# Solitary Wave Solutions of Nonlinear Integro-Partial Differential Equations of2+1-Dimensional and Its Models

Authors: Daba Meshesha Gusu; Shelama Diro
Journal: International Journal of Differential Equations
 (2022)

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## Abstract

The findings indicate an application of a new method of expansion of the formsZ′/Z and 1/Z to determine the solutions for wave of the solitary nature in the 2+1-dimensional modified form for nonlinear integro-partial differential equations. By using this strategy, we acquired solutions of wave which has a solitary nature that have been solved for three different kinds: hyperbolic, trigonometric, and rational functions. As a result, we obtained different forms of solutions which are new, effective, and powerful to illustrate the solitary nature of waves. The physical and geometrical interpretations have been shown using software in 2 and 3-dimensional surfaces. The obtained results have applications in mathematical and applied sciences. It can also solve different nonlinear integro-partial differential equations which have different applications in physical phenomena using this new method. It has many applications to solve the nonlinear nature of the physical world.


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## Body

## 1. Introduction

The real-world models can be illustrated by nonlinear partial differential equations (NLPDEs). They are decisive in phenomena of the world which is full of complexity in areas of applied mathematics, engineering, and physics. Solutions of nonlinear evolution equations (NLEEs) were acclimated to obtain phenomena of nonlinear wave nature. This event is an important and critical issue in occurrence of scientific nature, and hence critical analysis of solutions of NLEEs has a great advantage in understanding nonlinear physical phenomena. In nonlinear phenomena of physical systems, the solutions of NLIPDEs have important applications to describe the natural problems. These include traveling wave of solitary nature and propagation and vibrations in certain restricted space and time. Some scholars have studied the solitary wave solutions and their applications in nonlinear physical phenomena [1–5].Finding the exact solutions of NLIPDEs is not an easy way to determine, and hence certain scholars have constructed different methods to reduce the existing complexity: balancing homogeneous equation [6, 7], transform Darboux method [8, 9], first kind integral [10, 11], and function of tanh [12]. The methods like elliptc function of Jacobi and Kudryshov have been used for the presentation of truncated expansion to analysis nonlinear equations having analytical type solutions. One of the main advantages of determining exact solutions is that it allows scholars to create concrete physical world and innovate more experiments by creating suitable conditions with certain parameters. Hence, analyzing wave solutions of solitary type is a current issue in physical nonlinear systems. But, most of the equations of nonlinear physical phenomena of NLIPDES are easily solvable to illustrate physical models. As a result, many mathematicians have developed different methods. The solutions of NLIPDEs have a great applications in nonlinear sciences in wave dispresion, diffusion, reaction, dissipation, and convection. They provide us more information about the physical nature which can be advanced for better implementaions. These nonlinear integro-differential equations are solved by many different methods such as modification of simple equation [13], equations of auxiliary [14–16], G/G− methods of expansions [17, 18], F methods of expansion [19–21], and 2+1-dimensional space time of Kudryashov method of generalization. The method of f/g expansion for hierarchy equations of an integro differential determine solutions can be transformed to reduce differential nature Different scholars have investigated NLIPDEs to get solutions of such equations in an exact form [30–32]. Solitary wave and compaction type in a generalized form were investigated. Their exact solutions have different in nature: kink, complex, soliton, and solitary. The obtained solutions can also be incorporated in the form of periodic, solitary bell and kink [32, 35–37].As a result, some new exact solutions were obtained including triangular periodic wave solutions, exponential solutions, and complex traveling solutions. TheG˙/G-expansion method was used to construct some new traveling wave solutions including hyperbolic function, trigonometric function, and rational function solutions of the 2+1-dimensional Jaulent–Miodek equation formulated in [38, 39]. The technique factor of integrating to solve 2+1-dimensional space time was applied in [35, 40, 41]. Based on the previous results, the authors believe that these studies will be useful for understanding physical phenomena in many areas of applied mathematics and physics, especially nonlinear systems. Even if, those equations solved by many different methods as far as the researchers go through the existing literature, among those methods, we are concentrated in this article determining the wave solutions of solitary type in 2+1-dimensional space time of NLIPDEs.Generally, in this article, we are going to find the solitary wave solutions of NLIPDEs of2+1-dimensional which is given by(1)ut=uxxxx+5uuxx+53u3+uxyx−5∂x−1uyy+5uuy+5ux∂x−1uy,where ∂x−1=∫−∞xdx.


## 2. Mathematical Formulation and Method

### 2.1. Description of Special Expansion Method

In order to carry out this study successfully, we used two variablesZ′/Z,1/Z. Here, the following notes will be considered to proceed to the next steps.Remark 1.
If we see the second-order linear ordinary differential equation(2)Z″ξ+λZξ=α,(3)and setω=Z′Zandδ=1Z,then we getω′=−ω2+αδ−λ,δ′=−ωδ.Remark 2.
Ifλ<0, then the general solution of equation (2) has the following form:(4)Zξ=M1sinhξ−λ+M2coshξ−λ+αλ,where M1 and M2 are arbitrary constants.(5)Consequently,wehaveδ2=−ω2λ+2αδλ−λ2λ2σ+α2,where σ=M12−M22.Remark 3.
Ifλ>0, then the general solution of equation (2) has the following form:(6)Zξ=M1sinξλ+M2cosξλ+α2,(7)δ2=ω2λ−2αδλ+λ2λ2σ−α2,where σ = M12+M22Remark 4.
Ifλ=0, then the general solution of equation (2) has the following form: Z″+λZ=α, which becomes Z″=α.
Then, integrating both sides with respect toξ, it becomes the following.
Z′=αξ+M1, where M1 is constant of integration.
Again integrating with respect toξ, we get(8)Zξ=α2ξ2+M1ξ+M2,where M1 and M2 are constants. and subsequently, the relation between ω and δ is(9)δ2=ω2−2αδM12−2αM2.
Applying the transformation on equation (1),(10)ux,t=vxx,t,and hence (1) is transformed to(11)vxt=vxxxxxx+5vxvxxxx+53vx3x+vxxxy−5vyy+5vxvxy+5vxxvy.


## 2.1. Description of Special Expansion Method

In order to carry out this study successfully, we used two variablesZ′/Z,1/Z. Here, the following notes will be considered to proceed to the next steps.Remark 1.
If we see the second-order linear ordinary differential equation(2)Z″ξ+λZξ=α,(3)and setω=Z′Zandδ=1Z,then we getω′=−ω2+αδ−λ,δ′=−ωδ.Remark 2.
Ifλ<0, then the general solution of equation (2) has the following form:(4)Zξ=M1sinhξ−λ+M2coshξ−λ+αλ,where M1 and M2 are arbitrary constants.(5)Consequently,wehaveδ2=−ω2λ+2αδλ−λ2λ2σ+α2,where σ=M12−M22.Remark 3.
Ifλ>0, then the general solution of equation (2) has the following form:(6)Zξ=M1sinξλ+M2cosξλ+α2,(7)δ2=ω2λ−2αδλ+λ2λ2σ−α2,where σ = M12+M22Remark 4.
Ifλ=0, then the general solution of equation (2) has the following form: Z″+λZ=α, which becomes Z″=α.
Then, integrating both sides with respect toξ, it becomes the following.
Z′=αξ+M1, where M1 is constant of integration.
Again integrating with respect toξ, we get(8)Zξ=α2ξ2+M1ξ+M2,where M1 and M2 are constants. and subsequently, the relation between ω and δ is(9)δ2=ω2−2αδM12−2αM2.
Applying the transformation on equation (1),(10)ux,t=vxx,t,and hence (1) is transformed to(11)vxt=vxxxxxx+5vxvxxxx+53vx3x+vxxxy−5vyy+5vxvxy+5vxxvy.


## 3. Results and Discussion

The following are the main steps of the special expansion methods by using two variablesZ′/Z,1/Z.Step 1.
First we change the given equation nonlinear partial integro-differential equation to nonlinear partial differential equation.
Suppose we have the following NLPDEs:(12)Zv,vt,vx,vy,vxx,vyy,vxy,vxxx,vyyy,…=0,where Z is a polynomial of unknown function v and its total derivatives with respect to ξ. And its various partial derivatives in which the highest order derivatives and nonlinear terms are involved.Step 2.
We use wave transformation to convert NLPDES to ODEs:(13)vx,y,t=Vξ,ξ=x+y−kt,where k is arbitrary constant. The traveling wave transformation (13) and considering the resulting equation with respect to ξ, (12) transformed to ordinary differential equations.(14)Av,v′,v″,v′″,v″″,v″″′,v″″″,…=0.Step 3.
By assuming solutions of (14) transformed into variables of two polynomials of the form ω and δ,(15)Vξ=∑i=0Fciωi+∑i=1Fbiωi−1δ,where ci (i = 0, 1, 2, 3, …, N) and bi (i = 1, 2, 3, …, N) are arbitrary constants.
Also,ω and δ satisfy the following.(16)ωξ=Z′ξZξandδξ=1Z.Step 4.
Determine the positive integerF in equation (15) by using the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in equation (14). More precisely, if the degree of Vξ is deg Vξ = F, then the degree of other term will be expressed as follows:(17)degdgVξdξg=F+gand degVξhdrVξdξrs=Fh+sF+r,where F= degree highest order of the function, g= the highest order, h= is degree of coefficient of nonlinear term, s= is degree of nonlinear term, and r= is order of nonlinear term.Step 5.
From the above steps, we get the value off and we write the form of solution of equation (14) by equation (15).
Then, substitute (15) into (14) along with (3), and the left-hand side of (14) can be converted into the polynomial in ω and δ in which the degree of δ is no longer than 1.
Equating each coefficient of this polynomial to zero yields a system of algebraic equation which can be solved by using Maple or Mathematica to get the values of each coefficient (ai,bi,ci ) with leading coefficient variable t and the value of constant number k.
We useux,t  =  vxx,t and substitute it into (1) to reduce the NLIPDE (1) to the following NLPDEs of the equivalent to form of (18) as follows:(18)vxt=vxxxxxx+5vxvxxxx+53vx3x+vxxxy−5vyy+5vxvxy+5vxxvy.=0.
From the traveling wave transformation of (13), we have the following.
vx,y,t=Vξ, ξ=x+y−kt, where x,y represent the space variables, t⇒time variable, and k∈R − 0⇒ is the speed of traveling wave, which permits us to transform (12) into ODE:(19)vxxxxxx=dVxxxxxdξ.dξdx=dV″″dξ.dξdx=V″″″.1=V″″″(20)vxxxy=dVxxxdξ.dξdy=dV″′dξ.dξdy=V″″.1=V″″(21)vyy=dVydξ.dξdy=dV′dξ.dξdy=V″.1=V″(22)vxvxy=dVdξ.dξdx.dVxdξ.dξdy=V′.dV′dξ.dξdy,=V′V″.1=V′V″.(23)vxxvy=dVxdξ.dξdx.dVdξ.dξdy=dV′dξ.dξdx.V′=V″V′(24)vxvxxxx=dVdξ.dξdxdVdξ.dξdx.dVxxdξ.dξdx=dVdξ.dξdxV′.dV″dξ.dξdx=V′V″′(25)vx3x=dVdξ.dξdxdV3dξ.dξdx=dVdξ.dξdxV′3=V′3′(26)vxt=dVxdξ.dξdt=dV′dξ.dξdt=V″.−=−kV″(27)vx=dVdξ.dξdx=V′.1=V′.
Depending on (19)–(27), (18) becomes the following ODE forms:(28)−kV″=V″″″+5V′V″′′+53V′3+V″″−5V″+10V′V″,where ′  =  d/dξ. By integrating (28) with respect to ξ and taking the transformation W  =  V′ of equation (28), we obtain(29)W″″+5WW″+53W3+W″+5W2+k−5W+d,where d is the integration constant. Considering the homogeneous balance between the highest order derivative W″ and nonlinear term W3 in (29) by (17), we get the following.
f+2=f×0 + 3f+0 = f+4=3f2f=4f=2.
Consequently, the solution of (29) has the following form:(30)Wξ=∑i=02ciωi+∑i=12biωi−1δ,(31)orWξ=c0+c1ω+b1δ+c2ω2+b2ωδ,where c0,c1,c2,b1,b2 are constants.
To solve for (29), first we have to find W″″,WW″,W3,W″,W2, and W by using (31).
From (31), Wξ=c0+c1ω+b1δ+c2ω2+b2ωδ, and by using (3),(32)W′=c1−ω2+αδ−λ+b1−ωδ+2c2ω−ω2+αδ−λ+b2δ−ω2+αδ−λ+b2ω−ωδ.
Finally, substituting (3) in place of ω′ and δ′ in equation (31), we get the following equation.(33)W′=−c1ω2+c1αδ−c1λ−b1ωδ−2c2ω3+2c2αωδ−2c2λω−b2δω2+b2αδ2−b2λδ−b2ω2δ.
From (33), we can get W″ equal to the following results:(34)W″=2c1ω3−2c1αωδ+2c1λω−c1αωδ+b1ω2δ−b1αδ2+b1λδ+b1ω2δ+6c2ω4−6c2αω2δ+6c2λω2−2c2αω2δ−2c2αω2δ+2c2α2δ2−2c2αλδ+2c2λω2−2c2λαδ+2c2λ2+b2ω3δ+2b2δω3−2b2αδ2ω+2b2λωδ−2b2αδ2ω+b2λωδ+2b2ω3δ−2b2αωδ2+2b2λωδ+b2ω3δ.
From (33) and second partial derivatives of (31), by substituting into second partial derivative of equations in place of w’ as follows and finally adding according to the same power coefficient:(35)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αδ2−6b2αωδ2+2c2α2δ2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.
From (35), W′′′ is equivalent to(36)W′′′=24c2ω5+24c2αω3δ−24c2λω3−6c1ω4+6c1αω2δ−6c1λω2−18b2ω4δ+18b2αω2δ2−18b2λω2δ−6b2ω4δ−4b1ω3δ+4b1αωδ2−4b1λωδ−2b1ω3δ+20c2αω3δ−20c2α2ωδ2+20c2λαωδ+10c2αω3δ−16c2λω3+16c2λαωδ−16c2λ2ω+2b1αδ2ω+6b2αδ2ω2−6b2α2δ3+6b2αλδ2+12b2αδ2ω2−b1λωδ−5b2λω2δ+5b2λαδ2−5b2λ2δ−5b2λω2δ+3c1αω2δ−3c1α2δ2−4c2α2δ2ω+3c1αλδ+3c1αω2δ−2c1λω2+2c1λαδ−2c1λ2+4c2αλωδ.
From (35) and third partial derivatives of (31) with substitute (3) into third partial derivatives of equation in place of ω′ and δ′ equation (31) equivalent form of (34) as follows and finally adding according to the same power coefficient:(37)W″″=−24c2ω5−6c1ω4−24b2ω4δ+54c2αω3δ−40c2λω3−6b1ω3δ−6b2α2δ3+12c1αω2δ−8c1λω2+36b2αω2δ2−28b2λω2δ+6b1αδ2ω−24c2α2δ2ω+11b2αλδ2−3c1α2δ2−5b1λωδ+40c2λαωδ−16c2λ2ω−5b2λ2δ+5c1λαδ−2c1λ2.
From (37), W″″ is equivalent to(38)W″″=−120c2ω4−ω2+αδ−λ−24c1ω3−ω2+αδ−λ−96b2ω3δ−ω2+αδ−λ−24b2ω4−ωδ+162c2αω2δ−ω2+αδ−λ+54c2αω3−ωδ−120c2λω2−ω2+αδ−λ−6b1ω3−ωδ−18b1ω2δ−ω2+αδ−λ−18b2α2δ2−ωδ+24c1αωδ−ω2+αδ−λ+12c1αω2−ωδ−16c1λω−ω2+αδ−λ+72b2αωδ2−ω2+αδ−λ+72b2αω2δ−ωδ−28b2λω2−ωδ−56b2λωδ−ω2+αδ−λ+12b1αωδ−ωδ+6b1αδ2−ω2+αδ−λ−48c2α2ωδ−ωδ−24c2α2δ2−ω2+αδ−λ+22b2αλδ−ωδ−6c1α2δ−ωδ−5b1λδ−ω2+αδ−λ−5b1λω−ωδ+40c2λαδ−ω2+αδ−λ+40c2λαω−ωδ−16c2λ2−ω2+αδ−λ−5b2λ2−ωδ+5c1λα−ωδ=120c2ω6−120c2αω4δ+120c2λω4+24c1ω5−24c1αω3δ+24c1λω3+96b2ω5δ−96b2αω3δ2+96b2λω3δ+24b2ω5δ−162c2αω4δ+162c2α2ω2δ2−162c2λαω2δ−54c2αω4δ+120c2λω4−120c2λαω2δ+120c2λ2ω2+18b1ω4δ−18b1αω2δ2+18b1λω2δ+6b1ω4δ+18b2α2δ3ω−24c1αω3δ+24c1α2ωδ2−24c1λαωδ−12c1αω3δ+16c1λω3−16c1λαωδ+16c1λ2ω−72b2αω3δ2+72b2α2ωδ3−72b2αλωδ2−72b2αω3δ2+56b2λω3δ−56b2λαωδ2+56b2λ2ωδ+28b2λω3δ−12b1αδ2ω2−6b1αδ2ω2+6b1α2δ3−6b1λαδ2+48c2α2δ2ω2+24c2α2δ2ω2−24c2α3δ3+24c2α2λδ2−22b2αλδ2ω+6c1α2δ2ω+5b1λω2δ−5b1λαδ2+5b1λ2δ+5b1λω2δ−40c2λαδω2+40c2λα2δ2−40c2λ2αδ−40c2λαω2δ+16c2λ2ω2−16c2λ2αδ+16c2λ3+5b2λ2ωδ−5c1λαωδ.
From (37) and fourth partial derivatives of (31) with substitute (3) into fourth partial derivatives of equation in place of ω′ and δ′ equation (31) equivalent to form of (35) as follows and finally adding according to the same power coefficient:(39)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αω3δ2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2ω2δ2−362c2λαω2δ+136c2λ2ω2−36b1αω2δ2+28b1λω2δ+30c1α2ωδ2−150b2αλδ2ω−11b1λαδ2+64c2λα2δ2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ−56c2λ2αδ+16c2λ3.
When (31) and (33) are multiplied with each other, W×W″ is equivalent to(40)W×W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αδ2−6b2αωδ2+2c2α2δ2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2×c0+c1ω+b1δ+c2ω2+ωδ.
From (31) and (35), when they multiply each other, the result of them has equivalent form of (41) as follows according to the same power coefficient:(41)W×W″=6c22ω6+8c1c2ω5+12c2b2ω5δ+6c0c2ω4+2c12ω4+8b2c1ω4δ+8b1c2ω4δ−10c22αω4δ+8c22λω4+6b22ω4δ2+2c0c1ω3+6c0b2ω3δ+4b1c1ω3δ−10c1c2αω3δ+10c1c2λω3+8b1b2ω3δ2−16c2b2αω3δ2+13c2b2λω3δ−b2αδ3−7b2b1αωδ3+2b1c2α2δ3−6b22αω2δ3+2b2c2α2δ3ω+2c0b1ω2δ−10c0c2αω2δ+8c0c2λω2−9c1b2αω2δ2+7c1b2λω2δ−3c12αω2δ+2c12λω2+2b12ω2δ2−10b1c2αω2δ2+8b1c2λω2δ−c2b1αω2δ2+2c22α2ω2δ2+c2b1λω2δ+2c22λ2ω2−4c22αλω2δ+5b22λω2δ2−c0b1αδ2−6c0b2αωδ2+2c0c2α2δ2−4c1b1αωδ2+2c1c2α2δ2ω+b12λδ2+6b1b2λωδ2−4b1c2αλδ2−4c2b2αλδ2ω+5c0b2λωδ−3c0c1αωδ−4c1c2αλωδ+c1b1λωδ+2c2λ2ω+2b2c2λ2ωδ+c0b1λδ+2c0c1λω−4c0c2αλδ+2b1c2λ2δ+2c0c2λ2.
From (31), when multiplying two times itself by the form of W×W  =  W2, then(42)W2=W×W=c0+c1ω+b1δ+c2ω2+ωδ×c0+c1ω+b1δ+c2ω2+ωδ.
Finally,W2 are as follows:(43)W2=c22ω4+2c1c2ω3+2b2c2ω3δ+2c0c2ω2+c12ω2+b12δ2+2c2b1δω2+2c1b2ω2δ+2b1b2δ2ω+b22ω2δ2+2c0b1δ+2c0c1ω+2c0b2ωδ+2b1c1ωδ+c02.
From (31) and (43),(44)W3=W2×W=c22ω4+2c1c2ω3+2b2c2ω3δ+2c0c2ω2+c12ω2+b12δ2+2c2b1δω2+2c1b2ω2δ+2b1b2δ2ω+b22ω2δ2+2c0b1δ+2c0c1ω+2c0b2ωδ+2b1c1ωδ+c02×+2b1b2δ2ω+b22ω2δ2+2c0b1δ+2c0c1ω+2c0b2ωδ+2b1c1ωδ+c02.
The product of (31) and (43) is the following:(45)W3=c23ω6+3c1c22ω5+3b2c22ω5δ+3c0c22ω4+3c2c12ω4+6c1c2b2ω4δ+4b1c22ω4δ+3b22c2ω4δ2+6c0c1c2ω3+6c0c2b2ω3δ+c13ω3+6c2c1b1ω3δ+3b2c12ω3δ+3b22c1ω3δ2+6b2c2b1ω3δ2+b23ω3δ3+b13δ3+3b2b12δ3ω+3b1b22δ3ω2+3c02c2ω2+3c0c12ω2+6c0c2b1ω2δ+6c0c1b2ω2δ+3c0b22ω2δ2+6c1b1b2ω2δ2+3b1c12ω2δ+3c2b12ω2δ2+3c0b12δ2+6c0b1b2δ2ω+3b12c1ωδ2+3c02b1δ+3c02c1ω+3c02b2ωδ+6c0b1c1ωδ+c03.
Also, we havek−5W=kc0+kc1ω+kb1δ+kc2ω2+kb2ωδ−5c0−5c1ω−5b1δ−5c2ω2−5b2ωδ.


### 3.1. Case A:λ<0—Nature of Hyperbolic Solutions

By using (3), equation W′′′′ is equivalent to(46)W′′′′=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αω3−ω2λ+2αδλ−λ2λ2σ+α2+16c2λ3+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2ω2−ω2λ+2αδλ−λ2λ2σ+α2−362c2λαω2δ+136c2λ2ω2−56c2λ2αδ−36b1αω2−ω2λ+2αδλ−λ2λ2σ+α2+28b1λω2δ+30c1α2ω−ω2λ+2αδλ−λ2λ2σ+α2−150b2αλ−ω2λ+2αδλ−λ2λ2σ+α2ω−11b1λα−ω2λ+2αδλ−λ2λ2σ+α2+64c2λα2−ω2λ+2αδλ−λ2λ2σ+α2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ.Finally, by using equation (3), equation (35) becomes(47)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3+240b2αλω5λ2σ+α2−480b2α2λω3δλ2σ+α2+240b2αλ2ω3λ2σ+α2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3−234c2α2λω4λ2σ+α2+468c2α3λω2δλ2σ+α2−234c2α2λ2ω2λ2σ+α2−362c2λαω2δ+136c2λ2ω2+36b1αλω4λ2σ+α2−56c2λ2αδ−72b1α2λω2δλ2σ+α2+36b1αλ2ω2λ2σ+α2+28b1λω2δ−30c1α2λω3λ2σ+α2+60c1α3λωδλ2σ+α2−30c1α2λ2ωλ2σ+α2+150b2αλ2ω2λ2σ+α2−300b2α2λ2ωδλ2σ+α2+150b2αλ3ωλ2σ+α2+11b1λ2αω2λ2σ+α2−22b1λ2α2δλ2σ+α2+11b1αλ3λ2σ+α2−64c2λ2α2ω2λ2σ+α2+128c2λ2α3δλ2σ+α2−64c2α2λ3λ2σ+α2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ+16c2λ3.By using (3), equation 5×W×W′′ is equivalent to(48)5×W×W″=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4+30b22ω4−ω2λ+2αδλ−λ2λ2σ+α2+10c0c1ω3+30c0b2ω3δ−20c1c2αλωδ+20b1c1ω3δ−50c1c2αω3δ+50c1c2λω3+40b1b2ω3−ω2λ+2αδλ−λ2λ2σ+α2−80c2b2αω3−ω2λ+2αδλ−λ2λ2σ+α250c0c2αω2δ+65c2b2λω3δ−5b2αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−+40c0c2λω2−45c1b2αω2−ω2λ+2αδλ−λ2λ2σ+α2+35c1b2λω2δ−15c12αω2δ+10c12λω2+10b12ω2−ω2λ+2αδλ−λ2λ2σ+α2−50b1c2αω2+40b1c2λω2δ−5c2b1αω2−ω2λ+2αδλ−λ2λ2σ+α2+5c1b1λωδ+10c22α2ω2−ω2λ+2αδλ−λ2λ2σ+α2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λω2−ω2λ+2αδλ−λ2λ2σ+α2−5c0b1α−ω2λ+2αδλ−λ2λ2σ+α2−30c0b2αω−ω2λ+2αδλ−λ2λ2σ+α2+10c0c2α2−ω2λ+2αδλ−λ2λ2σ+α2−20c1b1αω−ω2λ+2αδλ−λ2λ2σ+α2+10c1c2α2−ω2λ+2αδλ−λ2λ2σ+α2ω+5b12λ−ω2λ+2αδλ−λ2λ2σ+α2+30b1b2λω−ω2λ+2αδλ−λ2λ2σ+α2−20b1c2αλ−ω2λ+2αδλ−λ2λ2σ+α2−20c2b2αλ−ω2λ+2αδλ−λ2λ2σ+α2ω+25c0b2λωδ−15c0c1αωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ+10b1c2λ2δ+10c0c2λ2.Finally, by using (3)(49)5×W×W″=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4−30b22λω6λ2σ+α2+60b22λαω4δλ2σ+α2−30b22λ2ω4λ2σ+α2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+50c1c2λω3−40b1b2λω5λ2σ+α2+80b1b2λω3δλ2σ+α2+10b1c2λ2δ−40b1b2λ2ω3λ2σ+α2+80c2b2αλω5λ2σ+α2−160c2b2α2λω3δλ2σ+α2+80c2b2αλ2ω3λ2σ+α2+65c2b2λω3δ−5b12αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2+45c1b2αλω4λ2σ+α2−90c1b2α2λω2δλ2σ+α2+45c1b2αλ2ω2λ2σ+α2+35c1b2λω2δ−15c12αω2δ+10c12λω2−10b12λω4λ2σ+α2+20b12λαω2δλ2σ+α2−10b12λ2ω2λ2σ+α2+50b1c2αλω4λ2σ+α2−100b1c2α2λω2δλ2σ+α2+50b1c2αλ2ω2λ2σ+α2+40b1c2λω2δ+5c2b1αλω4λ2σ+α2−10c2b1α2λω2δλ2σ+α2+5c2b1αλ2ω2λ2σ+α2−10c22α2λω4λ2σ+α2+20c22α3λω2δλ2σ+α2−10c22α2λ2ω2λ2σ+α2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ−25b22λ2ω4λ2σ+α2+50b22λ2αω2δλ2σ+α2−25b22λ3ω2λ2σ+α2+5c0b1αλω2λ2σ+α2−10c0b1α2λδλ2σ+α2+5c0b1αλ2λ2σ+α2+30c0b2αλω3λ2σ+α2−60c0b2α2λωδλ2σ+α2+30c0b2αλ2ωλ2σ+α2−10c0c2α2λω2λ2σ+α2+20c0c2α3λδλ2σ+α2−10c0c2α2λ2λ2σ+α2+20c1b1αλω3λ2σ+α2−40c1b1α2λωδλ2σ+α2+20c1b1αλ2ωλ2σ+α2−10c1c2α2λω3λ2σ+α240b1c2α2λ2δλ2σ+α2+20c1c2α3λωδλ2σ+α2−10c1c2α2λ2ωλ2σ+α2−5b12λ2ω2λ2σ+α2+10b12λ2αδλ2σ+α2−30b1b2λ2ω3λ2σ+α2+60b1b2λ2αωδλ2σ+α2−30b1b2λ3ωλ2σ+α2+20b1c2αλ2ω2λ2σ+α2+20b1c2αλ3λ2σ+α2+20c2b2λ2ω3λ2σ+α2−40c2b2α2λ2ωδλ2σ+α2+20c2b2αλ3ωλ2σ+α2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ−5b12λ3λ2σ+α2+5c1b1λωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ+10c0c2λ2.By using (3), equation of 5/3×W3 becomes(50)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2ω4−ω2λ+2αδλ−λ2λ2σ+α2+5c02b2ωδ+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+53c03+5b2c12ω3δ+5b22c1ω3−ω2λ+2αδλ−λ2λ2σ+α2+5b2b12δ3ω+10b2c2b1ω3−ω2λ+2αδλ−λ2λ2σ+α2+53b23ω3δ3+53b13δ3+5b1b22δ3ω2+5c02c2ω2+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22ω2−ω2λ+2αδλ−λ2λ2σ+α2+10c1b1b2ω2−ω2λ+2αδλ−λ2λ2σ+α2+5c02c1ω+5b1c12ω2δ+5c2b12ω2−ω2λ+2αδλ−λ2λ2σ+α2+5c0b12−ω2λ+2αδλ−λ2λ2σ+α2+10c0b1b2−ω2λ+2αδλ−λ2λ2σ+α2ω+5b12c1ω−ω2λ+2αδλ−λ2λ2σ+α2+5c02b1δ+10c0b1c1ωδ.Finally, by using equations (3)(51)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+203b1c22ω4δ−5b22c2λω6λ2σ+α2+10b22c2λαω4δλ2σ+α2−5b22c2λ2ω4λ2σ+α2+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ−5b22c1λω5λ2σ+α2+10b22c1λαω3δλ2σ+α2−5λ2b22c1ω3λ2σ+α2−10b2c2b1λω5λ2σ+α2+20b2c2b1λαω3δλ2σ+α2−10λ2b2c2b1ω3+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5b1b22δ3ω2+5c02c2ω2+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ−5c0b22λω4λ2σ+α2+10c0b22λαω2δλ2σ+α2−5c0b22λ2ω2λ2σ+α2−10c1b1b2ω4λ2σ+α2+20c1b1b2λαω2δλ2σ+α2−10c1b1b2λ2ω2λ2σ+α2+5b1c12ω2δ−5c2b12λω4λ2σ+α2+10c2b12αλω2δλ2σ+α2−5c2b12λ2ω2λ2σ+α2−5c0b12λω2λ2σ+α2+10c0b12λαδλ2σ+α2−5c0b12λ2λ2σ+α2−10c0b1b2λω3λ2σ+α2+20c0b1b2λαωδλ2σ+α2−10c0b1b2λ2ωλ2σ+α2−5b12c1λω3λ2σ+α2+10b12c1αλωδλ2σ+α2−5b12c1λ2ωλ2σ+α2+5c02b1δ+5c02c1ω+5c02b2ωδ+10c1c2b2ω4δ+10c0b1c1ωδ+53c03.By using (3) and (35), W′′ is equivalent to(52)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1α−ω2λ+2αδλ−λ2λ2σ+α2−6b2αω−ω2λ+2αδλ−λ2λ2σ+α2+2c2α2−ω2λ+2αδλ−λ2λ2σ+α2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.Finally, by using (3), (35) becomes(53)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2+b1αλω2λ2σ+α2−2b1α2λδλ2σ+α2+b1αλ2λ2σ+α2+6b2αλω3λ2σ+α2−12b2α2λωδλ2σ+α2+6b2αλ2ωλ2σ+α2−2c2α2λω2λ2σ+α2+4c2α3λδλ2σ+α2−2c2α2λ2λ2σ+α2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.By using (3) and (43), 5×W2 is equal to the following form:(54)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b12−ω2λ+2αδλ−λ2λ2σ+α2+10c2b1δω2+10c1b2ω2δ+10b1b2−ω2λ+2αδλ−λ2λ2σ+α2ω+5b22ω2−ω2λ+2αδλ−λ2λ2σ+α2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.By using equations (3) and (43), 5×W2 becomes(55)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2−5b22λω2λ2σ+α2+10b12αλδλ2σ+α2−5b12λ2λ2σ+α2+10c2b1ω2δ+10c1b2ω2δ−10b1b2λω3λ2σ+α2+20b1b2αλωδλ2σ+α2−10b1b2λ2ωλ2σ+α2−5b22λω4λ2σ+α2+10b22αλω2δλ2σ+α2−5b22λ2ω2λ2σ+α2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.By using equations (47), (49), (51), and (53), k−5W=kc0+kc1ω+kb1δ+kc2ω2+kb2ωδ−5c0−5c1ω−5b1δ−5c2ω2−5b2ωδ becomes the following:(56)120c2ω6+30c22ω6−30b22ω6λ2σ+α2+53c23ω6−5b22c2λω6λ2σ+α2+24c1ω5+240b2αλω5λ2σ+α2+40c1c2ω5−40b1b2λω5λ2σ+α2+80c2b2αλω5λ2σ+α2+5c1c22−5λb22c1ω5λ2σ+α2−10b2c2b1λω5λ2σ+α2+120b2ω5δ+60c2b2ω5δ+5b2c22ω5δ+240c2λω4−234c2α2λω4λ2σ+α2+36b1αλω4λ2σ+α2+30c0c2ω4+10c12ω4+40c22λω4−30b22λ2ω4λ2σ+α2+45c1b2αλω4λ2σ+α2−10b12λω4λ2σ+α2+50b1c2αλω4λ2σ+α2+5c2b1αλω4λ2σ+α2−30b1b2λ2ω3λ2σ+α2−10c22α2λω4λ2σ+α2+50c0c22ω4+5c2c12ω4−5b22c2λ2ω4λ2σ+α2−5λc0b22ω4λ2σ+α2−10c1b1b2λω4λ2σ+α2−5c2b12λω4λ2σ+α2+6c2ω4+5c22ω4−25b22λ2ω4λ2σ+α2−60c1αω3δ−5b22λω4λ2σ+α2−336c2αω4δ+24b1ω4δ+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+60b22λαω4δλ2σ+α2+10c1c2b2ω4δ+203b1c22ω4δ+10b22c2λαω4δλ2σ+α2+40c1λω3+240b2αλ2ω3λ2σ+α2+150b2αλ2ω3λ2σ+α2+10c0c1ω3+50c1c2λω3−40b1b2λ2ω3λ2σ+α2−30c1α2λω3λ2σ+α2+80c2b2αλ2ω3λ2σ+α2+20c1b1αλω3λ2σ+α2−10c1c2α2λω3λ2σ+α2+20c2b2λ2ω3λ2σ+α2+30c0b2αλω3λ2σ+α2+10c0c1c2ω3+53c13ω3−5b22c1λ2ω3λ2σ+α2+6b2αλω3λ2σ+α2−10b2c2b1λ2ω3λ2σ+α2+2c1ω3+10c1c2ω3−10b1b2λω3λ2σ+α2−10c0b1b2λω3λ2σ+α2−5b12c1λω3λ2σ+α2−480b2α2λω3δλ2σ+α2+180b2λω3δ+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+80b1b2αλω3δλ2σ+α2−160c2b2α2λω3δλ2σ+α2+65c2b2λω3δ+10c0c2b2ω3δ+10c2c1b1ω3δ+5b2c12+10b22c1αλω3δλ2σ+α2+20b2c2b1αλω3δ+6b2ω3δ+10b2c2ω3δ+53b23ω3δ3+53b23ω3δ3−30b22αδ3ω2+5b1b22δ3ω2+90b2α2δ3ω−35b2b1αδ3ω+10b2c2α2δ3ω+10c0c2ω2+5b2b12δ3ω−234c2α2λ2ω2λ2σ+α2+136c2λ2ω2+5c2b1αλ2ω2λ2σ+α2+36b1αλ2ω2λ2σ+α2+11b1λ2αω2λ2σ+α2−64c2λ2α2ω2λ2σ+α2+40c0c2λω2+45c1b2αλ2ω2λ2σ+α2+10c12λω2−10b12λ2ω2λ2σ+α2−10c22α2λ2ω2λ2σ+α2+10c22λ2ω2−25b22λ3ω2λ2σ+α2+5c0b1αλ2ω2λ2σ+α2−10c0c2α2λω2λ2σ+α2−5b12λ2ω2λ2σ+α2+20b1c2αλ2ω2λ2σ+α2+5c02c2ω2+5c0c12ω2−5c0b22λ2ω2λ2σ+α2+50b1c2αλ2ω2λ2σ+α2−10c1b1b2λ2ω2λ2σ+α2−5c2b12λ2ω2λ2σ+α2−5c0b12λω2λ2σ+α2+8c2λω2+b1αλω2λ2σ+α2−2c2α2λω2λ2σ+α2+5c12ω2−5b12λω2λ2σ+α2−5b22λ2ω2λ2σ+α2+c2kω2−5c2ω2+468c2α3λω2δλ2σ+α2−362c2αλω2δ−72b1α2λω2δλ2σ+α2+28b1λω2δ+10c0b1ω2δ−50c0c2αω2δ−90c1b2α2λω2δλ2σ+α2+35c1b2λω2δ−15c12αω2δ+20b12αλω2δλ2σ+α2−100b1c2α2λω2δλ2σ+α2+40b1c2λω2δ−10c2b1α2λω2δλ2σ+α2+20c22α3λω2δλ2σ+α2+5c2b1λω2δ−20c22αλω2δ+50b22αλ2ω2δλ2σ+α2+10c0c2b1ω2δ+10c0c1b2ω2δ+10c0b22αλω2δλ2σ+α2+20c1b1b2λαω2δλ2σ+α2+5b1c12ω2δ+10c2b12αλω2δλ2σ+α2+2b1ω2δ−10c2αω2δ+10c2b1ω2δ+10c1b2ω2δ+10b22λαω2δλ2σ+α2+60c1α3λωδλ2σ+α2−300b2α2λ2ωδλ2σ+α2+61b2λ2ωδ−60c0b2α2λωδλ2σ+α2−40c1b1α2λωδλ2σ+α2+20c1c2α3λωδλ2σ+α2+60b1b2αλ2ωδλ2σ+α2−40c2b2α2λ2ωδ+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+5c1b1λωδ+10b2c2λ2ωδ+20c0b1b2αλωδλ2σ+α2+10b12c1αλωδλ2σ+α2+5c02b2ωδ+10c0b1c1ωδ−12b2α2λωδλ2σ+α2+5b2λωδ−3c1αωδ+10c0b2ωδ+10b1c1ωδ+b2kωδ−5b2ωδ−45c1αλωδ+20b1b2αλωδλ2σ+α2+150b2αλ3ωλ2σ+α2−30c1α2λ2ωλ2σ+α2+16c1λ2ω+30c0b2αλ2ωλ2σ+α2+20c1b1αλ2ωλ2σ+α2−10c1c2α2λ2ωλ2σ+α2−30b1b2λ3ωλ2σ+α2+20c2b2αλ3ωλ2σ+α2+10c2λ2ω+10c0c1λω−10c0b1b2λ2ωλ2σ+α2−5b12c1λ2ωλ2σ+α2+5c02c1ω+6b2αλ2ωλ2σ+α2+2c1λω−10b1b2λ2ωλ2σ+α2+10c0c1ω+c1kω−5c1ω−22b1α2λ2δλ2σ+α2+128c2α3λ2δλ2σ+α2+5b1λ2δ−56c2αλ2δ−10c0b1α2λδλ2σ+α2+20c0c2α3λδλ2σ+α2+10b12αλ2δλ2σ+α2−40b1c2α2λ2δλ2σ+α2+5c0b1λδ−20c0c2αλδ+10b1c2λ2δ+10c0b12αλδλ2σ+α2+5c02b1δ−2b1α2λδλ2σ+α2+4c2α3λδλ2σ+α2+b1λδ−4c2αλδ+10b12αλδλ2σ+α2+10c0b1δ+b1kδ−5b1δ+11b1αλ3λ2σ+α2−64c2α2λ3λ2σ+α2+16c2λ3−15c0b12λ2λ2σ+α2+53c03+5c0αλ2+5c0b1αλ2λ2σ+α2−10c0c2α2λ2λ2σ+α2−5b12λ3λ2σ+α2+20b1c2αλ3λ2σ+α2+10c0c2λ2+b1αλ2λ2σ+α2−2c2α2λ2λ2σ+α2+2c2λ2+5c02−5b12λ2λ2σ+α2+kc0−5c0+d.Inserting (31) in (29) and applying (4) as well as (3), we have polynomials in the variable form of ω and δ. Then, we obtain equations of algebraic sets by applying the coefficients of the polynomials set to zero. Hence, the following algebraic expressions are obtained:(57)ω6:120c2+30c22−30b22λ2σ+α2+53c23−5b22c2λλ2σ+α2=0,ω5:24c1+240b2αλλ2σ+α2+40c1c2−40b1b2λλ2σ+α2+80c2b2αλλ2σ+α2+5c1c22−5λb22c1λ2σ+α2−10b2c2b1λλ2σ+α2=0,ω5δ:120b2+60c2b2+5b2c22=0,ω4:240c2λ−234c2α2λλ2σ+α2+36b1αλλ2σ+α2+30c0c2+10c12+40c22λ−30b22λ2λ2σ+α2+45c1b2αλλ2σ+α2−10b12λλ2σ+α2+50b1c2αλλ2σ+α2−10c22α2λλ2σ+α2+50c0c22+5c2c12−5b22c2λ2λ2σ+α2−5λc0b22λ2σ+α2−10c1b1b2λλ2σ+α2−5c2b12λλ2σ+α2+6c2+5c22−25b22λ2λ2σ+α2+5c2b1αλλ2σ+α2−5b22λλ2σ+α2=0,ω4δ:−336c2αω4δ+24b1ω4δ+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+60b22λαω4δλ2σ+α2+10c1c2b2ω4δ+203b1c22ω4δ+10b22c2λαω4δλ2σ+α2=0,ω3:+40c1λ+240b2αλ2λ2σ+α2−30c1α2λλ2σ+α2+150b2αλ2λ2σ+α2+10c0c1ω3+50c1c2λω3−40b1b2λ2λ2σ+α2+80c2b2αλ2λ2σ+α2+20c1b1αλλ2σ+α2−10c1c2α2λλ2σ+α2+53c13−30b1b2λ2λ2σ+α2+20c2b2λ2λ2σ+α2+30c0b2αλλ2σ+α2+10c0c1c2−5b22c1λ2λ2σ+α2+6b2αλλ2σ+α2−10b2c2b1λ2λ2σ+α2+2c1+10c1c2−10b1b2λλ2σ+α2−10c0b1b2λλ2σ+α2−5b12c1λλ2σ+α2=0,ω3δ:−60c1α−480b2α2λλ2σ+α2+180b2+30c0b2+20b1c1−50c1c2α+80b1b2αλλ2σ+α2−160c2b2α2λλ2σ+α2+65c2b2λ+10c0c2b2+10c2c1b1+5b2c12+10b22c1αλλ2σ+α2+20b2c2b1αλ+6b2+10b2c2=0,ω3δ3:53b23=0,δ3:6b1α2−24c2α3−5b12+10b1c2+53b13=0,δ3ω2:−30b22α+5b1b22=0,δ3ω:90b2α2−35b2b1α+10b2c2α2+5b2b12=0,ω2:−234c2α2λ2λ2σ+α2+136c2λ2+36b1αλ2λ2σ+α2+11b1λ2αλ2σ+α2−64c2λ2α2λ2σ+α2+40c0c2λ+45c1b2αλ2λ2σ+α2+10c12λ−10b12λ2λ2σ+α2+5c2b1αλ2λ2σ+α2−10c22α2λ2λ2σ+α2+10c22λ2−25b22λ3λ2σ+α2+5c0b1αλ2λ2σ+α2−10c0c2α2λλ2σ+α2−5b12λ2λ2σ+α2+20b1c2αλ2λ2σ+α2+5c02c2+5c0c12−5c0b22λ2λ2σ+α2+50b1c2αλ2λ2σ+α2−10c1b1b2λ2λ2σ+α2−5c2b12λ2λ2σ+α2−5c0b12λλ2σ+α2+8c2λ+b1αλλ2σ+α2−2c2α2λλ2σ+α2+10c0c2+5c12−5b12λλ2σ+α2−5b22λ2λ2σ+α2+c2k−5c2=0,ω2δ:468c2α3λλ2σ+α2−362c2αλ−72b1α2λλ2σ+α2+28b1λ+10c0b1−50c0c2α−90c1b2α2λλ2σ+α2+35c1b2λ−15c12α+20b12αλλ2σ+α2−100b1c2α2λλ2σ+α2+40b1c2λ−10c2b1α2λλ2σ+α2+20c22α3λλ2σ+α2+5c2b1λ−20c22αλ+50b22αλ2λ2σ+α2+10c0c2b1+10c0c1b2+10c0b22αλλ2σ+α2+20c1b1b2λαλ2σ+α2+5b1c12+10c2b12αλλ2σ+α2+2b1−10c2α+10c2b1+10c1b2+10b22λαλ2σ+α2=0,ωδ:60c1α3λλ2σ+α2−300b2α2λ2λ2σ+α2+61b2λ2−60c0b2α2λλ2σ+α2−40c1b1α2λλ2σ+α2+60b1b2αλ2λ2σ+α2−40c2b2α2λ2+25c0b2λ−15c0c1α−20c1c2αλ+5c1b1λ+10b2c2λ2+20c0b1b2αλλ2σ+α2+20c1c2α3λλ2σ+α2+10b12c1αλλ2σ+α2+5c02b2+10c0b1c1−12b2α2λλ2σ+α2+5b2λ−3c1α+10c0b2+10b1c1+b2k−5b2−45c1αλ+20b1b2αλλ2σ+α2=0,ω:150b2αλ3λ2σ+α2−30c1α2λ2λ2σ+α2+16c1λ2+30c0b2αλ2λ2σ+α2+20c1b1αλ2λ2σ+α2−30b1b2λ3λ2σ+α2+20c2b2αλ3λ2σ+α2+10c2λ2+10c0c1λ−10c0b1b2λ2λ2σ+α2−5b12c1λ2λ2σ+α2+5c02c1+6b2αλ2λ2σ+α2+2c1λ−10b1b2λ2λ2σ+α2−10c1c2α2λ2λ2σ+α2+10c0c1+c1k−5c1=0,δ:−22b1α2λ2λ2σ+α2+128c2α3λ2λ2σ+α2+5b1λ2−56c2αλ2−10c0b1α2λλ2σ+α2+20c0c2α3λλ2σ+α2−40b1c2α2λ2λ2σ+α2+5c0b1λ−20c0c2αλ+10b1c2λ2+10c0b12αλλ2σ+α2+5c02b1−2b1α2λλ2σ+α2+4c2α3λλ2σ+α2+b1λ−4c2αλ+10b12αλ2λ2σ+α2+10b12αλλ2σ+α2+10c0b1+b1k=0,ω0:−5b1δ+11b1αλ3λ2σ+α2−64c2α2λ3λ2σ+α2+16c2λ3−15c0b12λ2λ2σ+α2+53c03+5c0αλ2+5c0b1αλ2λ2σ+α2−10c0c2α2λ2λ2σ+α2−5b12λ3λ2σ+α2+20b1c2αλ3λ2σ+α2+10c0c2λ2+b1αλ2λ2σ+α2−2c2α2λ2λ2σ+α2+2c2λ2+5c02−5b12λ2λ2σ+α2+kc0−5c0+d=0.By solving the previous algebraic equations using the Mathematica software package 9.1, we get the following results.Result 1.
(58)c0=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ,c1=0,c2=−12,b1=44α7,b2=0,k=±416λ2−9d.
By substituting (58) into (31) with (13), (16), and (4), we get the exact solutions of (18) as follows:(59)W1=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ+44α7δ−12ω2.
By substituting the value ofω and δ from (3) we get  = −3969α2−14095α2λ−3969λ2σ−17640λ2σ/41895α2+λ2σ+44α/71/Z−12Z′/Z2.
When we put the value ofZ from equation (3), we get(60)=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ+44α71M1sinhξ−λ+M2coshξ−λ+α/λ−12M1sinhξ−λ+M2coshξ−λ+α/λ′M1sinhξ−λ+M2coshξ−λ+α/λ2,=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ+44α71M1sinhξ−λ+M2coshξ−λ+α/λ−12M1coshξ−λ−λ+M2sinhξ−λ−λM1sinhξ−λ+M2coshξ−λ+α/λ2.
Then, the exact solution of2+1-dimensional Sawada–Kotera equation of (18) is as follows:(61)=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ+44α71M1sinhξ−λ+M2coshξ−λ+α/λ−12M1coshξ−λ−λ+M2sinhξ−λ−λ2M1sinhξ−λ+M2coshξ−λ+α/λ2.
By lettingM1=0, α=0, and M2>0 in (61), the particular solution of (18) is(62)W1a=−3969λ2σ−17640λ2σ41895λ2σ−12−λM2sinhξ−λ2M2coshξ−λ2,(63)W1a=−3969λ2σ−17640λ2σ41895λ2σ+12λtanh2ξ−λ.
Then, by lettingλ=0.4, α=0, t=0, y=4, and σ=1, the graph of (63) has the following forms.
Figure1 indicates the soliton of (63) in three and two dimensions. If we put M2=0 and M1>0 in (61), we produce a solitary solution:(64)W1b=−3969λ2σ−17640λ2σ41895λ2σ−12−λM1coshξ−λ2M1sinhξ−λ2,(65)W1b=−3969λ2σ−17640λ2σ41895λ2σ+12λcoth2ξ−λ.
The particular solution of2+1-dimensional equation for hyperbolic function, if we put result on (61) when M2=0 and M1>0, has the following form.
The graph of the particular solution whereλ=−0.4, σ=1, t=0, y=0 is as follows.
Figure2 shows the soliton of (65) in three and 2 dimensions by considering certain parameters.Figure 1
The figures of equation (53) soliton in 3D and 2D.Figure 2
The figures of equation (55) soliton in 3D and 2D.Result 2.
(66)c0=−8α2−295α2λ−8λ2σ90α2+λ2σ,c1=0,c2=−6,b1=−9α,b2=0,k=λ2−2d.
By substituting (66) into (31) with (13) by using (4) and (16), we get the exact solutions of equation (18) as follows:(67)Wc=−8α2−295α2λ−8λ2σ90α2+λ2σ−9αδ−6ω2.
By substituting the value in place ofω and δ from (3), we get  =  −8α2−295α2λ−8λ2σ/90α2+λ2σ−9α1/Z−6Z′/Z2; when we put the value of Z from (3), we get(68)−8α2−295α2λ−8λ2σ90α2+λ2σ−9α1M1sinhξ−λ+M2coshξ−λ+α/λ−6M1coshξ−λ−λ+M2sinhξ−λ−λM1sinhξ−λ+M2coshξ−λ+α/λ2.
Then, the exact solution of2+1-dimensional equation from (18) is as follows:(69)=−8α2−295α2λ−8λ2σ90α2+λ2σ−9α1M1sinhξ−λ+M2coshξ−λ+α/λ−6M1coshξ−λ−λ+M2sinhξ−λ−λ2M1sinhξ−λ+M2coshξ−λ+α/λ2.
By lettingM1=0, α=0, and M2>0 in (69), the particular solution of (18) is(70)Wc=−8λ2σ90λ2σ−6−λM2sinhξ−λ2M2coshξ−λ2,(71)Wc=−8λ2σ90λ2σ+6λtanh2ξ−λ.
Then, by lettingλ=−0.3, α=0, t=0, y=2, and σ=−0.5, the graph of (71) has the following forms.
Figure3 represents the soliton of (71) in two and three dimensions using certain parameters.
If we putM2=0 and M1>0 in (61), we produce a solitary solution:(72)W1d=−8λ2σ90λ2σ−6−λM1coshξ−λ2M1sinhξ−λ2,(73)W1d=−8λ2σ90λ2σ+6λcoth2ξ−λ.
The particular solution of (2 + 1)-dimensional equation for hyperbolic function, if we put result on equation (51) when M2=0 and M1>0, has the following form.
The graph of the particular solution whereλ=−0.3, σ=−o.5, t=0, y=2,α=0 is as follows.
Figure4 illustrates the soliton of (73) in three-dimensional space and two-dimensional plane.Figure 3
The figures of (71) soliton in 3D and 2D.Figure 4
The figures of equation (63) soliton in 3D and 2D.


### 3.2. Case B:λ>0—Nature of Trigonometric Solutions

From (7), (29) and (39) become(74)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αω3ω2λ−2αδλ+λ2λ2σ−α2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2ω2ω2λ−2αδλ+λ2λ2σ−α2−362c2λαω2δ+136c2λ2ω2−36b1αω2ω2λ−2αδλ+λ2λ2σ−α2+61b2λ2ωδ+28b1λω2δ+30c1α2ωω2λ−2αδλ+λ2λ2σ−α2−150b2αλω2λ−2αδλ+λ2λ2σ−α2ω−11b1λαω2λ−2αδλ+λ2λ2σ−α2+64c2λα2ω2λ−2αδλ+λ2λ2σ−α2−45c1λαωδ+16c1λ2ω+5b1λ2δ−56c2λ2αδ+16c2λ3.Finally, by using (7), (29) and (35) become(75)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αλω5λ2σ−α2+480b2α2λω3δλ2σ−α2−240b2αλ2ω3λ2σ−α2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2λω4λ2σ−α2−468c2α3λω2δλ2σ−α2+234c2α2λ2ω2λ2σ−α2−362c2λαω2δ+136c2λ2ω2−36b1αλω4λ2σ−α2+72b1α2λω2δλ2σ−α2−36b1αλ2ω2λ2σ−α2+28b1λω2δ+30c1α2λω3λ2σ−α2−60c1α3λωδλ2σ−α2+30c1α2λ2ωλ2σ−α2−150b2αλ2ω2λ2σ−α2+300b2α2λ2ωδλ2σ−α2−150b2αλ3ωλ2σ−α2−11b1λ2αω2λ2σ−α2+22b1λ2α2δλ2σ−α2−11b1αλ3λ2σ−α2+64c2λ2α2ω2λ2σ−α2−128c2λ2α3δλ2σ−α2+64c2α2λ3λ2σ−α2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ−56c2λ2αδ+16c2λ3.From (7), (29) and (41) become(76)5×W×W′′=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4+30b22ω4ω2λ−2αδλ+λ2λ2σ−α2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+50c1c2λω3+10c12λω2+40b1b2ω3ω2λ−2αδλ+λ2λ2σ−α2−80c2b2αω3ω2λ−2αδλ+λ2λ2σ−α2+65c2b2λω3δ−5b2αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2−45c1b2αω2ω2λ−2αδλ+λ2λ2σ−α2+35c1b2λω2δ−15c12αω2δ+10b12ω2ω2λ−2αδλ+λ2λ2σ−α2−50b1c2αω2+40b1c2λω2δ−5c2b1αω2ω2λ−2αδλ+λ2λ2σ−α2+10c22α2ω2ω2λ−2αδλ+λ2λ2σ−α2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λω2ω2λ−2αδλ+λ2λ2σ−α2−5c0b1αω2λ−2αδλ+λ2λ2σ−α2−30c0b2αωω2λ−2αδλ+λ2λ2σ−α2+10c0c2α2ω2λ−2αδλ+λ2λ2σ−α2−20c1b1αωω2λ−2αδλ+λ2λ2σ−α2+10c1c2α2ω2λ−2αδλ+λ2λ2σ−α2ω+5b12λω2λ−2αδλ+λ2λ2σ−α2+30b1b2λωω2λ−2αδλ+λ2λ2σ−α2−20b1c2αλω2λ−2αδλ+λ2λ2σ−α2−20c2b2αλω2λ−2αδλ+λ2λ2σ−α2ω+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+5c1b1λωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ+10b1c2λ2δ+10c0c2λ2.Finally, by using (7), (29) and (41) become(77)5×W×W′′=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4+30b22λω6λ2σ−α2−60b22λαω4δλ2σ−α2+30b22λ2ω4λ2σ−α2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+40b1b2λω5λ2σ−α2−80b1b2λω3δλ2σ−α2+40b1b2λ2ω3λ2σ−α2−80c2b2αλω5λ2σ−α2+160c2b2α2λω3δλ2σ−α2−80c2b2αλ2ω3λ2σ−α2+65c2b2λω3δ−5b12αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2−45c1b2αλω4λ2σ−α2+90c1b2α2λω2δλ2σ−α2−45c1b2αλ2ω2λ2σ−α2+35c1b2λω2δ−15c12αω2δ+10c12λω2+10b12λω4λ2σ−α2−20b12λαω2δλ2σ−α2+10b12λ2ω2λ2σ−α2−50b1c2αω2λ2σ−α2+100b1c2α2λω2δλ2σ−α2−50b1c2αλ2ω2λ2σ−α2+10b2c2α2δ3ω+40b1c2λω2δ−5c2b1αλω4λ2σ−α2+10c2b1α2λω2δλ2σ−α2−5c2b1αλ2ω2λ2σ−α2+10c22α2λω4λ2σ−α2−20c22α3λω2δλ2σ−α2+10c22α2λ2ω2λ2σ−α2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λ2ω4λ2σ−α2−5c0b1αλω2λ2σ−α2+10c0b1α2λδλ2σ−α2−5c0b1αλ2λ2σ−α2−30c0b2αλω3λ2σ−α2+60c0b2α2λωδλ2σ−α2−30c0b2αλ2ωλ2σ−α2+10c0c2α2λω2λ2σ−α2−20c0c2α3λδλ2σ−α2+10c0c2α2λ2λ2σ−α2−20c1b1αλω3λ2σ−α2+10c1c2α2λω3λ2σ−α2−20c1c2α3λωδλ2σ−α2+10c1c2α2λ2ωλ2σ−α2+5b12λ2ω2λ2σ−α2−10b12λ2αδλ2σ−α2+40c1b1α2λωδλ2σ−α2−20c1b1αλ2ωλ2σ−α2+5b12λ3λ2σ−α2+30b1b2λ2ω3λ2σ−α2−60b1b2λ2αωδλ2σ−α2+30b1b2λ3ωλ2σ−α2−20b1c2αλ2ω2λ2σ−α2+40b1c2α2λ2δλ2σ−α2−20c2b2λ2ω3λ2σ−α2+40c2b2α2λ2ωδλ2σ−α2−20c2b2αλ3ωλ2σ−α2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+5c1b1λωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1−20b1c2αλ3λ2σ−α2λω−20c0c2αλδ+10b1c2λ2δ+10c0c2λ2.From (7), (29) and (45) become(78)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2ω4ω2λ−2αδλ+λ2λ2σ−α2+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ+5b22c1ω3ω2λ−2αδλ+λ2λ2σ−α2+10b2c2b1ω3ω2λ−2αδλ+λ2λ2σ−α2+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5b1b22δ3ω2+5c02c2ω2+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22ω2ω2λ−2αδλ+λ2λ2σ−α2+5c02c1ω+5c02b1δ+10c1b1b2ω2ω2λ−2αδλ+λ2λ2σ−α2+5b1c12ω2δ+5c2b12ω2ω2λ−2αδλ+λ2λ2σ−α2+5c0b12ω2λ−2αδλ+λ2λ2σ−α2+10c0b1b2ω2λ−2αδλ+λ2λ2σ−α2ω+5b12c1ωω2λ−2αδλ+λ2λ2σ−α2+5c02b2ωδ+10c0b1c1ωδ+53c03.By using (7), (29) and (45) become(79)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2λω6λ2σ−α2−10b22c2λαω4δλ2σ−α2+5b22c2λ2ω4λ2σ−α2+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ+5b22c1λω5λ2σ−α2−10b22c1λαω3δλ2σ−α2+5λ2b22c1ω3λ2σ−α2+10b2c2b1λω5λ2σ−α2−20b2c2b1λαω3δλ2σ−α2+5c02b1δ−10λ2b2c2b1ω3+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5b1b22δ3ω2+5c02c2ω2+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22λω4λ2σ−α2−10c0b22λαω2δλ2σ−α2+5c0b22λ2ω2λ2σ−α2+10c1b1b2ω4λ2σ−α2−20c1b1b2λαω2δλ2σ−α2+10c1b1b2λ2ω2λ2σ−α2+5b1c12ω2δ+5c2b12λω4λ2σ−α2−10c2b12αλω2δλ2σ−α2+5c2b12λ2ω2λ2σ−α2+5c0b12λω2λ2σ−α2−10c0b12λαδλ2σ−α2+5c0b12λ2λ2σ−α2+10c0b1b2λω3λ2σ−α2−20c0b1b2λαωδλ2σ−α2+10c0b1b2λ2ωλ2σ−α2+5b12c1λω3λ2σ−α2−10b12c1αλωδλ2σ−α2+5b12c1λ2ωλ2σ−α2+5c02c1ω+5c02b2ωδ+10c0b1c1ωδ+53c03.From (7), (29) and (35) become(80)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αω2λ−2αδλ+λ2λ2σ−α2−6b2αωω2λ−2αδλ+λ2λ2σ−α2+2c2α2ω2λ−2αδλ+λ2λ2σ−α2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.Again considering (7), (29) and (35) become(81)W′′=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αλω2λ2σ−α2+2b1α2λδλ2σ−α2−b1αλ2λ2σ−α2−6b2αλω3λ2σ−α2+12b2α2λωδλ2σ−α2−6b2αλ2ωλ2σ−α2+2c2α2λω2λ2σ−α2−4c2α3λδλ2σ−α2+2c2α2λ2λ2σ−α2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.From (7), (29) and (43) become(82)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b12ω2λ−2αδλ+λ2λ2σ−α2+10c2b1δω2+10c1b2ω2δ+10b1b2ω2λ−2αδλ+λ2λ2σ−α2ω+5b22ω2ω2λ−2αδλ+λ2λ2σ−α2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.Using (7), (29) and (43) become(83)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b22λω2λ2σ−α2−10b12αλδλ2σ−α2+5b12λ2λ2σ−α2+10c2b1ω2δ+10c1b2ω2δ+10b1b2λω3λ2σ−α2−20b1b2αλωδλ2σ−α2+10b1b2λ2ωλ2σ−α2+5b22λω4λ2σ−α2−10b22αλω2δλ2σ−α2+5b22λ2ω2λ2σ−α2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.By using (75), (77), (79), (81) and (83), k−5W=kc0+kc1ω+kb1δ+kc2ω2+kb2ωδ−5c0−5c1ω−5b1δ−5c2ω2−5b2ωδ becomes the following forms.(84)120c2ω6+30c22ω6+30b22λω6λ2σ−α2+53c23ω6+5b22c2λω6λ2σ−α2+24c1ω5−240b2αλω5λ2σ−α2+40c1c2ω5−80c2b2αλω5λ2σ−α2+40b1b2λω5λ2σ−α2+5c1c22ω5+5b22c1λω5λ2σ−α2+10b2c2b1λω5λ2σ−α2+120b2ω5δ+60c2b2ω5δ+5b2c22ω5δ+240c2λω4+234c2α2λω4λ2σ−α2−36b1αλω4λ2σ−α2+30c0c2ω4+10c12ω4+40c22λω4+30b22λ2ω4λ2σ−α2−45c1b2αλω4λ2σ−α2+10c22α2λω4λ2σ−α2−5c2b1αλω4λ2σ−α2+10c22α2λω4λ2σ−α2+25b22λ2ω4λ2σ−α2+5c0c22ω4+5c2c12ω4+5b22c2λ2ω4λ2σ−α2+5c0b22λω4λ2σ−α2+10c1b1b2ω4λ2σ−α2+5c2b12λω4λ2σ−α2+6c2ω4+5c22ω4+5b22λω4λ2σ−α2−336c2αω4δ+24b1ω4δ40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ−60b22λαω4δλ2σ−α2+10c1c2b2ω4δ+203b1c22ω4δ−10b22c2λαω4δλ2σ−α2+40c1λω3−240b2αλ2ω3λ2σ−α2+30c1α2λω3λ2σ−α2+10c0c1ω3+40b1b2λ2ω3λ2σ−α2−80c2b2αλ2ω3λ2σ−α2−30c0b2αλω3λ2σ−α2−20c1b1αλω3λ2σ−α2+10c1c2α2λω3λ2σ−α2+30b1b2λ2ω3λ2σ−α2−20c2b2λ2ω3λ2σ−α2+10c0c1c2ω3+53c13ω3+5λ2b22c1ω3λ2σ−α2−10λ2b2c2b1ω3+10c0b1b2λω3λ2σ−α2+5b12c1λω3λ2σ−α2+2c1ω3−6b2αλω3λ2σ−α2+10c1c2ω3+10b1b2λω3λ2σ−α2−60c1αω3δ+480b2α2λω3δλ2σ−α2+180b2ω3λδ+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ−80b1b2λω3δλ2σ−α2+160c2b2α2λω3δλ2σ−α2+65c2b2λω3δ+10c0c2b2ω3δ+10c2c1b1ω3δ+5b2c12ω3δ−10b22c1λαω3δλ2σ−α2−20b2c2b1λαω3δλ2σ−α2+6b2ω3δ+10b2c2ω3δ+53b23ω3δ3−30b22αω2δ3+5b1b22δ3ω2+90b2α2δ3ω−35b2b1αωδ3+10b2c2α2δ3ω+5b2b12δ3ω,+6b1α2δ3−24c2α3δ3−5b12αδ3+10b1c2α2δ3+53b13δ3+234c2α2λ2ω2λ2σ−α2+136c2λ2ω2−36b1αλ2ω2λ2σ−α2−150b2αλ2ω2λ2σ−α2−11b1λ2αω2λ2σ−α2+64c2λ2α2ω2λ2σ−α2+40c0c2λω2−45c1b2αλ2ω2λ2σ−α2+10c12λω2+10b12λ2ω2λ2σ−α2−50b1c2αω2λ2σ−α2−50b1c2αλ2ω2λ2σ−α2−5c2b1αλ2ω2λ2σ−α2+10c22α2λ2ω2λ2σ−α2+10c22λ2ω2−5c0b1αλω2λ2σ−α2+10c0c2α2λω2λ2σ−α2+5b12λ2λ2σ−α2−20b1c2αλ2ω2λ2σ−α2+5c02c2+5c0c12ω2+5c0b22λ2ω2λ2σ−α2+10c1b1b2λ2ω2λ2σ−α2+5c2b12λ2ω2λ2σ−α2+5c0b12λω2λ2σ−α2+8c2λω2−b1αλω2λ2σ−α2+2c2α2λω2λ2σ−α2+10c0c2ω2+5c12ω2+5b22λω2λ2σ−α2+5b22λ2ω2λ2σ−α2−468c2α3λω2δλ2σ−α2−362c2λαω2δ+72b1α2λω2δλ2σ−α2+28b1λω2δ+10c0b1ω2δ−50c0c2αω2δ+90c1b2α2λω2δλ2σ−α2+35c1b2λω2δ−15c12αω2δ−20b12λαω2δλ2σ−α2+100b1c2α2λω2δλ2σ−α2+40b1c2λω2δ+10c2b1α2λω2δλ2σ−α2−20c22α3λω2δλ2σ−α2+5c2b1λω2δ−20c22αλω2δ+10c0c2b1ω2δ+10c0c1b2ω2δ−10c0b22λαω2δλ2σ−α2−20c1b1b2λαω2δλ2σ−α2+5b1c12ω2δ−10c2b12αλω2δλ2σ−α2+2b1ω2δ−10c2αω2δ+10c2b1ω2δ+10c1b2ω2δ−10b22αλω2δλ2σ−α2−60c1α3λωδλ2σ−α2+300b2α2λ2ωδλ2σ−α2−45c1λαωδ+61b2λ2ωδ+60c0b2α2λωδλ2σ−α2+40c1b1α2λωδλ2σ−α2−20c1c2α3λωδλ2σ−α2−60b1b2λ2αωδλ2σ−α2+40c2b2α2λ2ωδλ2σ−α2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+5c1b1λωδ+10b2c2λ2ωδ−20c0b1b2λαωδλ2σ−α2−10b12c1αλωδλ2σ−α2+5c02b2ωδ+10c0b1c1ωδ+12b2α2λωδλ2σ−α2+5b2λωδ−3c1αωδ−20b1b2αλωδλ2σ−α2+10c0b2ωδ+10b1c1ωδ+30c1α2λ2ωλ2σ−α2−150b2αλ3ωλ2σ−α2+16c1λ2ω−30c0b2αλ2ωλ2σ−α2−20c1b1αλ2ωλ2σ−α2+10c1c2α2λ2ωλ2σ−α2+30b1b2λ3ωλ2σ−α2−20c2b2αλ3ωλ2σ−α2+10c2λ2ω+10c0c1λω2c2λ2,+10c0b1b2λ2ωλ2σ−α2+5b12c1λ2ωλ2σ−α2+5c02c1ω−6b2αλ2ωλ2σ−α2+2c1λω+10b1b2λ2ωλ2σ−α2+10c0c1ω+22b1λ2α2δλ2σ−α2−128c2λ2α3δλ2σ−α2+5b1λ2δ−56c2λ2αδ−10b12λ2αδλ2σ−α2+40b1c2α2λ2δλ2σ−α2+5c0b1λδ−20c0c2αλδ+10b1c2λ2δ−10c0b12λαδλ2σ−α2+5c02b1δ+2b1α2λδλ2σ−α2−4c2α3λδλ2σ−α2+b1λδ−4c2αλδ−10b12αλδλ2σ−α2+10c0b1δ−11b1αλ3λ2σ−α2+64c2α2λ3λ2σ−α2+16c2λ3+10c0c2α2λ2λ2σ−α2+5b12λ3λ2σ−α2+10c0c2λ2+5c0b12λ2λ2σ−α2+53c03−b1αλ2λ2σ−α2+2c2α2λ2λ2σ−α2+5b12λ2λ2σ−α2+5c02+d+kc0−5c0+d.Substituting (31) into (29) and using (4) with (3), we get the polynomials in ω and δ, setting the coefficients of this polynomial to zero to obtain the following set of algebraic equations:(85)ω6:120c2+30c22+30b22λλ2σ−α2+53c23+5b22c2λλ2σ−α2=0,ω5:24c1−240b2αλλ2σ−α2+40c1c2−80c2b2αλλ2σ−α2+40b1b2λλ2σ−α2+5c1c22+5b22c1λλ2σ−α2+10b2c2b1λλ2σ−α2=0,ω5δ:120b2+60c2b2+5b2c22=0,ω4:240c2λ+234c2α2λλ2σ−α2−36b1αλλ2σ−α2+30c0c2+10c12+40c22λ+30b22λ2λ2σ−α2−45c1b2αλλ2σ−α2+10c22α2λλ2σ−α2−5c2b1αλλ2σ−α2+10c22α2λλ2σ−α2+25b22λ2λ2σ−α2+5c0c22+5c2c12+5b22c2λ2λ2σ−α2+5c0b22λλ2σ−α2+10c1b1b2λ2σ−α2+5c2b12λλ2σ−α2+6c2+5c22+5b22λλ2σ−α2=0,ω4δ:−336c2α+24b140b2c1+40b1c2−50c22α−60b22λαλ2σ−α2+10c1c2b2+203b1c22−10b22c2λαλ2σ−α2=0,ω3:40c1λ−240b2αλ2λ2σ−α2+30c1α2λλ2σ−α2+10c0c1+40b1b2λ2λ2σ−α2−80c2b2αλ2λ2σ−α2−30c0b2αλλ2σ−α2−20c1b1αλλ2σ−α2+10c1c2α2λλ2σ−α2+30b1b2λ2λ2σ−α2−20c2b2λ2λ2σ−α2+10c0c1c2+53c13+5λ2b22c1λ2σ−α2−10λ2b2c2b1+10c0b1b2λλ2σ−α2+5b12c1λλ2σ−α2+2c1−6b2αλλ2σ−α2+10c1c2+10b1b2λλ2σ−α2=0,ω3δ:−60c1α+480b2α2λλ2σ−α2+180b2λ+30c0b2+20b1c1−50c1c2α−80b1b2λλ2σ−α2+160c2b2α2λλ2σ−α2+65c2b2λ+10c0c2b2+10c2c1b1+5b2c12−10b22c1λαλ2σ−α2−20b2c2b1λαλ2σ−α2+6b2+10b2c2=0,ω3δ3:53b23=0,δ3ω2:−30b22α+5b1b22=0,δ3ω:90b2−35b2b1α+10b2c2α2+5b2b12=0,δ3:6b1α2−24c2α3−5b12α+10b1c2α2+53b13=0,ω2:234c2α2λ2λ2σ−α2+136c2λ2−36b1αλ2λ2σ−α2−150b2αλ2λ2σ−α2−11b1λ2αλ2σ−α2+64c2λ2α2λ2σ−α2+40c0c2λ−45c1b2αλ2λ2σ−α2+10c12λ+10b12λ2λ2σ−α2−50b1c2αλ2σ−α2−50b1c2αλ2λ2σ−α2−5c2b1αλ2λ2σ−α2+10c22α2λ2λ2σ−α2+10c22λ2−5c0b1αλλ2σ−α2+10c0c2α2λλ2σ−α2+5c02c2+5c0c12+10c0c2α2λλ2σ−α2+5b12λ2λ2σ−α2−20b1c2αλ2λ2σ−α2+5c02c2+5c0c12+5c0b22λ2λ2σ−α2+10c1b1b2λ2λ2σ−α2+5c2b12λ2λ2σ−α2+5c0b12λλ2σ−α2+8c2λ−b1αλλ2σ−α2+2c2α2λλ2σ−α2+10c0c2ω2+5c12+5b22λλ2σ−α2+5b22λ2λ2σ−α2=0,ω2δ:−468c2α3λλ2σ−α2−362c2λα+72b1α2λλ2σ−α2+28b1λ+10c0b1−50c0c2α+90c1b2α2λλ2σ−α2+35c1b2λ−15c12α−20b12λαλ2σ−α2+100b1c2α2λλ2σ−α2+40b1c2λ+10c2b1α2λλ2σ−α2−20c22α3λλ2σ−α2+5c2b1λ−20c22αλ+10c0c2b1+10c0c1b2−10c0b22λαλ2σ−α2−20c1b1b2λαλ2σ−α2+5b1c12−10c2b12αλλ2σ−α2+2b1−10c2α+10c2b1+10c1b2−10b22αλλ2σ−α2=0,ω:30c1α2λ2λ2σ−α2−150b2αλ3λ2σ−α2+16c1λ2−30c0b2αλ2λ2σ−α2−20c1b1αλ2λ2σ−α2+10c1c2α2λ2λ2σ−α2+30b1b2λ3λ2σ−α2−20c2b2αλ3λ2σ−α2+10c2λ2+10c0c1λ+10c0b1b2λ2λ2σ−α2+5b12c1λ2λ2σ−α2+5c02c1−6b2αλ2λ2σ−α2+2c1λ+10b1b2λ2λ2σ−α2+10c0c1=0,δ:22b1λ2α2λ2σ−α2−128c2λ2α3λ2σ−α2+5b1λ2−56c2λ2α−10b12λ2αλ2σ−α2+40b1c2α2λ2λ2σ−α2+5c0b1λ−20c0c2αλ+10b1c2λ2−10c0b12λαλ2σ−α2+5c02b1+2b1α2λλ2σ−α2−4c2α3λλ2σ−α2+b1λ−4c2αλ−10b12αλλ2σ−α2+10c0b1=0,ω0:−11b1αλ3λ2σ−α2+64c2α2λ3λ2σ−α2+16c2λ3+10c0c2α2λ2λ2σ−α2+5b12λ3λ2σ−α2+10c0c2λ2+5c0b12λ2λ2σ−α2+53c03−b1αλ2λ2σ−α2+2c2α2λ2λ2σ−α2+2c2λ2+5b12λ2λ2σ−α2+5c02+c0+d=0.Solving the above equations of algebraic type by Mathematica software, we have.Result 3.
(86)c0=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ,c1=0,c2=−12,b1=6475763087,b2=0,α=−1618944851.
By substituting (86) into (31) with (13) by using (6) and (16), we get the exact solutions of (18) as follows:(87)W1=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+6475763087δ−12ω2.
By substituting the value ofω and δ from (3), we get(88)=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ/735−26209667236+23532201λ2σ+647576/30871/Z−12Z′/Z2.
When we put the value ofZ from (3), we get(89)=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+M2cosξ−λ+α/λ−12M1sinξλ+M2cosξλ+α/λ′M1sinξλ+M2cosξλ+α/λ2,=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+M2cosξλ+α/λ−12M1cosξλλ+M2sinξλλM1sinξλ+M2cosξλ+α/λ2.
Then, the exact solution of2+1-dimensional equation from (18) is as follows:(90)=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+M2cosξλ+α/λ−12M1cosξλλ+M2sinξλλ2M1sinξλ+M2coshξλ+α/λ2.
By lettingM1=0, α = −161894/4851 and M2>0 in (90), the particular solution of (18) is(91)We=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M2cosξλ+α/λ−12−λM2sinξλ2M2cosξλ2,(92)We=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M2cosξλ+α/λ+12λtan2ξλ.
Then, by lettingλ=0.4, t=0, y=3, α=−161894/4851, and σ=1, the graph of (92) has the following forms.
Figure5 shows periodic soliton in two and three dimensions for (92). When we put M2=0 and M1>0 in (89), we produce a solitary solution:(93)Wf=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+α/λ−12λM1coshξλ2M1sinhξλ2,(94)Wf=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+α/λ+12λcoth2ξλ.
The particular solution of2+1-dimensional equation for trigonometric function solutions, if we put result on (90) when M2=0 and M1>0, has the following form.
The graph of the particular solution whereλ=0.4, σ=1, t=0, y=3, α=−161894/4851 is as follows.
Figure6 represents a periodic soliton nature of (94) in three and two dimensions.Figure 5
The figures of (92) periodic in 3D and 2D.Figure 6
The figures of (94) periodic 3D and 2D.Result 4.
(95)c0=1/5−9−40λ,c1=0,c2=−12,b1=0,b2=0,α=0.
By substituting (95) into (31) with (6), (13), and (16), we get the exact solutions of (18) as follows:(96)W1=15−9−40λ−12ω2.
By substituting the value ofω and δ from (3), we get 1/5−9−40λ−12ω2.
When we put the value ofZ from (3), we get(97)1/5−9−40λ−12M1sinξλ+M2cosξλ+αλ′/M1sinξλ+M2cosξλ+αλ2,(98)=15−9−40λ−12M1cosξλλ+M2sinξλλM1sinξλ+M2cosξλ+α/λ2.
Then, the exact solution of2+1-dimensional equation from (18) is as follows:(99)=15−9−40λ−12M1cosξλλ+M2sinξλλ2M1sinξλ+M2coshξλ+α/λ2.
By lettingM1=0, α  = 0, and M2>0 in (99), the particular solution of (18) is(100)Wg=15−9−40λ−12−λM2sinξλ2M2cosξλ2,(101)Wg=15−9−40λ+12λtan2ξλ.
Then, by lettingλ=0.5, t=0, y=4, α=0, and σ=1, the graph of (101) has the following forms.
Figure7 shows the periodic nature for (101) in three and two dimensions. But, if we put M2=0 and M1>0 in (98), we produce a solitary solution:(102)Wh=15−9−40λ−12λM1coshξλ2M1sinhξλ2,(103)Wh=15−9−40λ+12λcoth2ξλ.
The particular solution of (2 + 1)-dimensional equation for trigonometric function solutions, if we put result on (103) when M2=0 and M1>0, has the following form.
The graph of the particular solution of (103) where λ=0.5, σ=1, t=0, y=4 is as follows.
Figure8 shows periodic nature for (101) in three-dimensional space and two-dimensional plane.Figure 7
The figures of (101) periodic in 3D and 2D.Figure 8
The figures of (103) periodic in 3D and 2D.


### 3.3. Case C:λ=0—Nature of Rational Solutions

From (9), (29) and (39) become(104)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αω3ω2−2αδM12−2αM2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2ω2ω2−2αδM12−2αM2−362c2λαω2δ+136c2λ2ω2−36b1αω2ω2−2αδM12−2αM2+28b1λω2δ+30c1α2ωω2−2αδM12−2αM2−150b2αλω2−2αδM12−2αM2ω−11b1λαω2−2αδM12−2αM2+64c2λα2ω2−2αδM12−2αM2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ−56c2λ2αδ+16c2λ3.Finally by using (9), (29 and (39) become(105)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αω5M12−2αM2+480b2α2ω3M12−2αM2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2ω4M12−2αM2−468c2α3ω2δM12−2αM2−362c2λαω2δ+136c2λ2ω2−36b1αω4M12−2αM2+72b1α2ω2δM12−2αM2+28b1λω2δ+30c1α2ω3M12−2αM2−60c1α3ωδM12−2αM2−150b2αλω3M12−2αM2+300b2α2λδωM12−2αM2−11b1λαω2M12−2αM2+22b1λα2δM12−2αM2+64c2λα2ω2M12−2αM2−128c2λα3δM12−2αM2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ−56c2λ2αδ+16c2λ3.From (9) and (29), (41) becomes(106)5×W×W″=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4+30b22ω4ω2−2αδM12−2αM2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+50c1c2λω3+40b1b2ω3ω2−2αδM12−2αM2r−80c2b2αω3ω2−2αδM12−2αM2+65c2b2λω3δ−5b2αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2−45c1b2αω2ω2−2αδM12−2αM2+35c1b2λω2δ−15c12αω2δ+10c12λω2+10b12ω2ω2−2αδM12−2αM2−50b1c2αω2+40b1c2λω2δ−5c2b1αω2ω2−2αδM12−2αM2+10c22α2ω2ω2−2αδM12−2αM2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λω2ω2−2αδM12−2αM2−5c0b1αω2−2αδM12−2αM2−30c0b2αωω2−2αδM12−2αM2+10c0c2α2ω2−2αδM12−2αM2−20c1b1αωω2−2αδM12−2αM2+10c1c2α2ω2−2αδM12−2αM2ω+5b12λω2−2αδM12−2αM2+10c0c2λ2+30b1b2λωω2−2αδM12−2αM2−20b1c2αλω2−2αδM12−2αM2−20c2b2αλω2−2αδM12−2αM2ω+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+10b1c2λ2δ+5c1b1λωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ.Finally, by using (9) and (29), (41) becomes(107)5×W×W″=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4−60b22ω4αδM12−2αM2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ+50c1c2λω3+40b1b2ω5M12−2αM2−80b1b2ω3αδM12−2αM2−80c2b2αω5M12−2αM2+360c2b2α2ω3δM12−2αM2+65c2b2λω3δ−5b2αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2−45c1b2αω4M12−2αM2+90c1b2α2ω2δM12−2αM2+35c1b2λω2δ−15c12αω2δ+10c12λω2+10b12ω4M12−2αM2−20b12ω2αδM12−2αM2−50b1c2αω2+40b1c2λω2δ−5c2b1αω4M12−2αM2+30b22ω6M12−2αM2+10c2b1α2ω2δM12−2αM2+10c22α2ω4M12−2αM2−50c1c2αω3δ−20c22α3ω2δM12−2αM2+30b22ω6M12−2αM2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λω4M12−2αM2−50b22λω2αδM12−2αM2−5c0b1αω2M12−2αM2+10c0b1α2δM12−2αM2−30c0b2αω3M12−2αM2+60c0b2α2ωδM12−2αM2+10c0c2α2ω2M12−2αM2−20c0c2α3δM12−2αM2−20c1b1αω3M12−2αM2+40c1b1α2ωδM12−2αM2+5c1b1λωδ+10c1c2α2ω3M12−2αM2−20c1c2α3δωM12−2αM2+5b12λω2M12−2αM2−10b12λαδM12−2αM2+30b1b2λω3M12−2αM2−60b1b2λωαδM12−2αM2−20b1c2αλω2M12−2αM2+40b1c2α2λδM12−2αM2−20c2b2αλω2M12−2αM2+40c2b2α2λδM12−2αM2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ+10b1c2λ2δ+10c0c2λ2.From (9), (45) becomes(108)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2ω4ω2−2αδM12−2αM2+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ+5b22c1ω3ω2−2αδM12−2αM2+10b2c2b1ω3ω2−2αδM12−2αM2+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5b1b22δ3ω2+5c02c2ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22ω2ω2−2αδM12−2αM2+10c1b1b2ω2ω2−2αδM12−2αM2+5c2b12ω2ω2−2αδM12−2αM2+10c0c1c2ω3+5c0b12ω2−2αδM12−2αM2+10c0b1b2ω2−2αδM12−2αM2ω+5c0c12ω2+5b12c1ωω2−2αδM12−2αM2+5c02b1δ+5c02c1ω+5b1c12ω2δ+5c02b2ωδ+10c0b1c1ωδ+53c03.Finally by using (9), (45) becomes(109)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2ω6M12−2αM2−105b22c2ω4αδM12−2αM2+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ+5b22c1ω5M12−2αM2−10b22c1ω3αδM12−2αM2+10b2c2b1ω5M12−2αM2−20b2c2b1ω3αδM12−2αM2+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22ω4M12−2αM2−10c0b22ω2αδM12−2αM2+10c1b1b2ω4M12−2αM2−20c1b1b2ω2αδM12−2αM2+5b1c12ω2δ+5c2b12ω4M12−2αM2−10c2b12ω2αδM12−2αM2+5c0b12ω2M12−2αM2−105c0b12αδM12−2αM2+10c0b1b2ω3M12−2αM2−20c0b1b2ωαδM12−2αM2+5b12c1ω3M12−2αM2−10b12c1ωαδM12−2αM2+5b1b22δ3ω2+5c02b1δ+5c02c1ω+5c02b2ωδ+5c02c2ω2+10c0b1c1ωδ+53c03.From (9) and (29), (35) becomes(110)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αω2−2αδM12−2αM2−6b2αωω2−2αδM12−2αM2+2c2α2ω2−2αδM12−2αM2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.By using (9), (29), and (35), we get(111)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αω2M12−2αM2+2b1α2δM12−2αM2−6b2αω3M12−2αM2+12b2α2ωδM12−2αM2+2c2α2ω2M12−2αM2−4c2α3δM12−2αM2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.From (9), (29), and (43),(112)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b12ω2−2αδM12−2αM2+10c2b1δω2+10c1b2ω2δ+10b1b2ω2−2αδM12−2αM2ω+5b22ω2ω2−2αδM12−2αM2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.At the end, by using (9) and (29) in 5×W2, we get(113)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b12ω2M12−2αM2−10b12αδM12−2αM2+10c2b1δω2+10c1b2ω2δ+10b1b2ω3M12−2αM2−20b1b2αδωM12−2αM2+5b22ω4M12−2αM2−10b22ω2αδM12−2αM2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.From (105), (107), (109), (111), and (113), equation k−5W=kc0+kc1ω+kb1δ+kc2ω2+kb2ωδ−5c0−5c1ω−5b1δ−5c2ω2−5b2ωδ is equivalent to(114)120c2ω6+30c22ω6+30b22ω6M12−2αM2+30b22ω6M12−2αM2+53c23ω6+5b22c2ω6M12−2αM2+24c1ω5−240b2αω5M12−2αM2+40c1c2ω5+40b1b2ω5M12−2αM2−80c2b2αω5M12−2αM2+5c1c22ω5+5b22c1ω5M12−2αM2+10b2c2b1ω5M12−2αM2+120b2ω5δ+60c2b2ω5δ+5b2c22ω5δ−336c2αω4δ+24b1ω4δ+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ−60b22ω4αδM12−2αM2+10c1c2b2ω4δ+203b1c22ω4δ−105b22c2ω4αδM12−2αM2+240c2λω4+234c2α2ω4M12−2αM2−36b1αω4M12−2αM2+30c0c2ω4+10c12ω4+40c22λω4−45c1b2αω4M12−2αM2+10b12ω4M12−2αM2−5c2b1αω4M12−2αM2+10c22α2ω4M12−2αM2+25b22λω4M12−2αM2+5c0c22ω4+5c2c12ω4+5c0b22ω4M12−2αM2+10c1b1b2ω4M12−2αM2+5c2b12ω4M12−2αM2+6c2ω4+5c22ω4+5b22ω4M12−2αM2−60c1αω3δ+180b2ω3λδ+30c0b2ω3δ+20b1c1ω3δ−80b1b2ω3αδM12−2αM2+360c2b2α2ω3δM12−2αM2+65c2b2λω3δ−50c1c2αω3δ+10c0c2b2ω3δ+10c2c1b1ω3δ+5b2c12ω3δ−10b22c1ω3αδM12−2αM2−20b2c2b1ω3αδM12−2αM2+6b2ω3δ+10b2c2ω3δ+40c1λω3+480b2α2ω3M12−2αM2+30c1α2ω3M12−2αM2−150b2αλω3M12−2αM2+10c0c1ω3+50c1c2λω3−30c0b2αω3M12−2αM2−20c1b1αω3M12−2αM2+10c1c2α2ω3M12−2αM2+30b1b2λω3M12−2αM2+10c0c1c2ω3+53c13ω3+10c0b1b2ω3M12−2αM2+5b12c1ω3M12−2αM22c1ω3+2c1ω3−6b2αω3M12−2αM2+10c1c2ω3+10b1b2ω3M12−2αM2+90b2α2δ3ω−35b2b1αωδ3+10b2c2α2δ3ω+5b2b12δ3ω+6b1α2δ3−24c2α3δ3−5b2αδ3+10b1c2α2δ3+53b13δ3−30b22αω2δ3+5b1b22δ3ω2+53b23ω3δ3−468c2α3ω2δM12−2αM2−362c2λαω2δ+72b1α2ω2δM12−2αM2+28b1λω2δ+10c0b1ω2δ,(115)−50c0c2αω2δ+90c1b2α2ω2δM12−2αM2+35c1b2λω2δ−15c12αω2δ−20b12ω2αδM12−2αM2+40b1c2λω2δ+10c2b1α2ω2δM12−2αM2−20c22α3ω2δM12−2αM2+5c2b1λω2δ−20c22αλω2δ−50b22λω2αδM12−2αM2−10c2αω2δ+10c1b2ω2δ−10b22ω2αδM12−2αM2+136c2λ2ω2−11b1λαω2M12−2αM2+64c2λα2ω2M12−2αM2+40c0c2λω2+10c12λω2−50b1c2αω2+10c22λ2ω2−5c0b1αω2M12−2αM2+10c0c2α2ω2M12−2αM2+5b12λω2M12−2αM2−20b1c2αλω2M12−2αM2−20c2b2αλω2M12−2αM2+5c0c12ω2+5c0b12ω2M12−2αM2+5c02c2ω2+8c2λω2−b1αω2M12−2αM2+2c2α2ω2M12−2αM2+10c0c2ω2+5c12ω2+5b12ω2M12−2αM2−60c1α3ωδM12−2αM2+300b2α2λδωM12−2αM2−45c1λαωδ+61b2λ2ωδ+60c0b2α2ωδM12−2αM2+40c1b1α2ωδM12−2αM2+5c1b1λωδ−20c1c2α3δωM12−2αM2−60b1b2λωαδM12−2αM2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ−20c1c2αλωδ+10b2c2λ2ωδ−20c0b1b2ωαδM12−2αM2−10b12c1ωαδM12−2αM2+5c02b2ωδ+10c0b1c1ωδ+12b2α2ωδM12−2αM2+5b2λωδ−3c1αωδ−20b1b2αδωM12−2αM2+10c0b2ωδ+10b1c1ωδ+16c1λ2ω+10c2λ2ω+10c0c1λω+5c02c1ω+2c1λω+10c0c1ω+22b1λα2δM12−2αM2−128c2λα3δM12−2αM2+5b1λ2δ−56c2λ2αδ+10c0b1α2δM12−2αM2−20c0c2α3δM12−2αM2−10b12λαδM12−2αM2+40b1c2α2λδM12−2αM2+40c2b2α2λδM12−2αM2+5c0b1λδ−20c0c2αλδ+10b1c2λ2δ−105c0b12αδM12−2αM2+5c02b1δ+2b1α2δM12−2αM2−4c2α3δM12−2αM2+b1λδ−4c2αλδ−10b12αδM12−2αM2+10c0b1δ+16c2λ3+10c0c2λ2+2c2λ2+5c02+d,(116)ω6:120c2+30c22+30b22M12−2αM2+30b22M12−2αM2+53c23+5b22c2M12−2αM2=0,ω5:24c1−240b2αM12−2αM2+40c1c2+40b1b2M12−2αM2−80c2b2αM12−2αM2+5c1c22+5b22c1M12−2αM2+10b2c2b1M12−2αM2=0,ω5δ:120b2+60c2b2+5b2c22=0,ω4δ:−336c2α+24b1+40b2c1+40b1c2−50c22α−60b22αM12−2αM2+10c1c2b2+203b1c22−105b22c2αM12−2αM2=0,ω4:240c2λ+234c2α2M12−2αM2−36b1αM12−2αM2+30c0c2+10c12+40c22λ−45c1b2αM12−2αM2+10b12M12−2αM2−5c2b1αM12−2αM2+10c22α2M12−2αM2+25b22λM12−2αM2+5c0c22+5c2c12+5c0b22M12−2αM2+10c1b1b2M12−2αM2+5c2b12M12−2αM2+6c2+5c22+5b22M12−2αM2=0,ω3δ:−60c1α+180b2λ+30c0b2+20b1c1−80b1b2αM12−2αM2+360c2b2α2M12−2αM2+65c2b2λ−50c1c2α+10c0c2b2+10c2c1b1+5b2c12−10b22c1αM12−2αM2−20b2c2b1αM12−2αM2+6b2+10b2c2=0,ω3:40c1λ+480b2α2M12−2αM2+30c1α2M12−2αM2−150b2αλM12−2αM2+10c0c1+50c1c2λ−30c0b2αM12−2αM2−20c1b1αM12−2αM2+10c1c2α2M12−2αM2+30b1b2λM12−2αM2+10c0c1c2+53c13+10c0b1b2M12−2αM2+5b12c1M12−2αM22c1+2c1−6b2αM12−2αM2+10c1c2+10b1b2M12−2αM2=0,(117)δ3ω:90b2α2−35b2b1α+10b2c2α2+5b2b12=0,δ2:6b1α2−24c2α3−5b2α+10b1c2α2+53b13=0,δ3ω2:−30b22α+5b1b22=0,ω3δ3:53b23=0,ω2δ:−468c2α3M12−2αM2−362c2λα+72b1α2M12−2αM2+28b1λ+10c0b1−50c0c2α+90c1b2α2M12−2αM2+35c1b2λ−15c12α−20b12αM12−2αM2+40b1c2λ+10c2b1α2M12−2αM2−20c22α3M12−2αM2+5c2b1λ−20c22αλ−50b22λαM12−2αM2−10c2α+10c1b2−10b22αM12−2αM2=0,ω2:136c2λ2−11b1λαM12−2αM2+64c2λα2M12−2αM2+40c0c2λ+10c12λ−50b1c2α+10c22λ2−5c0b1αM12−2αM2+10c0c2α2M12−2αM2+5b12λM12−2αM2−20b1c2αλM12−2αM2−20c2b2αλM12−2αM2+5c0c12+5c0b12M12−2αM2+5c02c2+8c2λ−b1αM12−2αM2+2c2α2M12−2αM2+10c0c2+5c12+5b12M12−2αM2=0,(118)ωδ:−60c1α3M12−2αM2+300b2α2λM12−2αM2−45c1λα+61b2λ2+60c0b2α2M12−2αM2+40c1b1α2M12−2αM2+5c1b1λ−20c1c2α3M12−2αM2−60b1b2λαM12−2αM2+25c0b2λ−15c0c1α−20c1c2αλ−20c1c2αλ+10b2c2λ2−20c0b1b2αM12−2αM2−10b12c1αM12−2αM2+5c02b2+10c0b1c1+12b2α2M12−2αM2+5b2λ−3c1α−20b1b2αM12−2αM2+10c0b2+10b1c1=0,ω:16c1λ2+10c2λ2+10c0c1λ+5c02c1+2c1λ+10c0c1=0,δ:22b1λα2M12−2αM2−128c2λα3M12−2αM2+5b1λ2−56c2λ2α+10c0b1α2M12−2αM2−20c0c2α3M12−2αM2−10b12λαM12−2αM2+40b1c2α2λM12−2αM2+40c2b2α2λM12−2αM2+5c0b1λ−20c0c2αλ+10b1c2λ2−105c0b12αM12−2αM2+5c02b1+2b1α2M12−2αM2−4c2α3M12−2αM2+b1λ−4c2αλ−10b12αM12−2αM2+10c0b1=0,ω0:16c2λ3+10c0c2λ2+2c2λ2+5c02+d=0.Solving the above equation of algebraic type by Mathematica software, we have(119)c0=15−9−40λ,c1=0,c2=−12,b1=0,b2=0,k=0.By substituting (119) into (31) with (8), (13), and (16), we get the exact solutions of (18) as follows:(120)W1=15−9−40λ−12ω2.By substituting the value ofω and δ from (3), we get 1/5−9−40λ−12ω2; when we put the value of Z from (3), we get(121)W1=15−9−40λ−12Z′Z2.When we put the value ofZ from (8) in (120),(122)W1=15−9−40λ−12α/2ξ2+M1ξ+M2′α/2ξ2+M1ξ+M22,W1=15−9−40λ−12αξ+M1α/2ξ2+M1ξ+M22.Hence, we get the exact solution of (2 + 1)-dimensional integro-partial equation with respect to result of (120) as follows: by letting α=0, M1>0, and M2>0 in (120), the particular solution of (1) is(123)W1=15−9−40λ−12M12M1ξ+M22.The particular solution of (2 + 1)-dimensional for rational function solutions and putting the result in equation (119) when M_2 > 0 and M_1 >0 have the form of graphical representation for the particular solution of Equation (123):Figure9 shows soliton behaviour for (120) in three-dimensional space and in two-dimensional plane.Figure 9
The figures of (120) soliton in 3D and 2D.


## 3.1. Case A:λ<0—Nature of Hyperbolic Solutions

By using (3), equation W′′′′ is equivalent to(46)W′′′′=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αω3−ω2λ+2αδλ−λ2λ2σ+α2+16c2λ3+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2ω2−ω2λ+2αδλ−λ2λ2σ+α2−362c2λαω2δ+136c2λ2ω2−56c2λ2αδ−36b1αω2−ω2λ+2αδλ−λ2λ2σ+α2+28b1λω2δ+30c1α2ω−ω2λ+2αδλ−λ2λ2σ+α2−150b2αλ−ω2λ+2αδλ−λ2λ2σ+α2ω−11b1λα−ω2λ+2αδλ−λ2λ2σ+α2+64c2λα2−ω2λ+2αδλ−λ2λ2σ+α2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ.Finally, by using equation (3), equation (35) becomes(47)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3+240b2αλω5λ2σ+α2−480b2α2λω3δλ2σ+α2+240b2αλ2ω3λ2σ+α2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3−234c2α2λω4λ2σ+α2+468c2α3λω2δλ2σ+α2−234c2α2λ2ω2λ2σ+α2−362c2λαω2δ+136c2λ2ω2+36b1αλω4λ2σ+α2−56c2λ2αδ−72b1α2λω2δλ2σ+α2+36b1αλ2ω2λ2σ+α2+28b1λω2δ−30c1α2λω3λ2σ+α2+60c1α3λωδλ2σ+α2−30c1α2λ2ωλ2σ+α2+150b2αλ2ω2λ2σ+α2−300b2α2λ2ωδλ2σ+α2+150b2αλ3ωλ2σ+α2+11b1λ2αω2λ2σ+α2−22b1λ2α2δλ2σ+α2+11b1αλ3λ2σ+α2−64c2λ2α2ω2λ2σ+α2+128c2λ2α3δλ2σ+α2−64c2α2λ3λ2σ+α2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ+16c2λ3.By using (3), equation 5×W×W′′ is equivalent to(48)5×W×W″=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4+30b22ω4−ω2λ+2αδλ−λ2λ2σ+α2+10c0c1ω3+30c0b2ω3δ−20c1c2αλωδ+20b1c1ω3δ−50c1c2αω3δ+50c1c2λω3+40b1b2ω3−ω2λ+2αδλ−λ2λ2σ+α2−80c2b2αω3−ω2λ+2αδλ−λ2λ2σ+α250c0c2αω2δ+65c2b2λω3δ−5b2αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−+40c0c2λω2−45c1b2αω2−ω2λ+2αδλ−λ2λ2σ+α2+35c1b2λω2δ−15c12αω2δ+10c12λω2+10b12ω2−ω2λ+2αδλ−λ2λ2σ+α2−50b1c2αω2+40b1c2λω2δ−5c2b1αω2−ω2λ+2αδλ−λ2λ2σ+α2+5c1b1λωδ+10c22α2ω2−ω2λ+2αδλ−λ2λ2σ+α2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λω2−ω2λ+2αδλ−λ2λ2σ+α2−5c0b1α−ω2λ+2αδλ−λ2λ2σ+α2−30c0b2αω−ω2λ+2αδλ−λ2λ2σ+α2+10c0c2α2−ω2λ+2αδλ−λ2λ2σ+α2−20c1b1αω−ω2λ+2αδλ−λ2λ2σ+α2+10c1c2α2−ω2λ+2αδλ−λ2λ2σ+α2ω+5b12λ−ω2λ+2αδλ−λ2λ2σ+α2+30b1b2λω−ω2λ+2αδλ−λ2λ2σ+α2−20b1c2αλ−ω2λ+2αδλ−λ2λ2σ+α2−20c2b2αλ−ω2λ+2αδλ−λ2λ2σ+α2ω+25c0b2λωδ−15c0c1αωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ+10b1c2λ2δ+10c0c2λ2.Finally, by using (3)(49)5×W×W″=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4−30b22λω6λ2σ+α2+60b22λαω4δλ2σ+α2−30b22λ2ω4λ2σ+α2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+50c1c2λω3−40b1b2λω5λ2σ+α2+80b1b2λω3δλ2σ+α2+10b1c2λ2δ−40b1b2λ2ω3λ2σ+α2+80c2b2αλω5λ2σ+α2−160c2b2α2λω3δλ2σ+α2+80c2b2αλ2ω3λ2σ+α2+65c2b2λω3δ−5b12αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2+45c1b2αλω4λ2σ+α2−90c1b2α2λω2δλ2σ+α2+45c1b2αλ2ω2λ2σ+α2+35c1b2λω2δ−15c12αω2δ+10c12λω2−10b12λω4λ2σ+α2+20b12λαω2δλ2σ+α2−10b12λ2ω2λ2σ+α2+50b1c2αλω4λ2σ+α2−100b1c2α2λω2δλ2σ+α2+50b1c2αλ2ω2λ2σ+α2+40b1c2λω2δ+5c2b1αλω4λ2σ+α2−10c2b1α2λω2δλ2σ+α2+5c2b1αλ2ω2λ2σ+α2−10c22α2λω4λ2σ+α2+20c22α3λω2δλ2σ+α2−10c22α2λ2ω2λ2σ+α2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ−25b22λ2ω4λ2σ+α2+50b22λ2αω2δλ2σ+α2−25b22λ3ω2λ2σ+α2+5c0b1αλω2λ2σ+α2−10c0b1α2λδλ2σ+α2+5c0b1αλ2λ2σ+α2+30c0b2αλω3λ2σ+α2−60c0b2α2λωδλ2σ+α2+30c0b2αλ2ωλ2σ+α2−10c0c2α2λω2λ2σ+α2+20c0c2α3λδλ2σ+α2−10c0c2α2λ2λ2σ+α2+20c1b1αλω3λ2σ+α2−40c1b1α2λωδλ2σ+α2+20c1b1αλ2ωλ2σ+α2−10c1c2α2λω3λ2σ+α240b1c2α2λ2δλ2σ+α2+20c1c2α3λωδλ2σ+α2−10c1c2α2λ2ωλ2σ+α2−5b12λ2ω2λ2σ+α2+10b12λ2αδλ2σ+α2−30b1b2λ2ω3λ2σ+α2+60b1b2λ2αωδλ2σ+α2−30b1b2λ3ωλ2σ+α2+20b1c2αλ2ω2λ2σ+α2+20b1c2αλ3λ2σ+α2+20c2b2λ2ω3λ2σ+α2−40c2b2α2λ2ωδλ2σ+α2+20c2b2αλ3ωλ2σ+α2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ−5b12λ3λ2σ+α2+5c1b1λωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ+10c0c2λ2.By using (3), equation of 5/3×W3 becomes(50)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2ω4−ω2λ+2αδλ−λ2λ2σ+α2+5c02b2ωδ+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+53c03+5b2c12ω3δ+5b22c1ω3−ω2λ+2αδλ−λ2λ2σ+α2+5b2b12δ3ω+10b2c2b1ω3−ω2λ+2αδλ−λ2λ2σ+α2+53b23ω3δ3+53b13δ3+5b1b22δ3ω2+5c02c2ω2+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22ω2−ω2λ+2αδλ−λ2λ2σ+α2+10c1b1b2ω2−ω2λ+2αδλ−λ2λ2σ+α2+5c02c1ω+5b1c12ω2δ+5c2b12ω2−ω2λ+2αδλ−λ2λ2σ+α2+5c0b12−ω2λ+2αδλ−λ2λ2σ+α2+10c0b1b2−ω2λ+2αδλ−λ2λ2σ+α2ω+5b12c1ω−ω2λ+2αδλ−λ2λ2σ+α2+5c02b1δ+10c0b1c1ωδ.Finally, by using equations (3)(51)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+203b1c22ω4δ−5b22c2λω6λ2σ+α2+10b22c2λαω4δλ2σ+α2−5b22c2λ2ω4λ2σ+α2+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ−5b22c1λω5λ2σ+α2+10b22c1λαω3δλ2σ+α2−5λ2b22c1ω3λ2σ+α2−10b2c2b1λω5λ2σ+α2+20b2c2b1λαω3δλ2σ+α2−10λ2b2c2b1ω3+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5b1b22δ3ω2+5c02c2ω2+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ−5c0b22λω4λ2σ+α2+10c0b22λαω2δλ2σ+α2−5c0b22λ2ω2λ2σ+α2−10c1b1b2ω4λ2σ+α2+20c1b1b2λαω2δλ2σ+α2−10c1b1b2λ2ω2λ2σ+α2+5b1c12ω2δ−5c2b12λω4λ2σ+α2+10c2b12αλω2δλ2σ+α2−5c2b12λ2ω2λ2σ+α2−5c0b12λω2λ2σ+α2+10c0b12λαδλ2σ+α2−5c0b12λ2λ2σ+α2−10c0b1b2λω3λ2σ+α2+20c0b1b2λαωδλ2σ+α2−10c0b1b2λ2ωλ2σ+α2−5b12c1λω3λ2σ+α2+10b12c1αλωδλ2σ+α2−5b12c1λ2ωλ2σ+α2+5c02b1δ+5c02c1ω+5c02b2ωδ+10c1c2b2ω4δ+10c0b1c1ωδ+53c03.By using (3) and (35), W′′ is equivalent to(52)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1α−ω2λ+2αδλ−λ2λ2σ+α2−6b2αω−ω2λ+2αδλ−λ2λ2σ+α2+2c2α2−ω2λ+2αδλ−λ2λ2σ+α2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.Finally, by using (3), (35) becomes(53)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2+b1αλω2λ2σ+α2−2b1α2λδλ2σ+α2+b1αλ2λ2σ+α2+6b2αλω3λ2σ+α2−12b2α2λωδλ2σ+α2+6b2αλ2ωλ2σ+α2−2c2α2λω2λ2σ+α2+4c2α3λδλ2σ+α2−2c2α2λ2λ2σ+α2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.By using (3) and (43), 5×W2 is equal to the following form:(54)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b12−ω2λ+2αδλ−λ2λ2σ+α2+10c2b1δω2+10c1b2ω2δ+10b1b2−ω2λ+2αδλ−λ2λ2σ+α2ω+5b22ω2−ω2λ+2αδλ−λ2λ2σ+α2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.By using equations (3) and (43), 5×W2 becomes(55)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2−5b22λω2λ2σ+α2+10b12αλδλ2σ+α2−5b12λ2λ2σ+α2+10c2b1ω2δ+10c1b2ω2δ−10b1b2λω3λ2σ+α2+20b1b2αλωδλ2σ+α2−10b1b2λ2ωλ2σ+α2−5b22λω4λ2σ+α2+10b22αλω2δλ2σ+α2−5b22λ2ω2λ2σ+α2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.By using equations (47), (49), (51), and (53), k−5W=kc0+kc1ω+kb1δ+kc2ω2+kb2ωδ−5c0−5c1ω−5b1δ−5c2ω2−5b2ωδ becomes the following:(56)120c2ω6+30c22ω6−30b22ω6λ2σ+α2+53c23ω6−5b22c2λω6λ2σ+α2+24c1ω5+240b2αλω5λ2σ+α2+40c1c2ω5−40b1b2λω5λ2σ+α2+80c2b2αλω5λ2σ+α2+5c1c22−5λb22c1ω5λ2σ+α2−10b2c2b1λω5λ2σ+α2+120b2ω5δ+60c2b2ω5δ+5b2c22ω5δ+240c2λω4−234c2α2λω4λ2σ+α2+36b1αλω4λ2σ+α2+30c0c2ω4+10c12ω4+40c22λω4−30b22λ2ω4λ2σ+α2+45c1b2αλω4λ2σ+α2−10b12λω4λ2σ+α2+50b1c2αλω4λ2σ+α2+5c2b1αλω4λ2σ+α2−30b1b2λ2ω3λ2σ+α2−10c22α2λω4λ2σ+α2+50c0c22ω4+5c2c12ω4−5b22c2λ2ω4λ2σ+α2−5λc0b22ω4λ2σ+α2−10c1b1b2λω4λ2σ+α2−5c2b12λω4λ2σ+α2+6c2ω4+5c22ω4−25b22λ2ω4λ2σ+α2−60c1αω3δ−5b22λω4λ2σ+α2−336c2αω4δ+24b1ω4δ+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+60b22λαω4δλ2σ+α2+10c1c2b2ω4δ+203b1c22ω4δ+10b22c2λαω4δλ2σ+α2+40c1λω3+240b2αλ2ω3λ2σ+α2+150b2αλ2ω3λ2σ+α2+10c0c1ω3+50c1c2λω3−40b1b2λ2ω3λ2σ+α2−30c1α2λω3λ2σ+α2+80c2b2αλ2ω3λ2σ+α2+20c1b1αλω3λ2σ+α2−10c1c2α2λω3λ2σ+α2+20c2b2λ2ω3λ2σ+α2+30c0b2αλω3λ2σ+α2+10c0c1c2ω3+53c13ω3−5b22c1λ2ω3λ2σ+α2+6b2αλω3λ2σ+α2−10b2c2b1λ2ω3λ2σ+α2+2c1ω3+10c1c2ω3−10b1b2λω3λ2σ+α2−10c0b1b2λω3λ2σ+α2−5b12c1λω3λ2σ+α2−480b2α2λω3δλ2σ+α2+180b2λω3δ+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+80b1b2αλω3δλ2σ+α2−160c2b2α2λω3δλ2σ+α2+65c2b2λω3δ+10c0c2b2ω3δ+10c2c1b1ω3δ+5b2c12+10b22c1αλω3δλ2σ+α2+20b2c2b1αλω3δ+6b2ω3δ+10b2c2ω3δ+53b23ω3δ3+53b23ω3δ3−30b22αδ3ω2+5b1b22δ3ω2+90b2α2δ3ω−35b2b1αδ3ω+10b2c2α2δ3ω+10c0c2ω2+5b2b12δ3ω−234c2α2λ2ω2λ2σ+α2+136c2λ2ω2+5c2b1αλ2ω2λ2σ+α2+36b1αλ2ω2λ2σ+α2+11b1λ2αω2λ2σ+α2−64c2λ2α2ω2λ2σ+α2+40c0c2λω2+45c1b2αλ2ω2λ2σ+α2+10c12λω2−10b12λ2ω2λ2σ+α2−10c22α2λ2ω2λ2σ+α2+10c22λ2ω2−25b22λ3ω2λ2σ+α2+5c0b1αλ2ω2λ2σ+α2−10c0c2α2λω2λ2σ+α2−5b12λ2ω2λ2σ+α2+20b1c2αλ2ω2λ2σ+α2+5c02c2ω2+5c0c12ω2−5c0b22λ2ω2λ2σ+α2+50b1c2αλ2ω2λ2σ+α2−10c1b1b2λ2ω2λ2σ+α2−5c2b12λ2ω2λ2σ+α2−5c0b12λω2λ2σ+α2+8c2λω2+b1αλω2λ2σ+α2−2c2α2λω2λ2σ+α2+5c12ω2−5b12λω2λ2σ+α2−5b22λ2ω2λ2σ+α2+c2kω2−5c2ω2+468c2α3λω2δλ2σ+α2−362c2αλω2δ−72b1α2λω2δλ2σ+α2+28b1λω2δ+10c0b1ω2δ−50c0c2αω2δ−90c1b2α2λω2δλ2σ+α2+35c1b2λω2δ−15c12αω2δ+20b12αλω2δλ2σ+α2−100b1c2α2λω2δλ2σ+α2+40b1c2λω2δ−10c2b1α2λω2δλ2σ+α2+20c22α3λω2δλ2σ+α2+5c2b1λω2δ−20c22αλω2δ+50b22αλ2ω2δλ2σ+α2+10c0c2b1ω2δ+10c0c1b2ω2δ+10c0b22αλω2δλ2σ+α2+20c1b1b2λαω2δλ2σ+α2+5b1c12ω2δ+10c2b12αλω2δλ2σ+α2+2b1ω2δ−10c2αω2δ+10c2b1ω2δ+10c1b2ω2δ+10b22λαω2δλ2σ+α2+60c1α3λωδλ2σ+α2−300b2α2λ2ωδλ2σ+α2+61b2λ2ωδ−60c0b2α2λωδλ2σ+α2−40c1b1α2λωδλ2σ+α2+20c1c2α3λωδλ2σ+α2+60b1b2αλ2ωδλ2σ+α2−40c2b2α2λ2ωδ+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+5c1b1λωδ+10b2c2λ2ωδ+20c0b1b2αλωδλ2σ+α2+10b12c1αλωδλ2σ+α2+5c02b2ωδ+10c0b1c1ωδ−12b2α2λωδλ2σ+α2+5b2λωδ−3c1αωδ+10c0b2ωδ+10b1c1ωδ+b2kωδ−5b2ωδ−45c1αλωδ+20b1b2αλωδλ2σ+α2+150b2αλ3ωλ2σ+α2−30c1α2λ2ωλ2σ+α2+16c1λ2ω+30c0b2αλ2ωλ2σ+α2+20c1b1αλ2ωλ2σ+α2−10c1c2α2λ2ωλ2σ+α2−30b1b2λ3ωλ2σ+α2+20c2b2αλ3ωλ2σ+α2+10c2λ2ω+10c0c1λω−10c0b1b2λ2ωλ2σ+α2−5b12c1λ2ωλ2σ+α2+5c02c1ω+6b2αλ2ωλ2σ+α2+2c1λω−10b1b2λ2ωλ2σ+α2+10c0c1ω+c1kω−5c1ω−22b1α2λ2δλ2σ+α2+128c2α3λ2δλ2σ+α2+5b1λ2δ−56c2αλ2δ−10c0b1α2λδλ2σ+α2+20c0c2α3λδλ2σ+α2+10b12αλ2δλ2σ+α2−40b1c2α2λ2δλ2σ+α2+5c0b1λδ−20c0c2αλδ+10b1c2λ2δ+10c0b12αλδλ2σ+α2+5c02b1δ−2b1α2λδλ2σ+α2+4c2α3λδλ2σ+α2+b1λδ−4c2αλδ+10b12αλδλ2σ+α2+10c0b1δ+b1kδ−5b1δ+11b1αλ3λ2σ+α2−64c2α2λ3λ2σ+α2+16c2λ3−15c0b12λ2λ2σ+α2+53c03+5c0αλ2+5c0b1αλ2λ2σ+α2−10c0c2α2λ2λ2σ+α2−5b12λ3λ2σ+α2+20b1c2αλ3λ2σ+α2+10c0c2λ2+b1αλ2λ2σ+α2−2c2α2λ2λ2σ+α2+2c2λ2+5c02−5b12λ2λ2σ+α2+kc0−5c0+d.Inserting (31) in (29) and applying (4) as well as (3), we have polynomials in the variable form of ω and δ. Then, we obtain equations of algebraic sets by applying the coefficients of the polynomials set to zero. Hence, the following algebraic expressions are obtained:(57)ω6:120c2+30c22−30b22λ2σ+α2+53c23−5b22c2λλ2σ+α2=0,ω5:24c1+240b2αλλ2σ+α2+40c1c2−40b1b2λλ2σ+α2+80c2b2αλλ2σ+α2+5c1c22−5λb22c1λ2σ+α2−10b2c2b1λλ2σ+α2=0,ω5δ:120b2+60c2b2+5b2c22=0,ω4:240c2λ−234c2α2λλ2σ+α2+36b1αλλ2σ+α2+30c0c2+10c12+40c22λ−30b22λ2λ2σ+α2+45c1b2αλλ2σ+α2−10b12λλ2σ+α2+50b1c2αλλ2σ+α2−10c22α2λλ2σ+α2+50c0c22+5c2c12−5b22c2λ2λ2σ+α2−5λc0b22λ2σ+α2−10c1b1b2λλ2σ+α2−5c2b12λλ2σ+α2+6c2+5c22−25b22λ2λ2σ+α2+5c2b1αλλ2σ+α2−5b22λλ2σ+α2=0,ω4δ:−336c2αω4δ+24b1ω4δ+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+60b22λαω4δλ2σ+α2+10c1c2b2ω4δ+203b1c22ω4δ+10b22c2λαω4δλ2σ+α2=0,ω3:+40c1λ+240b2αλ2λ2σ+α2−30c1α2λλ2σ+α2+150b2αλ2λ2σ+α2+10c0c1ω3+50c1c2λω3−40b1b2λ2λ2σ+α2+80c2b2αλ2λ2σ+α2+20c1b1αλλ2σ+α2−10c1c2α2λλ2σ+α2+53c13−30b1b2λ2λ2σ+α2+20c2b2λ2λ2σ+α2+30c0b2αλλ2σ+α2+10c0c1c2−5b22c1λ2λ2σ+α2+6b2αλλ2σ+α2−10b2c2b1λ2λ2σ+α2+2c1+10c1c2−10b1b2λλ2σ+α2−10c0b1b2λλ2σ+α2−5b12c1λλ2σ+α2=0,ω3δ:−60c1α−480b2α2λλ2σ+α2+180b2+30c0b2+20b1c1−50c1c2α+80b1b2αλλ2σ+α2−160c2b2α2λλ2σ+α2+65c2b2λ+10c0c2b2+10c2c1b1+5b2c12+10b22c1αλλ2σ+α2+20b2c2b1αλ+6b2+10b2c2=0,ω3δ3:53b23=0,δ3:6b1α2−24c2α3−5b12+10b1c2+53b13=0,δ3ω2:−30b22α+5b1b22=0,δ3ω:90b2α2−35b2b1α+10b2c2α2+5b2b12=0,ω2:−234c2α2λ2λ2σ+α2+136c2λ2+36b1αλ2λ2σ+α2+11b1λ2αλ2σ+α2−64c2λ2α2λ2σ+α2+40c0c2λ+45c1b2αλ2λ2σ+α2+10c12λ−10b12λ2λ2σ+α2+5c2b1αλ2λ2σ+α2−10c22α2λ2λ2σ+α2+10c22λ2−25b22λ3λ2σ+α2+5c0b1αλ2λ2σ+α2−10c0c2α2λλ2σ+α2−5b12λ2λ2σ+α2+20b1c2αλ2λ2σ+α2+5c02c2+5c0c12−5c0b22λ2λ2σ+α2+50b1c2αλ2λ2σ+α2−10c1b1b2λ2λ2σ+α2−5c2b12λ2λ2σ+α2−5c0b12λλ2σ+α2+8c2λ+b1αλλ2σ+α2−2c2α2λλ2σ+α2+10c0c2+5c12−5b12λλ2σ+α2−5b22λ2λ2σ+α2+c2k−5c2=0,ω2δ:468c2α3λλ2σ+α2−362c2αλ−72b1α2λλ2σ+α2+28b1λ+10c0b1−50c0c2α−90c1b2α2λλ2σ+α2+35c1b2λ−15c12α+20b12αλλ2σ+α2−100b1c2α2λλ2σ+α2+40b1c2λ−10c2b1α2λλ2σ+α2+20c22α3λλ2σ+α2+5c2b1λ−20c22αλ+50b22αλ2λ2σ+α2+10c0c2b1+10c0c1b2+10c0b22αλλ2σ+α2+20c1b1b2λαλ2σ+α2+5b1c12+10c2b12αλλ2σ+α2+2b1−10c2α+10c2b1+10c1b2+10b22λαλ2σ+α2=0,ωδ:60c1α3λλ2σ+α2−300b2α2λ2λ2σ+α2+61b2λ2−60c0b2α2λλ2σ+α2−40c1b1α2λλ2σ+α2+60b1b2αλ2λ2σ+α2−40c2b2α2λ2+25c0b2λ−15c0c1α−20c1c2αλ+5c1b1λ+10b2c2λ2+20c0b1b2αλλ2σ+α2+20c1c2α3λλ2σ+α2+10b12c1αλλ2σ+α2+5c02b2+10c0b1c1−12b2α2λλ2σ+α2+5b2λ−3c1α+10c0b2+10b1c1+b2k−5b2−45c1αλ+20b1b2αλλ2σ+α2=0,ω:150b2αλ3λ2σ+α2−30c1α2λ2λ2σ+α2+16c1λ2+30c0b2αλ2λ2σ+α2+20c1b1αλ2λ2σ+α2−30b1b2λ3λ2σ+α2+20c2b2αλ3λ2σ+α2+10c2λ2+10c0c1λ−10c0b1b2λ2λ2σ+α2−5b12c1λ2λ2σ+α2+5c02c1+6b2αλ2λ2σ+α2+2c1λ−10b1b2λ2λ2σ+α2−10c1c2α2λ2λ2σ+α2+10c0c1+c1k−5c1=0,δ:−22b1α2λ2λ2σ+α2+128c2α3λ2λ2σ+α2+5b1λ2−56c2αλ2−10c0b1α2λλ2σ+α2+20c0c2α3λλ2σ+α2−40b1c2α2λ2λ2σ+α2+5c0b1λ−20c0c2αλ+10b1c2λ2+10c0b12αλλ2σ+α2+5c02b1−2b1α2λλ2σ+α2+4c2α3λλ2σ+α2+b1λ−4c2αλ+10b12αλ2λ2σ+α2+10b12αλλ2σ+α2+10c0b1+b1k=0,ω0:−5b1δ+11b1αλ3λ2σ+α2−64c2α2λ3λ2σ+α2+16c2λ3−15c0b12λ2λ2σ+α2+53c03+5c0αλ2+5c0b1αλ2λ2σ+α2−10c0c2α2λ2λ2σ+α2−5b12λ3λ2σ+α2+20b1c2αλ3λ2σ+α2+10c0c2λ2+b1αλ2λ2σ+α2−2c2α2λ2λ2σ+α2+2c2λ2+5c02−5b12λ2λ2σ+α2+kc0−5c0+d=0.By solving the previous algebraic equations using the Mathematica software package 9.1, we get the following results.Result 1.
(58)c0=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ,c1=0,c2=−12,b1=44α7,b2=0,k=±416λ2−9d.
By substituting (58) into (31) with (13), (16), and (4), we get the exact solutions of (18) as follows:(59)W1=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ+44α7δ−12ω2.
By substituting the value ofω and δ from (3) we get  = −3969α2−14095α2λ−3969λ2σ−17640λ2σ/41895α2+λ2σ+44α/71/Z−12Z′/Z2.
When we put the value ofZ from equation (3), we get(60)=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ+44α71M1sinhξ−λ+M2coshξ−λ+α/λ−12M1sinhξ−λ+M2coshξ−λ+α/λ′M1sinhξ−λ+M2coshξ−λ+α/λ2,=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ+44α71M1sinhξ−λ+M2coshξ−λ+α/λ−12M1coshξ−λ−λ+M2sinhξ−λ−λM1sinhξ−λ+M2coshξ−λ+α/λ2.
Then, the exact solution of2+1-dimensional Sawada–Kotera equation of (18) is as follows:(61)=−3969α2−14095α2λ−3969λ2σ−17640λ2σ41895α2+λ2σ+44α71M1sinhξ−λ+M2coshξ−λ+α/λ−12M1coshξ−λ−λ+M2sinhξ−λ−λ2M1sinhξ−λ+M2coshξ−λ+α/λ2.
By lettingM1=0, α=0, and M2>0 in (61), the particular solution of (18) is(62)W1a=−3969λ2σ−17640λ2σ41895λ2σ−12−λM2sinhξ−λ2M2coshξ−λ2,(63)W1a=−3969λ2σ−17640λ2σ41895λ2σ+12λtanh2ξ−λ.
Then, by lettingλ=0.4, α=0, t=0, y=4, and σ=1, the graph of (63) has the following forms.
Figure1 indicates the soliton of (63) in three and two dimensions. If we put M2=0 and M1>0 in (61), we produce a solitary solution:(64)W1b=−3969λ2σ−17640λ2σ41895λ2σ−12−λM1coshξ−λ2M1sinhξ−λ2,(65)W1b=−3969λ2σ−17640λ2σ41895λ2σ+12λcoth2ξ−λ.
The particular solution of2+1-dimensional equation for hyperbolic function, if we put result on (61) when M2=0 and M1>0, has the following form.
The graph of the particular solution whereλ=−0.4, σ=1, t=0, y=0 is as follows.
Figure2 shows the soliton of (65) in three and 2 dimensions by considering certain parameters.Figure 1
The figures of equation (53) soliton in 3D and 2D.Figure 2
The figures of equation (55) soliton in 3D and 2D.Result 2.
(66)c0=−8α2−295α2λ−8λ2σ90α2+λ2σ,c1=0,c2=−6,b1=−9α,b2=0,k=λ2−2d.
By substituting (66) into (31) with (13) by using (4) and (16), we get the exact solutions of equation (18) as follows:(67)Wc=−8α2−295α2λ−8λ2σ90α2+λ2σ−9αδ−6ω2.
By substituting the value in place ofω and δ from (3), we get  =  −8α2−295α2λ−8λ2σ/90α2+λ2σ−9α1/Z−6Z′/Z2; when we put the value of Z from (3), we get(68)−8α2−295α2λ−8λ2σ90α2+λ2σ−9α1M1sinhξ−λ+M2coshξ−λ+α/λ−6M1coshξ−λ−λ+M2sinhξ−λ−λM1sinhξ−λ+M2coshξ−λ+α/λ2.
Then, the exact solution of2+1-dimensional equation from (18) is as follows:(69)=−8α2−295α2λ−8λ2σ90α2+λ2σ−9α1M1sinhξ−λ+M2coshξ−λ+α/λ−6M1coshξ−λ−λ+M2sinhξ−λ−λ2M1sinhξ−λ+M2coshξ−λ+α/λ2.
By lettingM1=0, α=0, and M2>0 in (69), the particular solution of (18) is(70)Wc=−8λ2σ90λ2σ−6−λM2sinhξ−λ2M2coshξ−λ2,(71)Wc=−8λ2σ90λ2σ+6λtanh2ξ−λ.
Then, by lettingλ=−0.3, α=0, t=0, y=2, and σ=−0.5, the graph of (71) has the following forms.
Figure3 represents the soliton of (71) in two and three dimensions using certain parameters.
If we putM2=0 and M1>0 in (61), we produce a solitary solution:(72)W1d=−8λ2σ90λ2σ−6−λM1coshξ−λ2M1sinhξ−λ2,(73)W1d=−8λ2σ90λ2σ+6λcoth2ξ−λ.
The particular solution of (2 + 1)-dimensional equation for hyperbolic function, if we put result on equation (51) when M2=0 and M1>0, has the following form.
The graph of the particular solution whereλ=−0.3, σ=−o.5, t=0, y=2,α=0 is as follows.
Figure4 illustrates the soliton of (73) in three-dimensional space and two-dimensional plane.Figure 3
The figures of (71) soliton in 3D and 2D.Figure 4
The figures of equation (63) soliton in 3D and 2D.


## 3.2. Case B:λ>0—Nature of Trigonometric Solutions

From (7), (29) and (39) become(74)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αω3ω2λ−2αδλ+λ2λ2σ−α2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2ω2ω2λ−2αδλ+λ2λ2σ−α2−362c2λαω2δ+136c2λ2ω2−36b1αω2ω2λ−2αδλ+λ2λ2σ−α2+61b2λ2ωδ+28b1λω2δ+30c1α2ωω2λ−2αδλ+λ2λ2σ−α2−150b2αλω2λ−2αδλ+λ2λ2σ−α2ω−11b1λαω2λ−2αδλ+λ2λ2σ−α2+64c2λα2ω2λ−2αδλ+λ2λ2σ−α2−45c1λαωδ+16c1λ2ω+5b1λ2δ−56c2λ2αδ+16c2λ3.Finally, by using (7), (29) and (35) become(75)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αλω5λ2σ−α2+480b2α2λω3δλ2σ−α2−240b2αλ2ω3λ2σ−α2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2λω4λ2σ−α2−468c2α3λω2δλ2σ−α2+234c2α2λ2ω2λ2σ−α2−362c2λαω2δ+136c2λ2ω2−36b1αλω4λ2σ−α2+72b1α2λω2δλ2σ−α2−36b1αλ2ω2λ2σ−α2+28b1λω2δ+30c1α2λω3λ2σ−α2−60c1α3λωδλ2σ−α2+30c1α2λ2ωλ2σ−α2−150b2αλ2ω2λ2σ−α2+300b2α2λ2ωδλ2σ−α2−150b2αλ3ωλ2σ−α2−11b1λ2αω2λ2σ−α2+22b1λ2α2δλ2σ−α2−11b1αλ3λ2σ−α2+64c2λ2α2ω2λ2σ−α2−128c2λ2α3δλ2σ−α2+64c2α2λ3λ2σ−α2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ−56c2λ2αδ+16c2λ3.From (7), (29) and (41) become(76)5×W×W′′=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4+30b22ω4ω2λ−2αδλ+λ2λ2σ−α2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+50c1c2λω3+10c12λω2+40b1b2ω3ω2λ−2αδλ+λ2λ2σ−α2−80c2b2αω3ω2λ−2αδλ+λ2λ2σ−α2+65c2b2λω3δ−5b2αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2−45c1b2αω2ω2λ−2αδλ+λ2λ2σ−α2+35c1b2λω2δ−15c12αω2δ+10b12ω2ω2λ−2αδλ+λ2λ2σ−α2−50b1c2αω2+40b1c2λω2δ−5c2b1αω2ω2λ−2αδλ+λ2λ2σ−α2+10c22α2ω2ω2λ−2αδλ+λ2λ2σ−α2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λω2ω2λ−2αδλ+λ2λ2σ−α2−5c0b1αω2λ−2αδλ+λ2λ2σ−α2−30c0b2αωω2λ−2αδλ+λ2λ2σ−α2+10c0c2α2ω2λ−2αδλ+λ2λ2σ−α2−20c1b1αωω2λ−2αδλ+λ2λ2σ−α2+10c1c2α2ω2λ−2αδλ+λ2λ2σ−α2ω+5b12λω2λ−2αδλ+λ2λ2σ−α2+30b1b2λωω2λ−2αδλ+λ2λ2σ−α2−20b1c2αλω2λ−2αδλ+λ2λ2σ−α2−20c2b2αλω2λ−2αδλ+λ2λ2σ−α2ω+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+5c1b1λωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ+10b1c2λ2δ+10c0c2λ2.Finally, by using (7), (29) and (41) become(77)5×W×W′′=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4+30b22λω6λ2σ−α2−60b22λαω4δλ2σ−α2+30b22λ2ω4λ2σ−α2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+40b1b2λω5λ2σ−α2−80b1b2λω3δλ2σ−α2+40b1b2λ2ω3λ2σ−α2−80c2b2αλω5λ2σ−α2+160c2b2α2λω3δλ2σ−α2−80c2b2αλ2ω3λ2σ−α2+65c2b2λω3δ−5b12αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2−45c1b2αλω4λ2σ−α2+90c1b2α2λω2δλ2σ−α2−45c1b2αλ2ω2λ2σ−α2+35c1b2λω2δ−15c12αω2δ+10c12λω2+10b12λω4λ2σ−α2−20b12λαω2δλ2σ−α2+10b12λ2ω2λ2σ−α2−50b1c2αω2λ2σ−α2+100b1c2α2λω2δλ2σ−α2−50b1c2αλ2ω2λ2σ−α2+10b2c2α2δ3ω+40b1c2λω2δ−5c2b1αλω4λ2σ−α2+10c2b1α2λω2δλ2σ−α2−5c2b1αλ2ω2λ2σ−α2+10c22α2λω4λ2σ−α2−20c22α3λω2δλ2σ−α2+10c22α2λ2ω2λ2σ−α2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λ2ω4λ2σ−α2−5c0b1αλω2λ2σ−α2+10c0b1α2λδλ2σ−α2−5c0b1αλ2λ2σ−α2−30c0b2αλω3λ2σ−α2+60c0b2α2λωδλ2σ−α2−30c0b2αλ2ωλ2σ−α2+10c0c2α2λω2λ2σ−α2−20c0c2α3λδλ2σ−α2+10c0c2α2λ2λ2σ−α2−20c1b1αλω3λ2σ−α2+10c1c2α2λω3λ2σ−α2−20c1c2α3λωδλ2σ−α2+10c1c2α2λ2ωλ2σ−α2+5b12λ2ω2λ2σ−α2−10b12λ2αδλ2σ−α2+40c1b1α2λωδλ2σ−α2−20c1b1αλ2ωλ2σ−α2+5b12λ3λ2σ−α2+30b1b2λ2ω3λ2σ−α2−60b1b2λ2αωδλ2σ−α2+30b1b2λ3ωλ2σ−α2−20b1c2αλ2ω2λ2σ−α2+40b1c2α2λ2δλ2σ−α2−20c2b2λ2ω3λ2σ−α2+40c2b2α2λ2ωδλ2σ−α2−20c2b2αλ3ωλ2σ−α2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+5c1b1λωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1−20b1c2αλ3λ2σ−α2λω−20c0c2αλδ+10b1c2λ2δ+10c0c2λ2.From (7), (29) and (45) become(78)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2ω4ω2λ−2αδλ+λ2λ2σ−α2+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ+5b22c1ω3ω2λ−2αδλ+λ2λ2σ−α2+10b2c2b1ω3ω2λ−2αδλ+λ2λ2σ−α2+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5b1b22δ3ω2+5c02c2ω2+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22ω2ω2λ−2αδλ+λ2λ2σ−α2+5c02c1ω+5c02b1δ+10c1b1b2ω2ω2λ−2αδλ+λ2λ2σ−α2+5b1c12ω2δ+5c2b12ω2ω2λ−2αδλ+λ2λ2σ−α2+5c0b12ω2λ−2αδλ+λ2λ2σ−α2+10c0b1b2ω2λ−2αδλ+λ2λ2σ−α2ω+5b12c1ωω2λ−2αδλ+λ2λ2σ−α2+5c02b2ωδ+10c0b1c1ωδ+53c03.By using (7), (29) and (45) become(79)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2λω6λ2σ−α2−10b22c2λαω4δλ2σ−α2+5b22c2λ2ω4λ2σ−α2+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ+5b22c1λω5λ2σ−α2−10b22c1λαω3δλ2σ−α2+5λ2b22c1ω3λ2σ−α2+10b2c2b1λω5λ2σ−α2−20b2c2b1λαω3δλ2σ−α2+5c02b1δ−10λ2b2c2b1ω3+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5b1b22δ3ω2+5c02c2ω2+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22λω4λ2σ−α2−10c0b22λαω2δλ2σ−α2+5c0b22λ2ω2λ2σ−α2+10c1b1b2ω4λ2σ−α2−20c1b1b2λαω2δλ2σ−α2+10c1b1b2λ2ω2λ2σ−α2+5b1c12ω2δ+5c2b12λω4λ2σ−α2−10c2b12αλω2δλ2σ−α2+5c2b12λ2ω2λ2σ−α2+5c0b12λω2λ2σ−α2−10c0b12λαδλ2σ−α2+5c0b12λ2λ2σ−α2+10c0b1b2λω3λ2σ−α2−20c0b1b2λαωδλ2σ−α2+10c0b1b2λ2ωλ2σ−α2+5b12c1λω3λ2σ−α2−10b12c1αλωδλ2σ−α2+5b12c1λ2ωλ2σ−α2+5c02c1ω+5c02b2ωδ+10c0b1c1ωδ+53c03.From (7), (29) and (35) become(80)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αω2λ−2αδλ+λ2λ2σ−α2−6b2αωω2λ−2αδλ+λ2λ2σ−α2+2c2α2ω2λ−2αδλ+λ2λ2σ−α2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.Again considering (7), (29) and (35) become(81)W′′=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αλω2λ2σ−α2+2b1α2λδλ2σ−α2−b1αλ2λ2σ−α2−6b2αλω3λ2σ−α2+12b2α2λωδλ2σ−α2−6b2αλ2ωλ2σ−α2+2c2α2λω2λ2σ−α2−4c2α3λδλ2σ−α2+2c2α2λ2λ2σ−α2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.From (7), (29) and (43) become(82)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b12ω2λ−2αδλ+λ2λ2σ−α2+10c2b1δω2+10c1b2ω2δ+10b1b2ω2λ−2αδλ+λ2λ2σ−α2ω+5b22ω2ω2λ−2αδλ+λ2λ2σ−α2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.Using (7), (29) and (43) become(83)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b22λω2λ2σ−α2−10b12αλδλ2σ−α2+5b12λ2λ2σ−α2+10c2b1ω2δ+10c1b2ω2δ+10b1b2λω3λ2σ−α2−20b1b2αλωδλ2σ−α2+10b1b2λ2ωλ2σ−α2+5b22λω4λ2σ−α2−10b22αλω2δλ2σ−α2+5b22λ2ω2λ2σ−α2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.By using (75), (77), (79), (81) and (83), k−5W=kc0+kc1ω+kb1δ+kc2ω2+kb2ωδ−5c0−5c1ω−5b1δ−5c2ω2−5b2ωδ becomes the following forms.(84)120c2ω6+30c22ω6+30b22λω6λ2σ−α2+53c23ω6+5b22c2λω6λ2σ−α2+24c1ω5−240b2αλω5λ2σ−α2+40c1c2ω5−80c2b2αλω5λ2σ−α2+40b1b2λω5λ2σ−α2+5c1c22ω5+5b22c1λω5λ2σ−α2+10b2c2b1λω5λ2σ−α2+120b2ω5δ+60c2b2ω5δ+5b2c22ω5δ+240c2λω4+234c2α2λω4λ2σ−α2−36b1αλω4λ2σ−α2+30c0c2ω4+10c12ω4+40c22λω4+30b22λ2ω4λ2σ−α2−45c1b2αλω4λ2σ−α2+10c22α2λω4λ2σ−α2−5c2b1αλω4λ2σ−α2+10c22α2λω4λ2σ−α2+25b22λ2ω4λ2σ−α2+5c0c22ω4+5c2c12ω4+5b22c2λ2ω4λ2σ−α2+5c0b22λω4λ2σ−α2+10c1b1b2ω4λ2σ−α2+5c2b12λω4λ2σ−α2+6c2ω4+5c22ω4+5b22λω4λ2σ−α2−336c2αω4δ+24b1ω4δ40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ−60b22λαω4δλ2σ−α2+10c1c2b2ω4δ+203b1c22ω4δ−10b22c2λαω4δλ2σ−α2+40c1λω3−240b2αλ2ω3λ2σ−α2+30c1α2λω3λ2σ−α2+10c0c1ω3+40b1b2λ2ω3λ2σ−α2−80c2b2αλ2ω3λ2σ−α2−30c0b2αλω3λ2σ−α2−20c1b1αλω3λ2σ−α2+10c1c2α2λω3λ2σ−α2+30b1b2λ2ω3λ2σ−α2−20c2b2λ2ω3λ2σ−α2+10c0c1c2ω3+53c13ω3+5λ2b22c1ω3λ2σ−α2−10λ2b2c2b1ω3+10c0b1b2λω3λ2σ−α2+5b12c1λω3λ2σ−α2+2c1ω3−6b2αλω3λ2σ−α2+10c1c2ω3+10b1b2λω3λ2σ−α2−60c1αω3δ+480b2α2λω3δλ2σ−α2+180b2ω3λδ+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ−80b1b2λω3δλ2σ−α2+160c2b2α2λω3δλ2σ−α2+65c2b2λω3δ+10c0c2b2ω3δ+10c2c1b1ω3δ+5b2c12ω3δ−10b22c1λαω3δλ2σ−α2−20b2c2b1λαω3δλ2σ−α2+6b2ω3δ+10b2c2ω3δ+53b23ω3δ3−30b22αω2δ3+5b1b22δ3ω2+90b2α2δ3ω−35b2b1αωδ3+10b2c2α2δ3ω+5b2b12δ3ω,+6b1α2δ3−24c2α3δ3−5b12αδ3+10b1c2α2δ3+53b13δ3+234c2α2λ2ω2λ2σ−α2+136c2λ2ω2−36b1αλ2ω2λ2σ−α2−150b2αλ2ω2λ2σ−α2−11b1λ2αω2λ2σ−α2+64c2λ2α2ω2λ2σ−α2+40c0c2λω2−45c1b2αλ2ω2λ2σ−α2+10c12λω2+10b12λ2ω2λ2σ−α2−50b1c2αω2λ2σ−α2−50b1c2αλ2ω2λ2σ−α2−5c2b1αλ2ω2λ2σ−α2+10c22α2λ2ω2λ2σ−α2+10c22λ2ω2−5c0b1αλω2λ2σ−α2+10c0c2α2λω2λ2σ−α2+5b12λ2λ2σ−α2−20b1c2αλ2ω2λ2σ−α2+5c02c2+5c0c12ω2+5c0b22λ2ω2λ2σ−α2+10c1b1b2λ2ω2λ2σ−α2+5c2b12λ2ω2λ2σ−α2+5c0b12λω2λ2σ−α2+8c2λω2−b1αλω2λ2σ−α2+2c2α2λω2λ2σ−α2+10c0c2ω2+5c12ω2+5b22λω2λ2σ−α2+5b22λ2ω2λ2σ−α2−468c2α3λω2δλ2σ−α2−362c2λαω2δ+72b1α2λω2δλ2σ−α2+28b1λω2δ+10c0b1ω2δ−50c0c2αω2δ+90c1b2α2λω2δλ2σ−α2+35c1b2λω2δ−15c12αω2δ−20b12λαω2δλ2σ−α2+100b1c2α2λω2δλ2σ−α2+40b1c2λω2δ+10c2b1α2λω2δλ2σ−α2−20c22α3λω2δλ2σ−α2+5c2b1λω2δ−20c22αλω2δ+10c0c2b1ω2δ+10c0c1b2ω2δ−10c0b22λαω2δλ2σ−α2−20c1b1b2λαω2δλ2σ−α2+5b1c12ω2δ−10c2b12αλω2δλ2σ−α2+2b1ω2δ−10c2αω2δ+10c2b1ω2δ+10c1b2ω2δ−10b22αλω2δλ2σ−α2−60c1α3λωδλ2σ−α2+300b2α2λ2ωδλ2σ−α2−45c1λαωδ+61b2λ2ωδ+60c0b2α2λωδλ2σ−α2+40c1b1α2λωδλ2σ−α2−20c1c2α3λωδλ2σ−α2−60b1b2λ2αωδλ2σ−α2+40c2b2α2λ2ωδλ2σ−α2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+5c1b1λωδ+10b2c2λ2ωδ−20c0b1b2λαωδλ2σ−α2−10b12c1αλωδλ2σ−α2+5c02b2ωδ+10c0b1c1ωδ+12b2α2λωδλ2σ−α2+5b2λωδ−3c1αωδ−20b1b2αλωδλ2σ−α2+10c0b2ωδ+10b1c1ωδ+30c1α2λ2ωλ2σ−α2−150b2αλ3ωλ2σ−α2+16c1λ2ω−30c0b2αλ2ωλ2σ−α2−20c1b1αλ2ωλ2σ−α2+10c1c2α2λ2ωλ2σ−α2+30b1b2λ3ωλ2σ−α2−20c2b2αλ3ωλ2σ−α2+10c2λ2ω+10c0c1λω2c2λ2,+10c0b1b2λ2ωλ2σ−α2+5b12c1λ2ωλ2σ−α2+5c02c1ω−6b2αλ2ωλ2σ−α2+2c1λω+10b1b2λ2ωλ2σ−α2+10c0c1ω+22b1λ2α2δλ2σ−α2−128c2λ2α3δλ2σ−α2+5b1λ2δ−56c2λ2αδ−10b12λ2αδλ2σ−α2+40b1c2α2λ2δλ2σ−α2+5c0b1λδ−20c0c2αλδ+10b1c2λ2δ−10c0b12λαδλ2σ−α2+5c02b1δ+2b1α2λδλ2σ−α2−4c2α3λδλ2σ−α2+b1λδ−4c2αλδ−10b12αλδλ2σ−α2+10c0b1δ−11b1αλ3λ2σ−α2+64c2α2λ3λ2σ−α2+16c2λ3+10c0c2α2λ2λ2σ−α2+5b12λ3λ2σ−α2+10c0c2λ2+5c0b12λ2λ2σ−α2+53c03−b1αλ2λ2σ−α2+2c2α2λ2λ2σ−α2+5b12λ2λ2σ−α2+5c02+d+kc0−5c0+d.Substituting (31) into (29) and using (4) with (3), we get the polynomials in ω and δ, setting the coefficients of this polynomial to zero to obtain the following set of algebraic equations:(85)ω6:120c2+30c22+30b22λλ2σ−α2+53c23+5b22c2λλ2σ−α2=0,ω5:24c1−240b2αλλ2σ−α2+40c1c2−80c2b2αλλ2σ−α2+40b1b2λλ2σ−α2+5c1c22+5b22c1λλ2σ−α2+10b2c2b1λλ2σ−α2=0,ω5δ:120b2+60c2b2+5b2c22=0,ω4:240c2λ+234c2α2λλ2σ−α2−36b1αλλ2σ−α2+30c0c2+10c12+40c22λ+30b22λ2λ2σ−α2−45c1b2αλλ2σ−α2+10c22α2λλ2σ−α2−5c2b1αλλ2σ−α2+10c22α2λλ2σ−α2+25b22λ2λ2σ−α2+5c0c22+5c2c12+5b22c2λ2λ2σ−α2+5c0b22λλ2σ−α2+10c1b1b2λ2σ−α2+5c2b12λλ2σ−α2+6c2+5c22+5b22λλ2σ−α2=0,ω4δ:−336c2α+24b140b2c1+40b1c2−50c22α−60b22λαλ2σ−α2+10c1c2b2+203b1c22−10b22c2λαλ2σ−α2=0,ω3:40c1λ−240b2αλ2λ2σ−α2+30c1α2λλ2σ−α2+10c0c1+40b1b2λ2λ2σ−α2−80c2b2αλ2λ2σ−α2−30c0b2αλλ2σ−α2−20c1b1αλλ2σ−α2+10c1c2α2λλ2σ−α2+30b1b2λ2λ2σ−α2−20c2b2λ2λ2σ−α2+10c0c1c2+53c13+5λ2b22c1λ2σ−α2−10λ2b2c2b1+10c0b1b2λλ2σ−α2+5b12c1λλ2σ−α2+2c1−6b2αλλ2σ−α2+10c1c2+10b1b2λλ2σ−α2=0,ω3δ:−60c1α+480b2α2λλ2σ−α2+180b2λ+30c0b2+20b1c1−50c1c2α−80b1b2λλ2σ−α2+160c2b2α2λλ2σ−α2+65c2b2λ+10c0c2b2+10c2c1b1+5b2c12−10b22c1λαλ2σ−α2−20b2c2b1λαλ2σ−α2+6b2+10b2c2=0,ω3δ3:53b23=0,δ3ω2:−30b22α+5b1b22=0,δ3ω:90b2−35b2b1α+10b2c2α2+5b2b12=0,δ3:6b1α2−24c2α3−5b12α+10b1c2α2+53b13=0,ω2:234c2α2λ2λ2σ−α2+136c2λ2−36b1αλ2λ2σ−α2−150b2αλ2λ2σ−α2−11b1λ2αλ2σ−α2+64c2λ2α2λ2σ−α2+40c0c2λ−45c1b2αλ2λ2σ−α2+10c12λ+10b12λ2λ2σ−α2−50b1c2αλ2σ−α2−50b1c2αλ2λ2σ−α2−5c2b1αλ2λ2σ−α2+10c22α2λ2λ2σ−α2+10c22λ2−5c0b1αλλ2σ−α2+10c0c2α2λλ2σ−α2+5c02c2+5c0c12+10c0c2α2λλ2σ−α2+5b12λ2λ2σ−α2−20b1c2αλ2λ2σ−α2+5c02c2+5c0c12+5c0b22λ2λ2σ−α2+10c1b1b2λ2λ2σ−α2+5c2b12λ2λ2σ−α2+5c0b12λλ2σ−α2+8c2λ−b1αλλ2σ−α2+2c2α2λλ2σ−α2+10c0c2ω2+5c12+5b22λλ2σ−α2+5b22λ2λ2σ−α2=0,ω2δ:−468c2α3λλ2σ−α2−362c2λα+72b1α2λλ2σ−α2+28b1λ+10c0b1−50c0c2α+90c1b2α2λλ2σ−α2+35c1b2λ−15c12α−20b12λαλ2σ−α2+100b1c2α2λλ2σ−α2+40b1c2λ+10c2b1α2λλ2σ−α2−20c22α3λλ2σ−α2+5c2b1λ−20c22αλ+10c0c2b1+10c0c1b2−10c0b22λαλ2σ−α2−20c1b1b2λαλ2σ−α2+5b1c12−10c2b12αλλ2σ−α2+2b1−10c2α+10c2b1+10c1b2−10b22αλλ2σ−α2=0,ω:30c1α2λ2λ2σ−α2−150b2αλ3λ2σ−α2+16c1λ2−30c0b2αλ2λ2σ−α2−20c1b1αλ2λ2σ−α2+10c1c2α2λ2λ2σ−α2+30b1b2λ3λ2σ−α2−20c2b2αλ3λ2σ−α2+10c2λ2+10c0c1λ+10c0b1b2λ2λ2σ−α2+5b12c1λ2λ2σ−α2+5c02c1−6b2αλ2λ2σ−α2+2c1λ+10b1b2λ2λ2σ−α2+10c0c1=0,δ:22b1λ2α2λ2σ−α2−128c2λ2α3λ2σ−α2+5b1λ2−56c2λ2α−10b12λ2αλ2σ−α2+40b1c2α2λ2λ2σ−α2+5c0b1λ−20c0c2αλ+10b1c2λ2−10c0b12λαλ2σ−α2+5c02b1+2b1α2λλ2σ−α2−4c2α3λλ2σ−α2+b1λ−4c2αλ−10b12αλλ2σ−α2+10c0b1=0,ω0:−11b1αλ3λ2σ−α2+64c2α2λ3λ2σ−α2+16c2λ3+10c0c2α2λ2λ2σ−α2+5b12λ3λ2σ−α2+10c0c2λ2+5c0b12λ2λ2σ−α2+53c03−b1αλ2λ2σ−α2+2c2α2λ2λ2σ−α2+2c2λ2+5b12λ2λ2σ−α2+5c02+c0+d=0.Solving the above equations of algebraic type by Mathematica software, we have.Result 3.
(86)c0=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ,c1=0,c2=−12,b1=6475763087,b2=0,α=−1618944851.
By substituting (86) into (31) with (13) by using (6) and (16), we get the exact solutions of (18) as follows:(87)W1=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+6475763087δ−12ω2.
By substituting the value ofω and δ from (3), we get(88)=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ/735−26209667236+23532201λ2σ+647576/30871/Z−12Z′/Z2.
When we put the value ofZ from (3), we get(89)=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+M2cosξ−λ+α/λ−12M1sinξλ+M2cosξλ+α/λ′M1sinξλ+M2cosξλ+α/λ2,=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+M2cosξλ+α/λ−12M1cosξλλ+M2sinξλλM1sinξλ+M2cosξλ+α/λ2.
Then, the exact solution of2+1-dimensional equation from (18) is as follows:(90)=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+M2cosξλ+α/λ−12M1cosξλλ+M2sinξλλ2M1sinξλ+M2coshξλ+α/λ2.
By lettingM1=0, α = −161894/4851 and M2>0 in (90), the particular solution of (18) is(91)We=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M2cosξλ+α/λ−12−λM2sinξλ2M2cosξλ2,(92)We=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M2cosξλ+α/λ+12λtan2ξλ.
Then, by lettingλ=0.4, t=0, y=3, α=−161894/4851, and σ=1, the graph of (92) has the following forms.
Figure5 shows periodic soliton in two and three dimensions for (92). When we put M2=0 and M1>0 in (89), we produce a solitary solution:(93)Wf=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+α/λ−12λM1coshξλ2M1sinhξλ2,(94)Wf=34675389753228+269042234177540λ−31133101923λ2σ−138369341880λ2σ735−26209667236+23532201λ2σ+64757630871M1sinξλ+α/λ+12λcoth2ξλ.
The particular solution of2+1-dimensional equation for trigonometric function solutions, if we put result on (90) when M2=0 and M1>0, has the following form.
The graph of the particular solution whereλ=0.4, σ=1, t=0, y=3, α=−161894/4851 is as follows.
Figure6 represents a periodic soliton nature of (94) in three and two dimensions.Figure 5
The figures of (92) periodic in 3D and 2D.Figure 6
The figures of (94) periodic 3D and 2D.Result 4.
(95)c0=1/5−9−40λ,c1=0,c2=−12,b1=0,b2=0,α=0.
By substituting (95) into (31) with (6), (13), and (16), we get the exact solutions of (18) as follows:(96)W1=15−9−40λ−12ω2.
By substituting the value ofω and δ from (3), we get 1/5−9−40λ−12ω2.
When we put the value ofZ from (3), we get(97)1/5−9−40λ−12M1sinξλ+M2cosξλ+αλ′/M1sinξλ+M2cosξλ+αλ2,(98)=15−9−40λ−12M1cosξλλ+M2sinξλλM1sinξλ+M2cosξλ+α/λ2.
Then, the exact solution of2+1-dimensional equation from (18) is as follows:(99)=15−9−40λ−12M1cosξλλ+M2sinξλλ2M1sinξλ+M2coshξλ+α/λ2.
By lettingM1=0, α  = 0, and M2>0 in (99), the particular solution of (18) is(100)Wg=15−9−40λ−12−λM2sinξλ2M2cosξλ2,(101)Wg=15−9−40λ+12λtan2ξλ.
Then, by lettingλ=0.5, t=0, y=4, α=0, and σ=1, the graph of (101) has the following forms.
Figure7 shows the periodic nature for (101) in three and two dimensions. But, if we put M2=0 and M1>0 in (98), we produce a solitary solution:(102)Wh=15−9−40λ−12λM1coshξλ2M1sinhξλ2,(103)Wh=15−9−40λ+12λcoth2ξλ.
The particular solution of (2 + 1)-dimensional equation for trigonometric function solutions, if we put result on (103) when M2=0 and M1>0, has the following form.
The graph of the particular solution of (103) where λ=0.5, σ=1, t=0, y=4 is as follows.
Figure8 shows periodic nature for (101) in three-dimensional space and two-dimensional plane.Figure 7
The figures of (101) periodic in 3D and 2D.Figure 8
The figures of (103) periodic in 3D and 2D.


## 3.3. Case C:λ=0—Nature of Rational Solutions

From (9), (29) and (39) become(104)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αω3ω2−2αδM12−2αM2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2ω2ω2−2αδM12−2αM2−362c2λαω2δ+136c2λ2ω2−36b1αω2ω2−2αδM12−2αM2+28b1λω2δ+30c1α2ωω2−2αδM12−2αM2−150b2αλω2−2αδM12−2αM2ω−11b1λαω2−2αδM12−2αM2+64c2λα2ω2−2αδM12−2αM2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ−56c2λ2αδ+16c2λ3.Finally by using (9), (29 and (39) become(105)W″″=120c2ω6+24c1ω5+120b2ω5δ−336c2αω4δ+240c2λω4+24b1ω4δ−60c1αω3δ+40c1λω3−240b2αω5M12−2αM2+480b2α2ω3M12−2αM2+180b2ω3λδ+90b2α2δ3ω+6b1α2δ3−24c2α3δ3+234c2α2ω4M12−2αM2−468c2α3ω2δM12−2αM2−362c2λαω2δ+136c2λ2ω2−36b1αω4M12−2αM2+72b1α2ω2δM12−2αM2+28b1λω2δ+30c1α2ω3M12−2αM2−60c1α3ωδM12−2αM2−150b2αλω3M12−2αM2+300b2α2λδωM12−2αM2−11b1λαω2M12−2αM2+22b1λα2δM12−2αM2+64c2λα2ω2M12−2αM2−128c2λα3δM12−2αM2−45c1λαωδ+16c1λ2ω+61b2λ2ωδ+5b1λ2δ−56c2λ2αδ+16c2λ3.From (9) and (29), (41) becomes(106)5×W×W″=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4+30b22ω4ω2−2αδM12−2αM2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ−50c1c2αω3δ+50c1c2λω3+40b1b2ω3ω2−2αδM12−2αM2r−80c2b2αω3ω2−2αδM12−2αM2+65c2b2λω3δ−5b2αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2−45c1b2αω2ω2−2αδM12−2αM2+35c1b2λω2δ−15c12αω2δ+10c12λω2+10b12ω2ω2−2αδM12−2αM2−50b1c2αω2+40b1c2λω2δ−5c2b1αω2ω2−2αδM12−2αM2+10c22α2ω2ω2−2αδM12−2αM2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λω2ω2−2αδM12−2αM2−5c0b1αω2−2αδM12−2αM2−30c0b2αωω2−2αδM12−2αM2+10c0c2α2ω2−2αδM12−2αM2−20c1b1αωω2−2αδM12−2αM2+10c1c2α2ω2−2αδM12−2αM2ω+5b12λω2−2αδM12−2αM2+10c0c2λ2+30b1b2λωω2−2αδM12−2αM2−20b1c2αλω2−2αδM12−2αM2−20c2b2αλω2−2αδM12−2αM2ω+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+10b1c2λ2δ+5c1b1λωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ.Finally, by using (9) and (29), (41) becomes(107)5×W×W″=30c22ω6+40c1c2ω5+60c2b2ω5δ+30c0c2ω4+10c12ω4+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ+40c22λω4−60b22ω4αδM12−2αM2+10c0c1ω3+30c0b2ω3δ+20b1c1ω3δ+50c1c2λω3+40b1b2ω5M12−2αM2−80b1b2ω3αδM12−2αM2−80c2b2αω5M12−2αM2+360c2b2α2ω3δM12−2αM2+65c2b2λω3δ−5b2αδ3−35b2b1αωδ3+10b1c2α2δ3−30b22αω2δ3+10b2c2α2δ3ω+10c0b1ω2δ−50c0c2αω2δ+40c0c2λω2−45c1b2αω4M12−2αM2+90c1b2α2ω2δM12−2αM2+35c1b2λω2δ−15c12αω2δ+10c12λω2+10b12ω4M12−2αM2−20b12ω2αδM12−2αM2−50b1c2αω2+40b1c2λω2δ−5c2b1αω4M12−2αM2+30b22ω6M12−2αM2+10c2b1α2ω2δM12−2αM2+10c22α2ω4M12−2αM2−50c1c2αω3δ−20c22α3ω2δM12−2αM2+30b22ω6M12−2αM2+5c2b1λω2δ+10c22λ2ω2−20c22αλω2δ+25b22λω4M12−2αM2−50b22λω2αδM12−2αM2−5c0b1αω2M12−2αM2+10c0b1α2δM12−2αM2−30c0b2αω3M12−2αM2+60c0b2α2ωδM12−2αM2+10c0c2α2ω2M12−2αM2−20c0c2α3δM12−2αM2−20c1b1αω3M12−2αM2+40c1b1α2ωδM12−2αM2+5c1b1λωδ+10c1c2α2ω3M12−2αM2−20c1c2α3δωM12−2αM2+5b12λω2M12−2αM2−10b12λαδM12−2αM2+30b1b2λω3M12−2αM2−60b1b2λωαδM12−2αM2−20b1c2αλω2M12−2αM2+40b1c2α2λδM12−2αM2−20c2b2αλω2M12−2αM2+40c2b2α2λδM12−2αM2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ+10c2λ2ω+10b2c2λ2ωδ+5c0b1λδ+10c0c1λω−20c0c2αλδ+10b1c2λ2δ+10c0c2λ2.From (9), (45) becomes(108)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2ω4ω2−2αδM12−2αM2+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ+5b22c1ω3ω2−2αδM12−2αM2+10b2c2b1ω3ω2−2αδM12−2αM2+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5b1b22δ3ω2+5c02c2ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22ω2ω2−2αδM12−2αM2+10c1b1b2ω2ω2−2αδM12−2αM2+5c2b12ω2ω2−2αδM12−2αM2+10c0c1c2ω3+5c0b12ω2−2αδM12−2αM2+10c0b1b2ω2−2αδM12−2αM2ω+5c0c12ω2+5b12c1ωω2−2αδM12−2αM2+5c02b1δ+5c02c1ω+5b1c12ω2δ+5c02b2ωδ+10c0b1c1ωδ+53c03.Finally by using (9), (45) becomes(109)53×W3=53c23ω6+5c1c22ω5+5b2c22ω5δ+5c0c22ω4+5c2c12ω4+10c1c2b2ω4δ+203b1c22ω4δ+5b22c2ω6M12−2αM2−105b22c2ω4αδM12−2αM2+10c0c1c2ω3+10c0c2b2ω3δ+53c13ω3+10c2c1b1ω3δ+5b2c12ω3δ+5b22c1ω5M12−2αM2−10b22c1ω3αδM12−2αM2+10b2c2b1ω5M12−2αM2−20b2c2b1ω3αδM12−2αM2+53b23ω3δ3+53b13δ3+5b2b12δ3ω+5c0c12ω2+10c0c2b1ω2δ+10c0c1b2ω2δ+5c0b22ω4M12−2αM2−10c0b22ω2αδM12−2αM2+10c1b1b2ω4M12−2αM2−20c1b1b2ω2αδM12−2αM2+5b1c12ω2δ+5c2b12ω4M12−2αM2−10c2b12ω2αδM12−2αM2+5c0b12ω2M12−2αM2−105c0b12αδM12−2αM2+10c0b1b2ω3M12−2αM2−20c0b1b2ωαδM12−2αM2+5b12c1ω3M12−2αM2−10b12c1ωαδM12−2αM2+5b1b22δ3ω2+5c02b1δ+5c02c1ω+5c02b2ωδ+5c02c2ω2+10c0b1c1ωδ+53c03.From (9) and (29), (35) becomes(110)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αω2−2αδM12−2αM2−6b2αωω2−2αδM12−2αM2+2c2α2ω2−2αδM12−2αM2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.By using (9), (29), and (35), we get(111)W″=6c2ω4+2c1ω3+6b2ω3δ+2b1ω2δ−10c2αω2δ+8c2λω2−b1αω2M12−2αM2+2b1α2δM12−2αM2−6b2αω3M12−2αM2+12b2α2ωδM12−2αM2+2c2α2ω2M12−2αM2−4c2α3δM12−2αM2+b1λδ+5b2λωδ−3c1αωδ+2c1λω−4c2αλδ+2c2λ2.From (9), (29), and (43),(112)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b12ω2−2αδM12−2αM2+10c2b1δω2+10c1b2ω2δ+10b1b2ω2−2αδM12−2αM2ω+5b22ω2ω2−2αδM12−2αM2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.At the end, by using (9) and (29) in 5×W2, we get(113)5×W2=5c22ω4+10c1c2ω3+10b2c2ω3δ+10c0c2ω2+5c12ω2+5b12ω2M12−2αM2−10b12αδM12−2αM2+10c2b1δω2+10c1b2ω2δ+10b1b2ω3M12−2αM2−20b1b2αδωM12−2αM2+5b22ω4M12−2αM2−10b22ω2αδM12−2αM2+10c0b1δ+10c0c1ω+10c0b2ωδ+10b1c1ωδ+5c02.From (105), (107), (109), (111), and (113), equation k−5W=kc0+kc1ω+kb1δ+kc2ω2+kb2ωδ−5c0−5c1ω−5b1δ−5c2ω2−5b2ωδ is equivalent to(114)120c2ω6+30c22ω6+30b22ω6M12−2αM2+30b22ω6M12−2αM2+53c23ω6+5b22c2ω6M12−2αM2+24c1ω5−240b2αω5M12−2αM2+40c1c2ω5+40b1b2ω5M12−2αM2−80c2b2αω5M12−2αM2+5c1c22ω5+5b22c1ω5M12−2αM2+10b2c2b1ω5M12−2αM2+120b2ω5δ+60c2b2ω5δ+5b2c22ω5δ−336c2αω4δ+24b1ω4δ+40b2c1ω4δ+40b1c2ω4δ−50c22αω4δ−60b22ω4αδM12−2αM2+10c1c2b2ω4δ+203b1c22ω4δ−105b22c2ω4αδM12−2αM2+240c2λω4+234c2α2ω4M12−2αM2−36b1αω4M12−2αM2+30c0c2ω4+10c12ω4+40c22λω4−45c1b2αω4M12−2αM2+10b12ω4M12−2αM2−5c2b1αω4M12−2αM2+10c22α2ω4M12−2αM2+25b22λω4M12−2αM2+5c0c22ω4+5c2c12ω4+5c0b22ω4M12−2αM2+10c1b1b2ω4M12−2αM2+5c2b12ω4M12−2αM2+6c2ω4+5c22ω4+5b22ω4M12−2αM2−60c1αω3δ+180b2ω3λδ+30c0b2ω3δ+20b1c1ω3δ−80b1b2ω3αδM12−2αM2+360c2b2α2ω3δM12−2αM2+65c2b2λω3δ−50c1c2αω3δ+10c0c2b2ω3δ+10c2c1b1ω3δ+5b2c12ω3δ−10b22c1ω3αδM12−2αM2−20b2c2b1ω3αδM12−2αM2+6b2ω3δ+10b2c2ω3δ+40c1λω3+480b2α2ω3M12−2αM2+30c1α2ω3M12−2αM2−150b2αλω3M12−2αM2+10c0c1ω3+50c1c2λω3−30c0b2αω3M12−2αM2−20c1b1αω3M12−2αM2+10c1c2α2ω3M12−2αM2+30b1b2λω3M12−2αM2+10c0c1c2ω3+53c13ω3+10c0b1b2ω3M12−2αM2+5b12c1ω3M12−2αM22c1ω3+2c1ω3−6b2αω3M12−2αM2+10c1c2ω3+10b1b2ω3M12−2αM2+90b2α2δ3ω−35b2b1αωδ3+10b2c2α2δ3ω+5b2b12δ3ω+6b1α2δ3−24c2α3δ3−5b2αδ3+10b1c2α2δ3+53b13δ3−30b22αω2δ3+5b1b22δ3ω2+53b23ω3δ3−468c2α3ω2δM12−2αM2−362c2λαω2δ+72b1α2ω2δM12−2αM2+28b1λω2δ+10c0b1ω2δ,(115)−50c0c2αω2δ+90c1b2α2ω2δM12−2αM2+35c1b2λω2δ−15c12αω2δ−20b12ω2αδM12−2αM2+40b1c2λω2δ+10c2b1α2ω2δM12−2αM2−20c22α3ω2δM12−2αM2+5c2b1λω2δ−20c22αλω2δ−50b22λω2αδM12−2αM2−10c2αω2δ+10c1b2ω2δ−10b22ω2αδM12−2αM2+136c2λ2ω2−11b1λαω2M12−2αM2+64c2λα2ω2M12−2αM2+40c0c2λω2+10c12λω2−50b1c2αω2+10c22λ2ω2−5c0b1αω2M12−2αM2+10c0c2α2ω2M12−2αM2+5b12λω2M12−2αM2−20b1c2αλω2M12−2αM2−20c2b2αλω2M12−2αM2+5c0c12ω2+5c0b12ω2M12−2αM2+5c02c2ω2+8c2λω2−b1αω2M12−2αM2+2c2α2ω2M12−2αM2+10c0c2ω2+5c12ω2+5b12ω2M12−2αM2−60c1α3ωδM12−2αM2+300b2α2λδωM12−2αM2−45c1λαωδ+61b2λ2ωδ+60c0b2α2ωδM12−2αM2+40c1b1α2ωδM12−2αM2+5c1b1λωδ−20c1c2α3δωM12−2αM2−60b1b2λωαδM12−2αM2+25c0b2λωδ−15c0c1αωδ−20c1c2αλωδ−20c1c2αλωδ+10b2c2λ2ωδ−20c0b1b2ωαδM12−2αM2−10b12c1ωαδM12−2αM2+5c02b2ωδ+10c0b1c1ωδ+12b2α2ωδM12−2αM2+5b2λωδ−3c1αωδ−20b1b2αδωM12−2αM2+10c0b2ωδ+10b1c1ωδ+16c1λ2ω+10c2λ2ω+10c0c1λω+5c02c1ω+2c1λω+10c0c1ω+22b1λα2δM12−2αM2−128c2λα3δM12−2αM2+5b1λ2δ−56c2λ2αδ+10c0b1α2δM12−2αM2−20c0c2α3δM12−2αM2−10b12λαδM12−2αM2+40b1c2α2λδM12−2αM2+40c2b2α2λδM12−2αM2+5c0b1λδ−20c0c2αλδ+10b1c2λ2δ−105c0b12αδM12−2αM2+5c02b1δ+2b1α2δM12−2αM2−4c2α3δM12−2αM2+b1λδ−4c2αλδ−10b12αδM12−2αM2+10c0b1δ+16c2λ3+10c0c2λ2+2c2λ2+5c02+d,(116)ω6:120c2+30c22+30b22M12−2αM2+30b22M12−2αM2+53c23+5b22c2M12−2αM2=0,ω5:24c1−240b2αM12−2αM2+40c1c2+40b1b2M12−2αM2−80c2b2αM12−2αM2+5c1c22+5b22c1M12−2αM2+10b2c2b1M12−2αM2=0,ω5δ:120b2+60c2b2+5b2c22=0,ω4δ:−336c2α+24b1+40b2c1+40b1c2−50c22α−60b22αM12−2αM2+10c1c2b2+203b1c22−105b22c2αM12−2αM2=0,ω4:240c2λ+234c2α2M12−2αM2−36b1αM12−2αM2+30c0c2+10c12+40c22λ−45c1b2αM12−2αM2+10b12M12−2αM2−5c2b1αM12−2αM2+10c22α2M12−2αM2+25b22λM12−2αM2+5c0c22+5c2c12+5c0b22M12−2αM2+10c1b1b2M12−2αM2+5c2b12M12−2αM2+6c2+5c22+5b22M12−2αM2=0,ω3δ:−60c1α+180b2λ+30c0b2+20b1c1−80b1b2αM12−2αM2+360c2b2α2M12−2αM2+65c2b2λ−50c1c2α+10c0c2b2+10c2c1b1+5b2c12−10b22c1αM12−2αM2−20b2c2b1αM12−2αM2+6b2+10b2c2=0,ω3:40c1λ+480b2α2M12−2αM2+30c1α2M12−2αM2−150b2αλM12−2αM2+10c0c1+50c1c2λ−30c0b2αM12−2αM2−20c1b1αM12−2αM2+10c1c2α2M12−2αM2+30b1b2λM12−2αM2+10c0c1c2+53c13+10c0b1b2M12−2αM2+5b12c1M12−2αM22c1+2c1−6b2αM12−2αM2+10c1c2+10b1b2M12−2αM2=0,(117)δ3ω:90b2α2−35b2b1α+10b2c2α2+5b2b12=0,δ2:6b1α2−24c2α3−5b2α+10b1c2α2+53b13=0,δ3ω2:−30b22α+5b1b22=0,ω3δ3:53b23=0,ω2δ:−468c2α3M12−2αM2−362c2λα+72b1α2M12−2αM2+28b1λ+10c0b1−50c0c2α+90c1b2α2M12−2αM2+35c1b2λ−15c12α−20b12αM12−2αM2+40b1c2λ+10c2b1α2M12−2αM2−20c22α3M12−2αM2+5c2b1λ−20c22αλ−50b22λαM12−2αM2−10c2α+10c1b2−10b22αM12−2αM2=0,ω2:136c2λ2−11b1λαM12−2αM2+64c2λα2M12−2αM2+40c0c2λ+10c12λ−50b1c2α+10c22λ2−5c0b1αM12−2αM2+10c0c2α2M12−2αM2+5b12λM12−2αM2−20b1c2αλM12−2αM2−20c2b2αλM12−2αM2+5c0c12+5c0b12M12−2αM2+5c02c2+8c2λ−b1αM12−2αM2+2c2α2M12−2αM2+10c0c2+5c12+5b12M12−2αM2=0,(118)ωδ:−60c1α3M12−2αM2+300b2α2λM12−2αM2−45c1λα+61b2λ2+60c0b2α2M12−2αM2+40c1b1α2M12−2αM2+5c1b1λ−20c1c2α3M12−2αM2−60b1b2λαM12−2αM2+25c0b2λ−15c0c1α−20c1c2αλ−20c1c2αλ+10b2c2λ2−20c0b1b2αM12−2αM2−10b12c1αM12−2αM2+5c02b2+10c0b1c1+12b2α2M12−2αM2+5b2λ−3c1α−20b1b2αM12−2αM2+10c0b2+10b1c1=0,ω:16c1λ2+10c2λ2+10c0c1λ+5c02c1+2c1λ+10c0c1=0,δ:22b1λα2M12−2αM2−128c2λα3M12−2αM2+5b1λ2−56c2λ2α+10c0b1α2M12−2αM2−20c0c2α3M12−2αM2−10b12λαM12−2αM2+40b1c2α2λM12−2αM2+40c2b2α2λM12−2αM2+5c0b1λ−20c0c2αλ+10b1c2λ2−105c0b12αM12−2αM2+5c02b1+2b1α2M12−2αM2−4c2α3M12−2αM2+b1λ−4c2αλ−10b12αM12−2αM2+10c0b1=0,ω0:16c2λ3+10c0c2λ2+2c2λ2+5c02+d=0.Solving the above equation of algebraic type by Mathematica software, we have(119)c0=15−9−40λ,c1=0,c2=−12,b1=0,b2=0,k=0.By substituting (119) into (31) with (8), (13), and (16), we get the exact solutions of (18) as follows:(120)W1=15−9−40λ−12ω2.By substituting the value ofω and δ from (3), we get 1/5−9−40λ−12ω2; when we put the value of Z from (3), we get(121)W1=15−9−40λ−12Z′Z2.When we put the value ofZ from (8) in (120),(122)W1=15−9−40λ−12α/2ξ2+M1ξ+M2′α/2ξ2+M1ξ+M22,W1=15−9−40λ−12αξ+M1α/2ξ2+M1ξ+M22.Hence, we get the exact solution of (2 + 1)-dimensional integro-partial equation with respect to result of (120) as follows: by letting α=0, M1>0, and M2>0 in (120), the particular solution of (1) is(123)W1=15−9−40λ−12M12M1ξ+M22.The particular solution of (2 + 1)-dimensional for rational function solutions and putting the result in equation (119) when M_2 > 0 and M_1 >0 have the form of graphical representation for the particular solution of Equation (123):Figure9 shows soliton behaviour for (120) in three-dimensional space and in two-dimensional plane.Figure 9
The figures of (120) soliton in 3D and 2D.


## 4. Conclusion

In this paper, the new special expansion of theZ′/Z and 1/Z method of expansion has been applied to discover solutions of solitary waves of NLIPDEs by the help of Mathematica software 9.1. We applied successfully the obtained soliton wave solutions (2+1) -dimensional space time using certain parameters. These parameters were taken as special values in which the solutions of wave of the solitary and periodic are developed from the exact solutions. The solution of NLIPDEs of (2+1) dimensional equation is analytically solved and it is an easy and direct way to solve nonlinear partial differential equations. To use the method effectively, we applied the combined expansion methods in the form of Z′/Z and 1/Z. The combined methods to solve the NLIPDEs for integro-differential equations led us to get exact solutions for three different cases of λ which has hyperbolic, trigonometric, and rational nature. We get different results for each case. The obtained results were illustrated by graphical plots. It can be concluded that the obtained solutions will serve as a very important milestone in the study of nonlinear phenomena arising in mathematical physics and engineering fields. Finally, the method is a powerful and effective method to solve soliton wave solutions for NLIPDEs.



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*Source: 9954649-2022-05-29.xml*