤5,Υi=x+νiy−ϕ1γi2−2iγiσi−σi2t,i=1,2,Ξj=κ1x+κ1γjy+ωjt,ω1=−ϕ1κ16+5κ14γ2+iσ2c1−5κ12γ2+iσ22c12κ1,ω2=−ϕ1κ16+5κ14γ2−iσ2c1−5κ12γ2−iσ22c12κ1,ω3=−ϕ1−5c12γ12κ22+5c1γ3κ24+κ26κ2.Option II-A: insertingν1=ν2∗=ϱ1+iσ1,a3=a4=ρ1,ν3=ν4∗=ϱ2+iσ2,a5=ρ2,ν5=ϱ3, the u received by equation (89) explains the structural property of special solutions. Again, considering γ1=1/3,σ1=2/3,κ1=2/3,γ2=0,σ2=3/5,κ2=1/2,γ3=−1/2, the formation of graphs at different times makes expressiveness and are shown in Figure 26 with density plot.Set II-B: substitutingν1=ν2∗=ϱ1+iσ1,a3=a4=ρ1,ν3=ϱ2,ν4=ϱ3,ν5=ϱ4,a5=ρ2, ux,y,t found by equation (89) transforms the structural property of solutions. Again, considering γ1=1,σ1=2/3,κ1=2/3,γ2=−1/2,γ3=−1/4,κ2=1/2,γ4=1, the formation of graphs at various times makes expressiveness and is shown in Figure 27.Set III: via insertingai=liε,ci=aiνi,i=1,…,4,b5=a5ν5,ι01=ι02∗=ι03=ι04∗=iπ,ι05=0 in equation (85) and providing limit as ε⟶0, the 5-soliton transforms(91)fx,y,t=Υ1Υ2Υ3Υ4+η34Υ1Υ2+η24Υ1Υ3+η23Υ1Υ4+η14Υ2Υ3+η13Υ2Υ4+η12Υ3Υ4+η12η34+η13η24+η14η23)l1l2l3l4ε4+Υ1Υ2Υ3Υ4+η45Υ1Υ2Υ3+η35Υ1Υ2Υ4+η25Υ1Υ3Υ4+η15Υ2Υ3Υ4+Υ1η35η45+η34Υ2+η25η45+η24Υ3+η25η35+η23Υ4+Υ2η15η45+η14Υ3+η15η35+η13Υ4+η15η25+η12Υ3Υ4+Υ1η25η35+η23η45+η24η35+η25η34+Υ2η15η35+η13η45+η14η35+η15η34Υ3η15η25+η12η45+η14η25+η15η24+Υ4η15η25+η12η35+η13η25+σ15η23+η13η24+η23η14+η34η15η25+η12+η35η14η25+η15η24+η45η15η25+η12η35+η13η25+η15η23expΞ5l1l2l3l4ε4+Oε5.Here,(92)η12=η13=η14=η23=η24=η34=1,Υi=x+νiy−ϕ1c1νi2t,i=1,2,3,4,Ξ5=κ1x+γ5y−ϕ1−5c12γ32κ12+5c1γ3κ14+κ16κ1t.Takingγ1=1/3,σ1=3/2,κ1=2/3,γ2=−1/3,σ2=2,γ3=1, the conformation of graphs at different times makes expressiveness and is shown in Figure 28.Figure 23
Plots of equation (86) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−15. (b) t=0. (c) t=15. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 24
Plots of equation (86) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−15. (b) t=0. (c) t=15. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 25
Plots of equation (86) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−30. (b) t=0. (c) t=30. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 26
Plots of equation (89) by considering ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−150. (b) t=0. (c) t=150. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 27
Plots of equation (89) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (a) t=−30. (b) t=0. (c) t=10. (a) f1. (b) f2. (c) f3.
(a)(b)(c)Figure 28
Plots of equation (91) by taking ϕ1t=1,ϕ9t=cost+3,c0=1,c1=1/10. (f1, f4) t=−30. (f2, f5) t=0. (f3, f6) t=30. (a) f1. (b) f2. (c) f3. (d) f4. (e) f5. (f) f6.
(a)(b)(c)(d)(e)(f)It is remarkable to detect that the found N-lump solutions of the above equation are general, and for the individual values of the involved parameters, some exacting solutions available in the proceeding literature are new and do not develop in the former study.


## 6. Conclusion

In this research, we solved the (2 + 1)-D VC Caudrey–Dodd–Gibbon–Kotera–Sawada model and obtained new exact various lump and its interaction solutions. This study shows that the variable-order evolution equations are quite efficient and accurate. Besides, the dynamic properties of these soliton outputs can be accurately explained by equations (28), (38), and (48). The most important thing is that equations (55) and (66), respectively, give specific steps to derive the 1-breather and 1-lump solution and the product 1-bell-shaped soliton with 1-lump or 1-breather solution of a higher-order breather solution. In addition, the method mentioned in this study can also be utilized to convey other integrable systems, such as the Caudrey–Dodd–Gibbon–Kotera–Sawada model. This contribution is effective and seems more natural in the literature. The implementation of the proposed method shows that the scheme is extremely efficient in finding smooth solutions to specified equations. The solution graphs for each solution showed which confirm that the method had good agreement with the exact result of the problem. The proposed method gives solution in the form of exact unction with high accuracy and minimal calculations. Finally, we may conclude that the suggested method is very efficient and accurate, which can be utilized to study any nonlinear problems arising in physical phenomena.



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