# Combined Damped Sinusoidal Oscillation Solutions to the (3 + 1)-D Variable-Coefficient Generalized NLW Equation in Liquid with Gas Bubbles

Authors: Xiuwang Wang; Jalil Manafian; Wayan Eka Mahendra; Azher M. Abed; Mustafa Z. Mahmoud; Guizhen Liang
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

This paper investigates the combined damped sinusoidal oscillation solutions to the3+1-D variable-coefficient (VC) generalized nonlinear wave equation. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize a binary Bell polynomial transformation for reducing the Cole-Hopf algorithm to get the exact solutions of the VC generalized NLW equation. The damped sinusoidal oscillations for two cases of the nonlinear wave ordinary differential equation will be studied. Using suitable mathematical assumptions, the novel kinds of solitary, periodic, and singular soliton solutions are derived and established in view of the trigonometric and rational functions of the governing equation. To achieve this, the illustrative example of the VC generalized nonlinear wave equation is provided to demonstrate the feasibility and reliability of the procedure used in this study. The trajectory solutions of the traveling waves are shown explicitly and graphically. The effect of the free parameters on the behavior of acquired figures of a few obtained solutions for two nonlinear rational exact cases was also discussed. By comparing the proposed method with the other existing methods, the results show that the execution of this method is concise, simple, and straightforward.


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## Body

## 1. Introduction

The advent of the concept of nonlinear partial differential equations (NLPDEs) has attracted the interest of many researchers due to their importance in accurately demonstrating the dynamics of abundant real-world systems in various fields of sciences such as physics, diffusion, biology, chaos theory, chemistry, engineering, economics, and commerce [1–3].In particular, more and much attention has been paid to constructing exact and approximate solutions, for example, a multiobjective optimization structure method [4], the differential transform method [5], an improved differential transform method [6], a new solitary periodic wave solution [7], the multiple Exp-function method [8], an expectation maximization algorithm [9], a multiobjective mixed integer linear programming model [10], Hirota’s bilinear method [11], a mean–semivariance approach [12], a genetic algorithm for preemptive scheduling of a single machine [13], solving absolute value problems [14], the inverse scattering transformation method [15], the multiple soliton solutions and fusion interaction phenomena [16], the truncated Painlevé series [17], a software product line engineering approach [18], a conceptual framework for SIDS alert system [19], the modified Pfaffian technique [20], the conserved quantities method [21], the Darboux transformation [22], and a computational intelligence approach [23].In [24], the N-soliton solutions, soliton molecules, and asymmetric solitons of the Korteweg-de Vries-Caudrey-Dodd-Gibbon equation were obtained by means of the velocity resonance method by Ma et al. Researchers analyzed the higher-order algebraic soliton solutions of the Gerdjikov-Ivanov equation by using the Darboux transformation and some limit technique, and according to the asymptotic balance between different algebraic terms, they obtained the asymptotic expressions of algebraic soliton solutions [25]. Wang studied the multisoliton solutions of the 2+1-dimensional PT-symmetric couplers with varying coefficients by using the homogeneous balance method [26]. The N-soliton solutions, M-breather solutions, and hybrid ones composed of solitons and breathers were constructed in a time-dependent KP equation by Wu [27]. Wang and Chen [28] investigated the higher-order Sawada-Kotera-type equation and the higher-order Lax-type equation in fluids. A 2+1-dimensional coupled nonlinear partial differential equation with variable coefficients in an inhomogeneous medium according to the Hirota bilinear form and symbolic computation, the breather wave solutions and lump solutions were constructed by using the extended homoclinic breather technique and the generalized positive quadratic function method in Ref. [29]. Moreover, Sadat et al. [30] got lump-type solutions and their interaction solutions with one- or two-stripe solutions through the Hirota bilinear scheme and the Cole-Hopf transformation for a generalized 3+1 shallow water-like equation. In addition, a kind of lump solution and two classes of interaction solutions were discussed to the 2+1-dimensional generalized KdV equation with the aid of the symbolic computation system Mathematica and Hirota bilinear scheme [31].The liquid with gas bubble problem has been made in the propagation of weakly nonlinear waves by Kudryashov and coauthors [32, 33]. In [34], a generalized 3+1-dimensional (3+1-D) nonlinear wave (NLW) equation has been investigated to find the first-order lump wave solution and second-order lump wave solution which is given
(1)Σt+ϕ1ΣΣx+ϕ2Σxxx+ϕ3Σxx+ϕ4Σyy+ϕ5Σzz=0,and also, ϕii=1,⋯,5 is the constant coefficient. With ϕ3=0, the above equation is transformed to the 3+1-dimensional generalized KP equation. N-soliton solutions and periodic wave solutions were studied for Equation (1) in liquid with gas bubbles in [35]. The 3+1-D variable-coefficient (VC) generalized NLW equation [36] is taken as follows:
(2)Σt+ϕ1tΣΣx+ϕ2tΣxxx+ϕ3tΣxx+ϕ4tΣyy+ϕ5tΣzz=0,and also, ϕiti=1,⋯,5 is the variable coefficient and Σ=Σx,y,z,t is the wave amplitude (unknown function) that should be searched. Equation (2) is a variable coefficient case of (1) that has been investigated in Refs. [34, 35]. The bilinear form, Bäcklund transformation, Lax pair, and infinitely many conservation laws were obtained via the binary Bell polynomials for a generalized 3+1-D VC NLW equation by Deng and Gao [36]. In [37], for Equation (2), a periodic–shape lump solution, a parabolic–shape lump solution, a cubic–shape lump solution, and interaction solutions between lump and one solitary wave and between lump and two solitary waves were investigated. Also, Guo and Chen [38] studied the multisoliton solutions and periodic solutions including X–periodic, Y–periodic, and 2–periodic wave solutions. A few of rational exact solution for the 3+1-D VC NLW equation has been studied in [39].Hirota’s bilinear method has always been a powerful tool for solving the N-soliton solution of the nonlinear evolution equation [40–42]. Since this method was proposed by Hirota, many scholars have continuously improved it to obtain the exact solution other than the N-soliton solution. Based on the bilinear method, Satsuma and Ablowitz proposed a long-wave limit method to obtain the lump solution, which decays algebraically in space [43]; Ohta and Yang found a way to get general rogue waves for Davey-Stewartson I system [44]; by means of bilinear approach, Lou discovered soliton solutions with even numbers and soliton solutions with odd numbers in nonlocal systems [45]; in recent years, the velocity resonance theory [46] and the resonance conditions mentioned in Refs. [47, 48] have further improved this method to better explain physical phenomena. Many researchers used various methods to study the nonlinear models by using the Hirota bilinear technique (HBT), such as a 3+1-dimensional nonlinear evolution equation [49], the generalized variable-coefficient Kadomtsev–Petviashvili equation [50], the new 3+1-dimensional generalized Kadomtsev–Petviashvili equation [51], and a generalized 3+1-D VC NLW equation [52]. Through these methods, some exact solutions of the nonlinear models of equations were obtained. In order to really understand these physical phenomena, it is of immense importance to solve nonlinear partial differential equations (NLPDEs) which govern these aforementioned phenomena. However, there is no general systematic theory that can be applied to NLPDEs so that their analytic solutions can be obtained. Nevertheless, in recent times, scientists have developed effective techniques to obtain viable analytical solutions to NLPDEs, such as some nonlinear equations, for example, the 3+1-dimensional BKP-Boussinesq equation [53], the generalized BKP equation [54], and new integrable Boussinesq equations of distinct dimensions [55]. In [56], the M-lump solution and N-soliton solution of the 2+1-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation were studied. In particular, more and much attention has been paid to constructing exact lump solutions to the third-order evolution equation [57] arising propagation of long waves over shallow water. In [58], a weight number was utilized in an algorithm to convey the Hirota condition while transforming the Hirota function in wave vectors to a homogeneous polynomial. Also, a generalized algorithm to prove the Hirota conditions was suggested by comparing degrees of the multivariate polynomials derived from the Hirota function in wave vectors by Ma [59]. In addition, N-soliton solutions in the case of nonlocal integrable equations have been presented by Riemann-Hilbert problems containing nonlocal reverse–space nonlinear Schrödinger hierarchies [60] and nonlocal real reverse–space time matrix AKNS hierarchies [61]. The simplified Hirota’s technique with new complex forms was developed suitably to construct multiple–soliton solutions with a complex structure for modified KdV-Sine-Gordon equation in integrable form by Wazwaz and Kaur [62]. According to the bilinear forms, the novel N-soliton solutions of 1+1 and 2+1-dimensional generalized Broer–Kaup systems were obtained by using Hirota’s bilinear method in [63].Motivated by the above studies, we apply the proposed analytical method which is presented by [39] to solve the mentioned problems above. The advantage of the proposed method is that it can be applied to the integrable model and will be obtained plenty of combined damped sinusoidal oscillation solutions with three kinds of plotted graphs.The rest of the paper is organized as follows: Section2 presents the definition of the binary Bell polynomials and their properties. Also, the bilinear form of the 3+1-D VC generalized NLW equation is constructed in Section 3. The proposed method HBT, for obtaining the combined damped sinusoidal oscillation solutions of Equation (2), is presented in Section 4. Finally, in Section 5, we briefly summarize and discuss the results in a conclusion.


## 2. Binary Bell Polynomials

Through Refs. [39, 64], take λ=λx1,x2,⋯,xn be a C∞ function with multivariables, the general form can be written as
(3)Yn1x1,⋯,njxjλ≡Yn1,⋯,njλd1x1,⋯,djxj=e−λ∂x1n1⋯∂xjnjeλ,it is named the multi-D Bell polynomials as follows:
(4)λd1x1,⋯,djxj=∂x1d1⋯∂xjdjλ,λ0xi≡λ,d1=0,⋯,n1;⋯;dj=0,⋯,nj,and we have
(5)Y1λ=λx,Y2λ=λ2x+λx2,Y3λ=λ3x+3λxλ2x+λx3,⋯,λ=λx,t,Yx,tλ=λx,t+λxλt,Y2x,tλ=λ2x,t+λ2xλt+2λx,tλx+λx2λt,....The multidimensional binary Bell polynomials can be written as(6)Σn1x1,⋯,njxjμ1,μ2=Yn1,⋯,njλλd1x1,⋯,djxj=μ1d1x1,⋯,djxj,d1+d2+⋯+dj,isodd,μ2d1x1,⋯,djxj,d1+d2+⋯+dj,iseven.We have the following conditions:(7)Σxμ1=μ1x,Σ2xμ1,μ2=μ22x+μ1x2,Σx,tμ1,μ2=μ2x,t+μ1xμ1t,....Proposition 1.
Letμ1=lnΩ1/Ω2,μ2=lnΩ1Ω2, then the relations between binary Bell polynomials and Hirota D-operator read
(8)Σn1x1,⋯,njxjμ1,μ2μ1=lnΩ1/Ω2,μ2=lnΩ1Ω2=Ω1Ω2−1Dx1n1⋯DxjnjΩ1Ω2,with Hirota operator
(9)∏i=1jDxinig.η=∏i=1j∂∂xi−∂∂xi′niΩ1x1,⋯,xjΩ2x1′,⋯,xj′x1=x1′,⋯,xj=xj′.Proposition 2.
TakeΞγ=∑iδiPd1x1,⋯,djxj=0 and μ1=lnΩ1/Ω2,μ1=lnΩ1Ω2, we have
(10)∑iδ1iYn1x1,⋯,njxjμ1,μ2=0,∑iδ1iYd1x1,⋯,djxjμ1,μ2=0,which need to satisfy
(11)Rγ′,γ=Rγ′−Rγ=Rμ2+μ1−Rμ2−μ1=0.The generalized Bell polynomialsYn1x1,⋯,njxjξ is as
(12)Ω1Ω2−1Dx1n1⋯DxjnjΩ1Ω2=Σn1x1,⋯,njxjμ1,μ2μ1=lnΩ1/Ω2,μ2=lnΩ1Ω2=Σn1x1,⋯,njxjμ1,μ1+γμ1=lnΩ1/Ω2,γ=lnΩ1Ω2=∑k1n1⋯∑kjnj∏i=1jnikiPk1x1,⋯,kjxjγYn1−k1x1,⋯,nj−kjxjμ1.The Cole-Hopf relation is as follows:(13)Yk1x1,⋯,kjxjμ1=lnφ=φn1x1,⋯,njxjφ,Ω1Ω2−1Dx1n1⋯DxjnjΩ1Ω2Ω2=expγ2,Ω1/Ω2=φ=φ−1∑k1n1⋯∑kjnj∏l=1jnlklPk1x1,⋯,klxlγφn1−k1x1,⋯,nd−klxl,with
(14)Ytμ1=φtφ,Y2xμ1,β=γ2x+φ2xφ,Y2x,yμ1,μ2=γ2xφyφ+2γx,yφxφ+φ2x,yφ.


## 3. Bilinear Form Equation to the Equation in Liquid with Gas Bubble

To discover the linearizing representation, take the following form:(15)u=ctπxx+u0,π=πx,y,z,t.Inserting Equation (15) into Equation (2), one obtains
(16)Rπ=ddtct∂∂xπx,y,z,t+ct∂2∂x∂tπx,y,z,t+ϕ1tct2∂∂xπx,y,z,t2+ϕ1tct2πx,y,z,t∂2∂x2πx,y,z,t+ϕ2tct∂4∂x4πx,y,z,t+ϕ3tct∂2∂x2πx,y,z,t+ϕ4tct∂2∂y2πx,y,z,t+ϕ5tct∂2∂z2πx,y,z,t=0,with
(17)ct=12ϕ2tϕ1t.The novel equationRπ is as
(18)Rπ=Px,t+ϕ2tP4x+3P2x2+ϕ3tP2x+ϕ4tP2y+ϕ5tP2z=0.Applying a change of dependent variable,(19)π=lng⇔u=12ϕ2tϕ1tlngxx.Theorem 3.
With the following relations:(20)π=lng⇔u=12ϕ2tϕ1tlngxx,into Equation (2), the 3+1-D VC NLW equation can be stated as follows:
(21)Rg=ggxt−gxgt+ϕ2tgg4x−4gxg3x+3gxx2+ϕ3tggxx−gx2ϕ4tggyy−gy2+ϕ5tggzz−gz2=12DxDt+ϕ2tDx4+ϕ3tDx2+ϕ4tDy2+ϕ5tDz2g.g=0,where g=gx,y,z,t and π=πx,y,z,t.


## 4. Damped Oscillation Solutions for Generalized CV NLW Equation

The two subsections including damped sinusoidal oscillations wave solutions and combined damped sinusoidal oscillation wave solutions are investigated in the following subsections.


### 4.1. Damped Oscillation Wave Solutions

Here, we utilize to formulate the new exact solutions to the3+1-dimensional generalized CV NLW equation. Consider the below function for studying the damped oscillation wave solutions as
(22)g=ea1σ4tcosa2+σ5tsina3,al=αlx+βly+δlz+σlt,l=1,2,3.Afterwards, the valuesαl,βl,δl,σlt,l=1,2,3 will be found. By making use of Equation (22) into (21) and taking the coefficients, each power of cosa2 and sina2 to zero yields a system of equations (algebraic) (these are not collected here for the minimalist) for αl,βl,δl,σlt,l=1,2,3. These algebraic equations by using the emblematic computation software like Maple give the solutions in the following with using u=12ϕ2/ϕ1lngxx as follows:
(23)4α24ϕ2s42+4α34ϕ2s52−α22ϕ3s42−α32ϕ3s52−β22ϕ4s42−β32ϕ4s52−δ22ϕ5s42−δ32ϕ5s52−S2α2s42−S3α3s52=0,−S4α3s5+S5α3s4=0,α24ϕ2s4s5+6α22α32ϕ2s4s5+α34ϕ2s4s5−α22ϕ3s4s5−α32ϕ3s4s5−β22ϕ4s4s5−β32ϕ4s4s5−δ22ϕ5s4s5−δ32ϕ5s4s5−S2α2s4s5−S3α3s4s5=0,−4α23α3ϕ2s4s5−4α2α33ϕ2s4s5+2α2α3ϕ3s4s5+2β2β3ϕ4s4s5+2δ2δ3ϕ5s4s5+S2α3s4s5+S3α2s4s5=0,−S4α2s5+S5α2s4=0,where sl=σit and Sl=d/dtσlt,l=1,⋯,5 and the obtained solutions are as follows:


#### 4.1.1. Set I Solutions

(24)σ5t=3α32ϕ2α2−α32α2+α32+A13α22ϕ2α2−α32α2+α32+A1σ4t3α32ϕ2α2−α32α2+α32+A1,σ2t=α2α22ϕ2+3α32ϕ2−ϕ3α22−α32+A2tα22−α32+C1,σ3t=α33α22ϕ2+α32ϕ2−ϕ3α22−α32+A3tα22−α32+C2,A1=ϕ4α2β3−α3β22+ϕ5α2δ3−α3δ22,A2=−ϕ4α2β22+α2β32−2α3β2β3−ϕ5α2δ22+α2δ32−2α3δ2δ3,A3=−ϕ42α2β2β3−α3β22−α3β32−ϕ52α2δ2δ3−α3δ22−α3δ32,where ϕl=ϕlt,l=1,⋯,5 and σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(25)Σ1=12ϕ2ϕ1lngx,y,z,txx=2∂2/∂x2gx,y,z,tgx,y,z,t−∂/∂xgx,y,z,t2gx,y,z,t2,(26)gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosα2x+β2y+δ2z+σ2t+σ5tsinα3x+β3y+δ3z+σ3t.Figures1–3, respectively, show the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(27)C1=C2=1,α1=β2=δ1=0.2,α2=β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=σ4t=ϕ1=ϕ2=ϕ3=ϕ4=t,ϕ5=2t,z=1,y=2,C1=C2=α1=β2=δ1=α2=β1=β3=δ2=δ3=1,α3=2,σ1t=1,σ4t=2,ϕ1=cost,ϕ2=cos2t,ϕ3=cos2t+1,ϕ4=cos2t+2,ϕ5=cos2t+3,z=1,y=1,C1=C2=α1=β2=δ1=α2=β1=β3=δ2=δ3=1,α3=2,σ1t=1,σ4t=2,ϕ1=1,ϕ2=1,ϕ3=1,ϕ4=1,ϕ5=1,z=1,y=−1,in Equation (25).Figure 1
Plot of periodic wave solution (25) Σ1 (3D plot, density plot, and 2D plot t), respectively.Figure 2
Plot of periodic wave solution (25) Σ1 (3D plot, density plot, and 2D plot t), respectively.Figure 3
Plot of periodic wave solution (25) Σ1 (3D plot, density plot, and 2D plot t), respectively.


#### 4.1.2. Set II Solutions

(28)β3=α3β2α2,σ5t=3α22ϕ2α22−α322+ϕ5α2δ3−α3δ22σ4t3α32ϕ2α22−α322+ϕ5α2δ3−α3δ22,σ2t=α22ϕ2α22+3α32α22−α32+A1tα2α22−α32+C1,σ3t=α22α3ϕ23α22+α32α22−α32+A2tα22α22−α32+C2,A1=−α22−α32α22ϕ3+β22ϕ4−α2ϕ5α2δ22+α2δ32−2α3δ2δ3,A2=−α3α22ϕ3+β22ϕ4α22−α32−α22ϕ52α2δ2δ3−α3δ22−α3δ32,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(29)Σ2=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosα2x+β2y+δ2z+σ2t+σ5tsinα3x+α3β2yα2+δ3z+σ3t.


#### 4.1.3. Set III Solutions

(30)σ5t=±3α34ϕ2+δ22ϕ5ϕ5δ2σ4t3α34ϕ2+δ22ϕ5,σ2t=−2δ3δ2ϕ5tα3+C1,σ3t=α34ϕ2−α32ϕ3−β32ϕ4−δ22ϕ5−δ32ϕ5tα3+C2,α2=β2=0,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(31)Σ3=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosδ2z+σ2t+σ5tsinα3x+β3y+δ3z+σ3t.


#### 4.1.4. Set IV Solutions

(32)σ2t=4α24ϕ2−α22ϕ3−β32ϕ4−δ32ϕ5tα2+C1,σ3t=4α24ϕ2−α22ϕ3−β32ϕ4−δ32ϕ5tα2+C2,σ5t=C3σ4t,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(33)Σ4=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosα2x+β3y+δ3z+σ2t+σ5tsinα2x+β3y+δ3z+σ3t.


#### 4.1.5. Set V Solutions

(34)σ2t=∫−−σ5t2α34ϕ23σ4t2+σ5t2+α32ϕ3σ4t2σ5t2+σ4t4β22ϕ4+δ22ϕ5α3σ4t3σ5tdt+C1,σ3t=∫−−σ5t2α34ϕ2σ4t2+3σ5t2+α32ϕ3σ4t2σ5t2+σ4t4β22ϕ4+δ22ϕ5σ4t2σ5t2α3dt+C2,σ5t=C3σ4t,α2=C3α3,β3=β2C3,δ3=δ2C3,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(35)Σ5=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosC3α3x+β2y+δ2z+σ2t+σ5tsinα3x+β2yC3+δ2zC3+σ3t.Figures4–6, respectively, show the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(36)C1=C2=1,C3=2,α1=β2=δ1=0.2,β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=σ4t=ϕ1=ϕ2=ϕ3=ϕ4=t,ϕ5=2t,z=1,y=2,C1=C2=1,C3=2,α1=β2=δ1=0.2,β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=t,σ4t=tcost,ϕ1=t+1,ϕ2=sint,ϕ3=t+t2,ϕ4=t−t3,ϕ5=2t+1,z=1,y=2,C1=C2=1,C3=2,α1=β2=δ1=0.2,β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=t,σ4t=sint2,ϕ1=1,ϕ2=2,ϕ3=3,ϕ4=3,ϕ5=2,z=1,y=1,in Equation (35).Figure 4
Plot of periodic wave solution (35) Σ5 (3D plot, density plot, and 2D plot t), respectively.Figure 5
Plot of periodic wave solution (35) Σ5 (3D plot, density plot, and 2D plot t), respectively.Figure 6
Plot of periodic wave solution (35) Σ5 (3D plot, density plot, and 2D plot t), respectively.


### 4.2. Combined Damped Oscillation Wave Solutions

Here, we utilize to formulate the new exact solutions to the3+1-dimensional generalized CV NLW equation. Consider the below function for studying the combined damped oscillation wave solutions as
(37)g=ea1σ5tcosa2+σ6tsina2+ea3σ7tcosa4+σ8tsina5,al=αlx+βly+δlz+σlt,l=1,2,3,4.Afterwards, the valuesαl,βl,δl,σlt,l=1,2,3,4 will be found. By making use of Equation (37) into (21) and taking the coefficients, each power of cosa2 and sina2, cosa4 and sina4 to zero yields a system of equations (algebraic) (these are not collected here for the minimalist) for αl,βl,δl,σlt,l=1,2,3,4. These algebraic equations by using the emblematic computation software like Maple give the solutions in the following with using u=12ϕ2/ϕ1lngxx as follows:


#### 4.2.1. Set I Solutions

(38)σ7t=0,σ4t=C3,σ2t=−arctanσ5tσ6t+2δ2ϕ5β1δ2−β2δ1+β2δ3−β3δ2tα1β2−α3β2+C1,σ3t=∫Mtdt+C2,α2=α4=β4=δ4=0,ϕ4=−δ22ϕ5β22,Mt=−σ8tA5+A6−A7+A1+σ8tA2+A3−A4σ8tβ22σ5t2+σ6t2α1−α3,A1=σ8tddtσ1tβ22σ5t2+σ6t2α1−α3+σ8tβ22α1−α3ddtσ6tσ6t+σ5tddtσ5t,A2=σ6t2α14β22ϕ2+α34β22ϕ2+α12β22ϕ3+α32β22ϕ3−β12δ22ϕ5+β22δ12ϕ5+β22δ32ϕ5−β32δ22ϕ5,A3=σ5t2α14α34β24ϕ22+α32β22ϕ3+1,A4=ddtσ8tβ22σ5t2+σ6t2α1−α3,A5=2α1α3β22ϕ22α12−3α1α3+2α32σ5t2+σ6t2,A6=2σ6t2α1α3β22ϕ3−β1β3δ22ϕ5+β22δ1δ3ϕ5,A7=σ5t2α12β22ϕ3−2α1α3β22ϕ3−β12δ22ϕ5+2β1β3δ22ϕ5+β22δ12ϕ5−2β22δ1δ3ϕ5+β22δ32ϕ5−β32δ22ϕ5,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(39)Σ1=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+δ2z+σ2t+σ6tsinβ2y+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set I-I:(40)σ6t=0,σ8t=−tan−σ4t+C3σ7t,σ2t=2δ2ϕ5β1δ2−β2δ1+β2δ3−β3δ2tβ2α1−α3+C1,σ3t=∫Mtdt+C2,α2=α4=β4=δ4=0,ϕ4=−δ22ϕ5β22,Mt=−A1+A2σ7tσ5tβ22α1−α3+C2,A1=−σ5tσ7tβ22ϕ2α1−α34+β22ϕ3α1−α32−ϕ5β1δ2−β2δ1+β2δ3−β3δ2β1δ2+β2δ1−β2δ3−β3δ2,A2=β22α1−α3σ8tddtσ4tσ5t−σ5tσ7tddtσ1t+σ5tddtσ7t−σ7tddtσ5t,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(41)Σ2=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+δ2z+σ2t+σ6tsinβ2y+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set I-II:(42)σ8t=−tan−σ4t+C3σ7t,σ2t=−arctanσ5tσ6t+2δ2ϕ5β1δ2−β2δ1+β2δ3−β3δ2tα1β2−α3β2+C1,σ3t=∫Mtdt+C2,α2=α4=β4=δ4=0,ϕ4=−δ22ϕ5β22,Mt=−A1+A2+A3σ7tβ22σ5t2+σ6t2α1−α3+C2,A1=2σ7tσ5t2+σ6t22α13α3β22ϕ2−3α12α32β22ϕ2+2α1α33β22ϕ2+α1α3β22ϕ3−β1β3δ22ϕ5+β22δ1δ3ϕ5,A2=β22α1−α3σ5t2+σ6t2σ8tddtσ4t−σ7tddtσ1t+σ6t2ddtσ7t−σ6tddtσ6tσ7t+σ5t2ddtσ7t−σ5tσ7tddtσ5t,A3=−σ7tσ5t2+σ6t2α14β22ϕ2+α34β22ϕ2+α12β22ϕ3+α32β22ϕ3−β12δ22ϕ5+β22δ12ϕ5+β22δ32ϕ5−β32δ22ϕ5,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3,4, are free values. It concludes along with the bilinear equation; the exact solution will be as
(43)Σ3=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+δ2z+σ2t+σ6tsinβ2y+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.


#### 4.2.2. Set II Solutions

(44)σ4t=C3,σ7t=0,σ2t=−arctanσ5tσ6t+C1,α2=β2=δ2=δ4=0,σ3t=∫A4+A1+A2+A3σ8tσ5t2+σ6t2α1−α3dt+C2,A1=σ8tσ5t2+σ6t2α14ϕ2+α34ϕ2+α12ϕ3+α32ϕ3+β12ϕ4+β32ϕ4+δ12ϕ5+ϕ5δ32,A2=−ddtσ8tσ5t2+σ6t2α1−α3,A3=σ8tα1−α3σ5t2ddtσ1t+ddtσ1tσ6t2+σ5tddtσ5t+σ6tddtσ6t,A4=−2σ8tσ5t2+σ6t22α13α3ϕ2−3α12α32ϕ2+2α1α33ϕ2+α1α3ϕ3+β1β3ϕ4+δ1ϕ5δ3,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3,4, are free values. It concludes along with the bilinear equation; the exact solution will be as
(45)Σ4=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set II-I:(46)σ2t=C1,σ5t=0,σ8t=−tan−σ4t+C3σ7t,α2=β2=δ2=δ4=0,σ3t=∫−A1+A2σ6tσ7tα1−α3dt+C2,A1=−σ6tσ7tϕ2α1−α34+ϕ3α1−α32+ϕ4β1−β32+ϕ5δ1−δ32,A2=−α1−α3ddtσ1tσ6tσ7t−σ8tσ6tddtσ4t−σ6tddtσ7t+ddtσ6tσ7t,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(47)Σ5=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set II-II:(48)σ2t=−arctanσ5tσ6t+C1,σ8t=−tan−σ4t+C3σ7t,α2=β2=δ2=δ4=0,σ3t=∫−A3+A1+A2σ7tσ5t2+σ6t2α1−α3dt+C2,A1=ddtσ7tσ5t2+σ6t2α1−α3,A2=−α1−α3σ5t2+σ6t2ddtσ1tσ7t−σ8tddtσ4t+σ5tddtσ5tσ7t+σ6tddtσ6tσ7t,A3=−σ7tσ5t2+σ6t2ϕ2α1−α34+ϕ3α1−α32+ϕ4β1−β32+ϕ5δ1−δ32,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3,4, are free values. It concludes along with the bilinear equation; the exact solution will be as
(49)Σ6=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Figures7 and 8, respectively, show the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(50)C1=C2=1,C3=0.2,α1=δ1=0.2,β1=α3=0.1,β3=0.3,δ3=0.4,σ1t=σ5t=σ6t=σ7t=ϕ1=ϕ2=ϕ3=t,ϕ4=t+1,ϕ5=2t,z=1,y=1,C1=C2=1,C3=0.2,α1=δ1=0.2,β1=α3=0.1,β3=0.3,δ3=0.4,σ1t=sint,σ4t=sin2t,σ5t=sin3t,σ6t=sin4t,σ7t=sint,ϕ1=3,ϕ2=2,ϕ3=2,ϕ4=1,ϕ5=2,z=1,y=1,in equation (49).Figure 7
Plot of periodic wave solution (49) Σ6 (3D plot, density plot, and 2D plot t), respectively.Figure 8
Plot of periodic wave solution (49) Σ6 (3D plot, density plot, and 2D plot t), respectively.


#### 4.2.3. Set III Solutions

III-I:(51)Σ7=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ6tsinβ2y+β2δ4zβ4+2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ42α1−α3+C1+eα3x+β3y+δ3z−∫A1+A2/σ8tσ6tβ42α1−α3dtσ8tsinβ4y+δ4z+2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ4α1−α3+C3,A1=−σ6tσ8tβ42ϕ2α1−α34+β42ϕ3α1−α32−ϕ5β1δ4−β3δ42−β4δ1−β4δ32,A2=−β42α1−α3σ6tσ8tddtσ1t−σ6tddtσ8t+σ8tddtσ6t.III-II:(52)Σ8=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+β2δ4zβ4−arctanσ5tσ6t+A1+σ6tsinβ2y+β2δ4zβ4−arctanσ5tσ6t+A1+eα3x+β3y+δ3z−∫A2+A3/σ8tβ42σ5t2+σ6t2α1−α3dtσ8tsinβ4y+δ4z+2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ4α1−α3+C3,A1=2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ42α1−α3,A2=−β42α1−α3σ5t2+σ6t2σ8tddtσ1t−ddtσ8t+σ6tσ8tddtσ6t+σ8tσ5tddtσ5t,A3=−σ8tσ5t2+σ6t2β42ϕ2α1−α34+β42ϕ3α1−α32−ϕ5β1δ4−β3δ42−β4δ1−β4δ32.III-III:(53)Σ9=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ6tsinβ2y+β2δ4zβ4+2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ4A0+C1+eα3x+β3y+δ3z−∫A2+A3/σ6tσ7tβ4A0dtσ7tcosβ4y+δ4z+σ4t−tanC3−∫A1A0dtσ7tsinβ4y+δ4z+σ4t,A0=β4α1−α3,A1=−2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3+A0ddtσ4t,A2=−β42α1−α3σ7tσ6tddtσ1t−ddtσ4tσ6tσ8t+σ7tddtσ6t−σ6tddtσ7t,A3=−σ6tσ7tβ42ϕ2α1−α34+σ7tβ42ϕ3α1−α32−ϕ5β1δ4−β3δ4−β4δ1+β4δ3σ7tβ1δ4−σ7tβ3δ4+σ7tβ4δ1−σ7tβ4δ3−2σ8tβ4δ4.III-IV:(54)Σ10=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosX1+σ6tsinX1+eα3x+β3y+δ3z−∫A3+A4/σ7tβ42σ5t2+σ6t2α1−α3dtσ7tcosX2−tanC3−∫A1β4α1−α3dtσ7tsinX2,X1=β2y+β2δ4zβ4−arctanσ5tσ6t+A2t,X2=β4y+δ4z+σ4t,A1=−2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3+β4α1−α3ddtσ4t,A2=2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3β42α1−α3,A3=−σ5t2+σ6t2σ7tβ42ϕ2α1−α34+σ7tβ42ϕ3α1−α32−ϕ5β1δ4−β3δ4−β4δ1+β4δ3σ7tβ1δ4−σ7tβ3δ4+σ7tβ4δ1−σ7tβ4δ3−2σ8tβ4δ4,A4=−β42α1−α3σ5t2+σ6t2σ7tddtσ1t−ddtσ4tσ8t−ddtσ7t+ddtσ5tσ7tσ5t+σ7tσ6tddtσ6t.Figure9 shows the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(55)C1=C2=1,C3=0.2,α1=δ1=0.2,β1=α3=0.1,β3=0.3,δ3=0.4,in Equation (54).Figure 9
Plot of periodic wave solution (54) Σ10 (3D plot, density plot, and 2D plot t), respectively.


#### 4.2.4. Set IV Solutions

IV-I:(56)Σ11=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ6tsinC1+eα3x+β3y+δ3z+2tβ1ϕ4β4−β3ϕ4β4+δ1δ4ϕ5−δ3δ4ϕ5/α4+σ1t+lnσ6t/σ8tσ8t×sinα4x+β4y+δ4z−3ϕ3α42−ϕ44β12−8β1β3+4β32−3β42−ϕ54δ12−8δ1δ3+4δ32−3δ42t3α4,ϕ2=13ϕ4β1−β32+ϕ5δ1−δ32α44.IV-II:(57)Σ12=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ5tcosC1−arctanσ5tσ6t+σ6tsinC1−arctanσ5tσ6t+eα3x+β3y+δ3z+∫A2+A1/σ8tα4σ5t2+σ6t2dtσ8t×sinα4x+β4y+δ4z−3ϕ3α42−ϕ44β12−8β1β3+4β32−3β42−ϕ54δ12−8δ1δ3+4δ32−3δ42t3α4,A1=α4σ5t2+σ6t2σ8tddtσ1t−ddtσ8t+σ8tσ5tddtσ5t+σ6tddtσ6t,A2=2σ8tσ5t2+σ6t2β1ϕ4β4−β3ϕ4β4+δ1δ4ϕ5−δ3δ4ϕ5,ϕ2=ϕ4β1−β32+ϕ5δ1−δ323α44.IV-III:(58)Σ13=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosα4x+β4y+δ4z+σ4t+σ8tsinα4x+β4y+δ4z+σ4t,σ2t=C1,σ3t=∫−13A2+A3σ7tσ8tσ6tα4σ7t2+σ8t2dt+C2,σ8t=∫A1−3α4σ7t2+σ8t2σ8t2d/dtσ4t+σ7t2d/dtσ4t+σ8td/dtσ7t−3σ7t3α4−3σ7tσ8t2α4dt,A1=−σ8t43ϕ3α42−4β12ϕ4+8β1ϕ4β3−4ϕ4β32+3β42ϕ4−4δ12ϕ5+8δ1ϕ5δ3−4ϕ5δ32+3δ42ϕ5+2σ8t2σ7t4σ8tβ1β4ϕ4−4σ8tβ3β4ϕ4+4σ8tδ1δ4ϕ5−4σ8tδ3δ4ϕ5−3σ7tα42ϕ3−σ7tβ42ϕ4−σ7tδ42ϕ5−3σ7t4ϕ3α42+β42ϕ4+δ42ϕ5,A2=3σ8tα4σ7t2+σ8t2σ8tσ6tddtσ4t−σ7tddtσ1tσ6t−σ7tddtσ6t+σ6tddtσ7t,A3=σ6t3α42ϕ3σ7t2+σ8t22−ϕ4σ8t44β12−8β1β3+4β32−3β42+14σ8t3σ7tβ4β1−β3−4σ8t2σ7t2−β3+β1−β4−β3+β1+β4−2σ8tσ7t3β4β1−β3−β42σ7t4−ϕ5σ8t44δ12−8δ1δ3+4δ32−3δ42+14σ8t3σ7tδ4δ1−δ3−4σ8t2σ7t2−δ3+δ1−δ4−δ3+δ1+δ4−2σ8tσ7t3δ4δ1−δ3−δ42σ7t4,ϕ2=13σ8t2ϕ4β1σ8t−β3σ8t+β4σ7t2+ϕ5δ1σ8t−δ3σ8t+δ4σ7t2α44σ7t4+2σ7t2σ8t2+σ8t4.IV-IV:(59)Σ14=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosα4x+β4y+δ4z+σ4t+σ8tsinα4x+β4y+δ4z+σ4t,σ2t=−arctanσ5tσ6t+C1,σ8t=∫A2+A1−3σ7t3α4−3σ7tσ8t2α4dt,σ3t=∫−13A3+A4σ7tσ8tα4σ5t2+σ6t2σ7t2+σ8t2dt+C2,A1=−3α4σ7t2+σ8t2σ8t2ddtσ4t+σ7t2ddtσ4t+σ8tddtσ7t,A2=−3α42ϕ3σ7t2+σ8t22+ϕ4σ8t44β12−8β1β3+4β32−3β42+σ7tβ48σ8t3β1−8σ8t3β3−2σ8t2σ7tβ4−3σ7t3β4+ϕ5σ8t44δ12−8δ1δ3+4δ32−3δ42+σ7tδ48σ8t3δ1−8σ8t3δ3−2σ8t2σ7tδ4−3σ7t3δ4,A3=3σ8tα4σ7t2+σ8t2σ5t2+σ6t2σ8tddtσ4t−σ7tddtσ1t+ddtσ7t−σ7tσ5tddtσ5t+σ6tddtσ6t,A4=σ5t2+σ6t2σ8t43ϕ3α42−4β12ϕ4+8β1ϕ4β3−4ϕ4β32+3β42ϕ4−4δ12ϕ5+8δ1ϕ5δ3−4ϕ5δ32+3δ42ϕ5+σ7t43ϕ3α42+β42ϕ4+δ42ϕ5+2σ8t2σ7t23ϕ3α42+2β12ϕ4+2ϕ4β32−2β42ϕ4+2δ12ϕ5−2δ42ϕ5+2σ8t2σ7t23ϕ3α42+2β12ϕ4+2ϕ4β32−2β42ϕ4+2δ12ϕ5−4δ1ϕ5δ3+2ϕ5δ32−2δ42ϕ5−14σ8t3σ7tβ1ϕ4β4−β3ϕ4β4+δ1δ4ϕ5−δ3δ4ϕ5−2σ8tσ7t24σ8tβ1β3ϕ4−σ7tβ1β4ϕ4+σ7tβ3β4ϕ4−σ7tδ1δ4ϕ5,ϕ2=13σ8t2ϕ4β1σ8t−β3σ8t+β4σ7t2+ϕ5δ1σ8t−δ3σ8t+δ4σ7t2α44σ7t4+2σ7t2σ8t2+σ8t4.


#### 4.2.5. Set V Solutions

(60)Σ15=12ϕ2ϕ1lngxx,g=eX1+σ1tσ5tcosα2x+α2β4yα4+δ2z+σ2t+σ6tsinα2x+α2β4yα4+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosα4x+β4y+δ4z+σ4t+σ8tsinα4x+β4y+δ4z+σ4t,X1=σ8tα2σ5t+α3σ6t−α4σ6tσ7txσ6tσ8t+σ8tα2β4σ5t+α4β3σ6t−α4β4σ6tσ7tyα4σ6tσ8t+12B1+B2+B3zα2α4σ6tσ8tα2σ5tσ8t−α4σ6tσ7t,B1=α2δ4+α4δ2σ5t2+2σ5tσ6tα4δ3+σ6t2α2δ4−α4δ2σ8t2α22,B2=−2σ7tσ5tσ8tσ6tα2α4α2δ4+α4δ2,B3=σ6t2α42σ7t2α2δ4+α4δ2−2σ7tσ8tα2δ3−σ8t2α2δ4−α4δ2,σ2t=∫−A1+A2σ6t2σ8t2α42α2δ4−α4δ22σ6t2+σ5t2dt+C1,A1=σ8t2σ6t2α42α2δ4−α4δ22ddtσ5tσ6t−ddtσ6tσ5t,A2=α2σ6t2+σ5t212σ7tσ6tα45δ22ϕ2σ7tσ6tα4−2σ5tσ8tα2+σ8t24α22α42ϕ23σ5t2α42δ22−σ6t2α22δ42+2σ6t2α2α4δ2δ4−σ6t2α42δ22+σ6t2α2δ4−α4δ22α42ϕ3+β42ϕ4,A3=−σ8t2σ6t2α4α2δ4−α4δ22σ7tddtσ8t−σ8tddtσ7t,A4=σ8t2+σ7t212α22α42δ42ϕ2σ7tσ6tα4−σ5tσ8tα22−σ8t2σ6t2α2δ4−α4δ224α44ϕ2−α42ϕ3−β42ϕ4,σ4t=∫−A3+A4σ6t2σ8t2α4α2δ4−α4δ22σ8t2+σ7t2dt,σ3t=∫−A5+A6σ6t3σ8t3α42α2δ4−α4δ22σ8t2+σ7t2σ6t2+σ5t2dt,A5=S1S2σ7tσ6tα4−σ5tσ8tα24σ7tσ6tα43ϕ2S3S4+4σ5t2σ8t2α22α42ϕ2S3+σ8t2σ6t2α2δ4−α4δ212α2α43ϕ2α2δ2−α4δ4+α42ϕ3+β42ϕ4α2δ4−α4δ2,A6=−σ8t3σ6t3α42α2δ4−α4δ22ddtσ1tS1S2+S2ddtσ5tσ5t+ddtσ6tσ6t−S1σ7tddtσ7t+σ8tddtσ8t,S1=σ6t2+σ5t2,S2=σ8t2+σ7t2,S3=α22δ42+α2α4δ2δ4+α42δ22,S4=σ7tσ6tα4−2σ5tσ8tα2,α1=σ5tσ8tα2+α3σ6tσ8t−σ7tσ6tα4σ6tσ8t,ϕ5=12σ5tσ8tα2−σ7tσ6tα42α42α22ϕ2α2δ4−α4δ22σ6t2σ8t2,δ1=1/2σ8t2α22+σ6t2α42σ7t2α2δ4+σ7t2α4δ2−σ8t2α2δ4+σ8t2α4δ2α2α4σ6tσ8tσ5tσ8tα2−σ7tσ6tα4,H1=σ5t2α2δ4+σ5t2α4δ2+σ6t2α2δ4−σ6t2α4δ2.


#### 4.2.6. Set VI Solutions

(61)Σ16=12ϕ2ϕ1lngxx,g=eα3x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.


## 4.1. Damped Oscillation Wave Solutions

Here, we utilize to formulate the new exact solutions to the3+1-dimensional generalized CV NLW equation. Consider the below function for studying the damped oscillation wave solutions as
(22)g=ea1σ4tcosa2+σ5tsina3,al=αlx+βly+δlz+σlt,l=1,2,3.Afterwards, the valuesαl,βl,δl,σlt,l=1,2,3 will be found. By making use of Equation (22) into (21) and taking the coefficients, each power of cosa2 and sina2 to zero yields a system of equations (algebraic) (these are not collected here for the minimalist) for αl,βl,δl,σlt,l=1,2,3. These algebraic equations by using the emblematic computation software like Maple give the solutions in the following with using u=12ϕ2/ϕ1lngxx as follows:
(23)4α24ϕ2s42+4α34ϕ2s52−α22ϕ3s42−α32ϕ3s52−β22ϕ4s42−β32ϕ4s52−δ22ϕ5s42−δ32ϕ5s52−S2α2s42−S3α3s52=0,−S4α3s5+S5α3s4=0,α24ϕ2s4s5+6α22α32ϕ2s4s5+α34ϕ2s4s5−α22ϕ3s4s5−α32ϕ3s4s5−β22ϕ4s4s5−β32ϕ4s4s5−δ22ϕ5s4s5−δ32ϕ5s4s5−S2α2s4s5−S3α3s4s5=0,−4α23α3ϕ2s4s5−4α2α33ϕ2s4s5+2α2α3ϕ3s4s5+2β2β3ϕ4s4s5+2δ2δ3ϕ5s4s5+S2α3s4s5+S3α2s4s5=0,−S4α2s5+S5α2s4=0,where sl=σit and Sl=d/dtσlt,l=1,⋯,5 and the obtained solutions are as follows:


### 4.1.1. Set I Solutions

(24)σ5t=3α32ϕ2α2−α32α2+α32+A13α22ϕ2α2−α32α2+α32+A1σ4t3α32ϕ2α2−α32α2+α32+A1,σ2t=α2α22ϕ2+3α32ϕ2−ϕ3α22−α32+A2tα22−α32+C1,σ3t=α33α22ϕ2+α32ϕ2−ϕ3α22−α32+A3tα22−α32+C2,A1=ϕ4α2β3−α3β22+ϕ5α2δ3−α3δ22,A2=−ϕ4α2β22+α2β32−2α3β2β3−ϕ5α2δ22+α2δ32−2α3δ2δ3,A3=−ϕ42α2β2β3−α3β22−α3β32−ϕ52α2δ2δ3−α3δ22−α3δ32,where ϕl=ϕlt,l=1,⋯,5 and σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(25)Σ1=12ϕ2ϕ1lngx,y,z,txx=2∂2/∂x2gx,y,z,tgx,y,z,t−∂/∂xgx,y,z,t2gx,y,z,t2,(26)gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosα2x+β2y+δ2z+σ2t+σ5tsinα3x+β3y+δ3z+σ3t.Figures1–3, respectively, show the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(27)C1=C2=1,α1=β2=δ1=0.2,α2=β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=σ4t=ϕ1=ϕ2=ϕ3=ϕ4=t,ϕ5=2t,z=1,y=2,C1=C2=α1=β2=δ1=α2=β1=β3=δ2=δ3=1,α3=2,σ1t=1,σ4t=2,ϕ1=cost,ϕ2=cos2t,ϕ3=cos2t+1,ϕ4=cos2t+2,ϕ5=cos2t+3,z=1,y=1,C1=C2=α1=β2=δ1=α2=β1=β3=δ2=δ3=1,α3=2,σ1t=1,σ4t=2,ϕ1=1,ϕ2=1,ϕ3=1,ϕ4=1,ϕ5=1,z=1,y=−1,in Equation (25).Figure 1
Plot of periodic wave solution (25) Σ1 (3D plot, density plot, and 2D plot t), respectively.Figure 2
Plot of periodic wave solution (25) Σ1 (3D plot, density plot, and 2D plot t), respectively.Figure 3
Plot of periodic wave solution (25) Σ1 (3D plot, density plot, and 2D plot t), respectively.


### 4.1.2. Set II Solutions

(28)β3=α3β2α2,σ5t=3α22ϕ2α22−α322+ϕ5α2δ3−α3δ22σ4t3α32ϕ2α22−α322+ϕ5α2δ3−α3δ22,σ2t=α22ϕ2α22+3α32α22−α32+A1tα2α22−α32+C1,σ3t=α22α3ϕ23α22+α32α22−α32+A2tα22α22−α32+C2,A1=−α22−α32α22ϕ3+β22ϕ4−α2ϕ5α2δ22+α2δ32−2α3δ2δ3,A2=−α3α22ϕ3+β22ϕ4α22−α32−α22ϕ52α2δ2δ3−α3δ22−α3δ32,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(29)Σ2=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosα2x+β2y+δ2z+σ2t+σ5tsinα3x+α3β2yα2+δ3z+σ3t.


### 4.1.3. Set III Solutions

(30)σ5t=±3α34ϕ2+δ22ϕ5ϕ5δ2σ4t3α34ϕ2+δ22ϕ5,σ2t=−2δ3δ2ϕ5tα3+C1,σ3t=α34ϕ2−α32ϕ3−β32ϕ4−δ22ϕ5−δ32ϕ5tα3+C2,α2=β2=0,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(31)Σ3=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosδ2z+σ2t+σ5tsinα3x+β3y+δ3z+σ3t.


### 4.1.4. Set IV Solutions

(32)σ2t=4α24ϕ2−α22ϕ3−β32ϕ4−δ32ϕ5tα2+C1,σ3t=4α24ϕ2−α22ϕ3−β32ϕ4−δ32ϕ5tα2+C2,σ5t=C3σ4t,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(33)Σ4=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosα2x+β3y+δ3z+σ2t+σ5tsinα2x+β3y+δ3z+σ3t.


### 4.1.5. Set V Solutions

(34)σ2t=∫−−σ5t2α34ϕ23σ4t2+σ5t2+α32ϕ3σ4t2σ5t2+σ4t4β22ϕ4+δ22ϕ5α3σ4t3σ5tdt+C1,σ3t=∫−−σ5t2α34ϕ2σ4t2+3σ5t2+α32ϕ3σ4t2σ5t2+σ4t4β22ϕ4+δ22ϕ5σ4t2σ5t2α3dt+C2,σ5t=C3σ4t,α2=C3α3,β3=β2C3,δ3=δ2C3,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(35)Σ5=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosC3α3x+β2y+δ2z+σ2t+σ5tsinα3x+β2yC3+δ2zC3+σ3t.Figures4–6, respectively, show the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(36)C1=C2=1,C3=2,α1=β2=δ1=0.2,β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=σ4t=ϕ1=ϕ2=ϕ3=ϕ4=t,ϕ5=2t,z=1,y=2,C1=C2=1,C3=2,α1=β2=δ1=0.2,β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=t,σ4t=tcost,ϕ1=t+1,ϕ2=sint,ϕ3=t+t2,ϕ4=t−t3,ϕ5=2t+1,z=1,y=2,C1=C2=1,C3=2,α1=β2=δ1=0.2,β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=t,σ4t=sint2,ϕ1=1,ϕ2=2,ϕ3=3,ϕ4=3,ϕ5=2,z=1,y=1,in Equation (35).Figure 4
Plot of periodic wave solution (35) Σ5 (3D plot, density plot, and 2D plot t), respectively.Figure 5
Plot of periodic wave solution (35) Σ5 (3D plot, density plot, and 2D plot t), respectively.Figure 6
Plot of periodic wave solution (35) Σ5 (3D plot, density plot, and 2D plot t), respectively.


## 4.1.1. Set I Solutions

(24)σ5t=3α32ϕ2α2−α32α2+α32+A13α22ϕ2α2−α32α2+α32+A1σ4t3α32ϕ2α2−α32α2+α32+A1,σ2t=α2α22ϕ2+3α32ϕ2−ϕ3α22−α32+A2tα22−α32+C1,σ3t=α33α22ϕ2+α32ϕ2−ϕ3α22−α32+A3tα22−α32+C2,A1=ϕ4α2β3−α3β22+ϕ5α2δ3−α3δ22,A2=−ϕ4α2β22+α2β32−2α3β2β3−ϕ5α2δ22+α2δ32−2α3δ2δ3,A3=−ϕ42α2β2β3−α3β22−α3β32−ϕ52α2δ2δ3−α3δ22−α3δ32,where ϕl=ϕlt,l=1,⋯,5 and σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(25)Σ1=12ϕ2ϕ1lngx,y,z,txx=2∂2/∂x2gx,y,z,tgx,y,z,t−∂/∂xgx,y,z,t2gx,y,z,t2,(26)gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosα2x+β2y+δ2z+σ2t+σ5tsinα3x+β3y+δ3z+σ3t.Figures1–3, respectively, show the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(27)C1=C2=1,α1=β2=δ1=0.2,α2=β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=σ4t=ϕ1=ϕ2=ϕ3=ϕ4=t,ϕ5=2t,z=1,y=2,C1=C2=α1=β2=δ1=α2=β1=β3=δ2=δ3=1,α3=2,σ1t=1,σ4t=2,ϕ1=cost,ϕ2=cos2t,ϕ3=cos2t+1,ϕ4=cos2t+2,ϕ5=cos2t+3,z=1,y=1,C1=C2=α1=β2=δ1=α2=β1=β3=δ2=δ3=1,α3=2,σ1t=1,σ4t=2,ϕ1=1,ϕ2=1,ϕ3=1,ϕ4=1,ϕ5=1,z=1,y=−1,in Equation (25).Figure 1
Plot of periodic wave solution (25) Σ1 (3D plot, density plot, and 2D plot t), respectively.Figure 2
Plot of periodic wave solution (25) Σ1 (3D plot, density plot, and 2D plot t), respectively.Figure 3
Plot of periodic wave solution (25) Σ1 (3D plot, density plot, and 2D plot t), respectively.


## 4.1.2. Set II Solutions

(28)β3=α3β2α2,σ5t=3α22ϕ2α22−α322+ϕ5α2δ3−α3δ22σ4t3α32ϕ2α22−α322+ϕ5α2δ3−α3δ22,σ2t=α22ϕ2α22+3α32α22−α32+A1tα2α22−α32+C1,σ3t=α22α3ϕ23α22+α32α22−α32+A2tα22α22−α32+C2,A1=−α22−α32α22ϕ3+β22ϕ4−α2ϕ5α2δ22+α2δ32−2α3δ2δ3,A2=−α3α22ϕ3+β22ϕ4α22−α32−α22ϕ52α2δ2δ3−α3δ22−α3δ32,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(29)Σ2=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosα2x+β2y+δ2z+σ2t+σ5tsinα3x+α3β2yα2+δ3z+σ3t.


## 4.1.3. Set III Solutions

(30)σ5t=±3α34ϕ2+δ22ϕ5ϕ5δ2σ4t3α34ϕ2+δ22ϕ5,σ2t=−2δ3δ2ϕ5tα3+C1,σ3t=α34ϕ2−α32ϕ3−β32ϕ4−δ22ϕ5−δ32ϕ5tα3+C2,α2=β2=0,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(31)Σ3=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosδ2z+σ2t+σ5tsinα3x+β3y+δ3z+σ3t.


## 4.1.4. Set IV Solutions

(32)σ2t=4α24ϕ2−α22ϕ3−β32ϕ4−δ32ϕ5tα2+C1,σ3t=4α24ϕ2−α22ϕ3−β32ϕ4−δ32ϕ5tα2+C2,σ5t=C3σ4t,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(33)Σ4=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosα2x+β3y+δ3z+σ2t+σ5tsinα2x+β3y+δ3z+σ3t.


## 4.1.5. Set V Solutions

(34)σ2t=∫−−σ5t2α34ϕ23σ4t2+σ5t2+α32ϕ3σ4t2σ5t2+σ4t4β22ϕ4+δ22ϕ5α3σ4t3σ5tdt+C1,σ3t=∫−−σ5t2α34ϕ2σ4t2+3σ5t2+α32ϕ3σ4t2σ5t2+σ4t4β22ϕ4+δ22ϕ5σ4t2σ5t2α3dt+C2,σ5t=C3σ4t,α2=C3α3,β3=β2C3,δ3=δ2C3,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(35)Σ5=12ϕ2ϕ1lngx,y,z,txx,gx,y,z,t=eα1x+β1y+δ1z+σ1tσ4tcosC3α3x+β2y+δ2z+σ2t+σ5tsinα3x+β2yC3+δ2zC3+σ3t.Figures4–6, respectively, show the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(36)C1=C2=1,C3=2,α1=β2=δ1=0.2,β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=σ4t=ϕ1=ϕ2=ϕ3=ϕ4=t,ϕ5=2t,z=1,y=2,C1=C2=1,C3=2,α1=β2=δ1=0.2,β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=t,σ4t=tcost,ϕ1=t+1,ϕ2=sint,ϕ3=t+t2,ϕ4=t−t3,ϕ5=2t+1,z=1,y=2,C1=C2=1,C3=2,α1=β2=δ1=0.2,β1=0.1,α3=β3=δ2=0.3,δ3=0.4,σ1t=t,σ4t=sint2,ϕ1=1,ϕ2=2,ϕ3=3,ϕ4=3,ϕ5=2,z=1,y=1,in Equation (35).Figure 4
Plot of periodic wave solution (35) Σ5 (3D plot, density plot, and 2D plot t), respectively.Figure 5
Plot of periodic wave solution (35) Σ5 (3D plot, density plot, and 2D plot t), respectively.Figure 6
Plot of periodic wave solution (35) Σ5 (3D plot, density plot, and 2D plot t), respectively.


## 4.2. Combined Damped Oscillation Wave Solutions

Here, we utilize to formulate the new exact solutions to the3+1-dimensional generalized CV NLW equation. Consider the below function for studying the combined damped oscillation wave solutions as
(37)g=ea1σ5tcosa2+σ6tsina2+ea3σ7tcosa4+σ8tsina5,al=αlx+βly+δlz+σlt,l=1,2,3,4.Afterwards, the valuesαl,βl,δl,σlt,l=1,2,3,4 will be found. By making use of Equation (37) into (21) and taking the coefficients, each power of cosa2 and sina2, cosa4 and sina4 to zero yields a system of equations (algebraic) (these are not collected here for the minimalist) for αl,βl,δl,σlt,l=1,2,3,4. These algebraic equations by using the emblematic computation software like Maple give the solutions in the following with using u=12ϕ2/ϕ1lngxx as follows:


### 4.2.1. Set I Solutions

(38)σ7t=0,σ4t=C3,σ2t=−arctanσ5tσ6t+2δ2ϕ5β1δ2−β2δ1+β2δ3−β3δ2tα1β2−α3β2+C1,σ3t=∫Mtdt+C2,α2=α4=β4=δ4=0,ϕ4=−δ22ϕ5β22,Mt=−σ8tA5+A6−A7+A1+σ8tA2+A3−A4σ8tβ22σ5t2+σ6t2α1−α3,A1=σ8tddtσ1tβ22σ5t2+σ6t2α1−α3+σ8tβ22α1−α3ddtσ6tσ6t+σ5tddtσ5t,A2=σ6t2α14β22ϕ2+α34β22ϕ2+α12β22ϕ3+α32β22ϕ3−β12δ22ϕ5+β22δ12ϕ5+β22δ32ϕ5−β32δ22ϕ5,A3=σ5t2α14α34β24ϕ22+α32β22ϕ3+1,A4=ddtσ8tβ22σ5t2+σ6t2α1−α3,A5=2α1α3β22ϕ22α12−3α1α3+2α32σ5t2+σ6t2,A6=2σ6t2α1α3β22ϕ3−β1β3δ22ϕ5+β22δ1δ3ϕ5,A7=σ5t2α12β22ϕ3−2α1α3β22ϕ3−β12δ22ϕ5+2β1β3δ22ϕ5+β22δ12ϕ5−2β22δ1δ3ϕ5+β22δ32ϕ5−β32δ22ϕ5,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(39)Σ1=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+δ2z+σ2t+σ6tsinβ2y+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set I-I:(40)σ6t=0,σ8t=−tan−σ4t+C3σ7t,σ2t=2δ2ϕ5β1δ2−β2δ1+β2δ3−β3δ2tβ2α1−α3+C1,σ3t=∫Mtdt+C2,α2=α4=β4=δ4=0,ϕ4=−δ22ϕ5β22,Mt=−A1+A2σ7tσ5tβ22α1−α3+C2,A1=−σ5tσ7tβ22ϕ2α1−α34+β22ϕ3α1−α32−ϕ5β1δ2−β2δ1+β2δ3−β3δ2β1δ2+β2δ1−β2δ3−β3δ2,A2=β22α1−α3σ8tddtσ4tσ5t−σ5tσ7tddtσ1t+σ5tddtσ7t−σ7tddtσ5t,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(41)Σ2=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+δ2z+σ2t+σ6tsinβ2y+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set I-II:(42)σ8t=−tan−σ4t+C3σ7t,σ2t=−arctanσ5tσ6t+2δ2ϕ5β1δ2−β2δ1+β2δ3−β3δ2tα1β2−α3β2+C1,σ3t=∫Mtdt+C2,α2=α4=β4=δ4=0,ϕ4=−δ22ϕ5β22,Mt=−A1+A2+A3σ7tβ22σ5t2+σ6t2α1−α3+C2,A1=2σ7tσ5t2+σ6t22α13α3β22ϕ2−3α12α32β22ϕ2+2α1α33β22ϕ2+α1α3β22ϕ3−β1β3δ22ϕ5+β22δ1δ3ϕ5,A2=β22α1−α3σ5t2+σ6t2σ8tddtσ4t−σ7tddtσ1t+σ6t2ddtσ7t−σ6tddtσ6tσ7t+σ5t2ddtσ7t−σ5tσ7tddtσ5t,A3=−σ7tσ5t2+σ6t2α14β22ϕ2+α34β22ϕ2+α12β22ϕ3+α32β22ϕ3−β12δ22ϕ5+β22δ12ϕ5+β22δ32ϕ5−β32δ22ϕ5,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3,4, are free values. It concludes along with the bilinear equation; the exact solution will be as
(43)Σ3=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+δ2z+σ2t+σ6tsinβ2y+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.


### 4.2.2. Set II Solutions

(44)σ4t=C3,σ7t=0,σ2t=−arctanσ5tσ6t+C1,α2=β2=δ2=δ4=0,σ3t=∫A4+A1+A2+A3σ8tσ5t2+σ6t2α1−α3dt+C2,A1=σ8tσ5t2+σ6t2α14ϕ2+α34ϕ2+α12ϕ3+α32ϕ3+β12ϕ4+β32ϕ4+δ12ϕ5+ϕ5δ32,A2=−ddtσ8tσ5t2+σ6t2α1−α3,A3=σ8tα1−α3σ5t2ddtσ1t+ddtσ1tσ6t2+σ5tddtσ5t+σ6tddtσ6t,A4=−2σ8tσ5t2+σ6t22α13α3ϕ2−3α12α32ϕ2+2α1α33ϕ2+α1α3ϕ3+β1β3ϕ4+δ1ϕ5δ3,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3,4, are free values. It concludes along with the bilinear equation; the exact solution will be as
(45)Σ4=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set II-I:(46)σ2t=C1,σ5t=0,σ8t=−tan−σ4t+C3σ7t,α2=β2=δ2=δ4=0,σ3t=∫−A1+A2σ6tσ7tα1−α3dt+C2,A1=−σ6tσ7tϕ2α1−α34+ϕ3α1−α32+ϕ4β1−β32+ϕ5δ1−δ32,A2=−α1−α3ddtσ1tσ6tσ7t−σ8tσ6tddtσ4t−σ6tddtσ7t+ddtσ6tσ7t,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(47)Σ5=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set II-II:(48)σ2t=−arctanσ5tσ6t+C1,σ8t=−tan−σ4t+C3σ7t,α2=β2=δ2=δ4=0,σ3t=∫−A3+A1+A2σ7tσ5t2+σ6t2α1−α3dt+C2,A1=ddtσ7tσ5t2+σ6t2α1−α3,A2=−α1−α3σ5t2+σ6t2ddtσ1tσ7t−σ8tddtσ4t+σ5tddtσ5tσ7t+σ6tddtσ6tσ7t,A3=−σ7tσ5t2+σ6t2ϕ2α1−α34+ϕ3α1−α32+ϕ4β1−β32+ϕ5δ1−δ32,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3,4, are free values. It concludes along with the bilinear equation; the exact solution will be as
(49)Σ6=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Figures7 and 8, respectively, show the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(50)C1=C2=1,C3=0.2,α1=δ1=0.2,β1=α3=0.1,β3=0.3,δ3=0.4,σ1t=σ5t=σ6t=σ7t=ϕ1=ϕ2=ϕ3=t,ϕ4=t+1,ϕ5=2t,z=1,y=1,C1=C2=1,C3=0.2,α1=δ1=0.2,β1=α3=0.1,β3=0.3,δ3=0.4,σ1t=sint,σ4t=sin2t,σ5t=sin3t,σ6t=sin4t,σ7t=sint,ϕ1=3,ϕ2=2,ϕ3=2,ϕ4=1,ϕ5=2,z=1,y=1,in equation (49).Figure 7
Plot of periodic wave solution (49) Σ6 (3D plot, density plot, and 2D plot t), respectively.Figure 8
Plot of periodic wave solution (49) Σ6 (3D plot, density plot, and 2D plot t), respectively.


### 4.2.3. Set III Solutions

III-I:(51)Σ7=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ6tsinβ2y+β2δ4zβ4+2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ42α1−α3+C1+eα3x+β3y+δ3z−∫A1+A2/σ8tσ6tβ42α1−α3dtσ8tsinβ4y+δ4z+2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ4α1−α3+C3,A1=−σ6tσ8tβ42ϕ2α1−α34+β42ϕ3α1−α32−ϕ5β1δ4−β3δ42−β4δ1−β4δ32,A2=−β42α1−α3σ6tσ8tddtσ1t−σ6tddtσ8t+σ8tddtσ6t.III-II:(52)Σ8=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+β2δ4zβ4−arctanσ5tσ6t+A1+σ6tsinβ2y+β2δ4zβ4−arctanσ5tσ6t+A1+eα3x+β3y+δ3z−∫A2+A3/σ8tβ42σ5t2+σ6t2α1−α3dtσ8tsinβ4y+δ4z+2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ4α1−α3+C3,A1=2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ42α1−α3,A2=−β42α1−α3σ5t2+σ6t2σ8tddtσ1t−ddtσ8t+σ6tσ8tddtσ6t+σ8tσ5tddtσ5t,A3=−σ8tσ5t2+σ6t2β42ϕ2α1−α34+β42ϕ3α1−α32−ϕ5β1δ4−β3δ42−β4δ1−β4δ32.III-III:(53)Σ9=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ6tsinβ2y+β2δ4zβ4+2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ4A0+C1+eα3x+β3y+δ3z−∫A2+A3/σ6tσ7tβ4A0dtσ7tcosβ4y+δ4z+σ4t−tanC3−∫A1A0dtσ7tsinβ4y+δ4z+σ4t,A0=β4α1−α3,A1=−2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3+A0ddtσ4t,A2=−β42α1−α3σ7tσ6tddtσ1t−ddtσ4tσ6tσ8t+σ7tddtσ6t−σ6tddtσ7t,A3=−σ6tσ7tβ42ϕ2α1−α34+σ7tβ42ϕ3α1−α32−ϕ5β1δ4−β3δ4−β4δ1+β4δ3σ7tβ1δ4−σ7tβ3δ4+σ7tβ4δ1−σ7tβ4δ3−2σ8tβ4δ4.III-IV:(54)Σ10=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosX1+σ6tsinX1+eα3x+β3y+δ3z−∫A3+A4/σ7tβ42σ5t2+σ6t2α1−α3dtσ7tcosX2−tanC3−∫A1β4α1−α3dtσ7tsinX2,X1=β2y+β2δ4zβ4−arctanσ5tσ6t+A2t,X2=β4y+δ4z+σ4t,A1=−2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3+β4α1−α3ddtσ4t,A2=2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3β42α1−α3,A3=−σ5t2+σ6t2σ7tβ42ϕ2α1−α34+σ7tβ42ϕ3α1−α32−ϕ5β1δ4−β3δ4−β4δ1+β4δ3σ7tβ1δ4−σ7tβ3δ4+σ7tβ4δ1−σ7tβ4δ3−2σ8tβ4δ4,A4=−β42α1−α3σ5t2+σ6t2σ7tddtσ1t−ddtσ4tσ8t−ddtσ7t+ddtσ5tσ7tσ5t+σ7tσ6tddtσ6t.Figure9 shows the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(55)C1=C2=1,C3=0.2,α1=δ1=0.2,β1=α3=0.1,β3=0.3,δ3=0.4,in Equation (54).Figure 9
Plot of periodic wave solution (54) Σ10 (3D plot, density plot, and 2D plot t), respectively.


### 4.2.4. Set IV Solutions

IV-I:(56)Σ11=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ6tsinC1+eα3x+β3y+δ3z+2tβ1ϕ4β4−β3ϕ4β4+δ1δ4ϕ5−δ3δ4ϕ5/α4+σ1t+lnσ6t/σ8tσ8t×sinα4x+β4y+δ4z−3ϕ3α42−ϕ44β12−8β1β3+4β32−3β42−ϕ54δ12−8δ1δ3+4δ32−3δ42t3α4,ϕ2=13ϕ4β1−β32+ϕ5δ1−δ32α44.IV-II:(57)Σ12=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ5tcosC1−arctanσ5tσ6t+σ6tsinC1−arctanσ5tσ6t+eα3x+β3y+δ3z+∫A2+A1/σ8tα4σ5t2+σ6t2dtσ8t×sinα4x+β4y+δ4z−3ϕ3α42−ϕ44β12−8β1β3+4β32−3β42−ϕ54δ12−8δ1δ3+4δ32−3δ42t3α4,A1=α4σ5t2+σ6t2σ8tddtσ1t−ddtσ8t+σ8tσ5tddtσ5t+σ6tddtσ6t,A2=2σ8tσ5t2+σ6t2β1ϕ4β4−β3ϕ4β4+δ1δ4ϕ5−δ3δ4ϕ5,ϕ2=ϕ4β1−β32+ϕ5δ1−δ323α44.IV-III:(58)Σ13=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosα4x+β4y+δ4z+σ4t+σ8tsinα4x+β4y+δ4z+σ4t,σ2t=C1,σ3t=∫−13A2+A3σ7tσ8tσ6tα4σ7t2+σ8t2dt+C2,σ8t=∫A1−3α4σ7t2+σ8t2σ8t2d/dtσ4t+σ7t2d/dtσ4t+σ8td/dtσ7t−3σ7t3α4−3σ7tσ8t2α4dt,A1=−σ8t43ϕ3α42−4β12ϕ4+8β1ϕ4β3−4ϕ4β32+3β42ϕ4−4δ12ϕ5+8δ1ϕ5δ3−4ϕ5δ32+3δ42ϕ5+2σ8t2σ7t4σ8tβ1β4ϕ4−4σ8tβ3β4ϕ4+4σ8tδ1δ4ϕ5−4σ8tδ3δ4ϕ5−3σ7tα42ϕ3−σ7tβ42ϕ4−σ7tδ42ϕ5−3σ7t4ϕ3α42+β42ϕ4+δ42ϕ5,A2=3σ8tα4σ7t2+σ8t2σ8tσ6tddtσ4t−σ7tddtσ1tσ6t−σ7tddtσ6t+σ6tddtσ7t,A3=σ6t3α42ϕ3σ7t2+σ8t22−ϕ4σ8t44β12−8β1β3+4β32−3β42+14σ8t3σ7tβ4β1−β3−4σ8t2σ7t2−β3+β1−β4−β3+β1+β4−2σ8tσ7t3β4β1−β3−β42σ7t4−ϕ5σ8t44δ12−8δ1δ3+4δ32−3δ42+14σ8t3σ7tδ4δ1−δ3−4σ8t2σ7t2−δ3+δ1−δ4−δ3+δ1+δ4−2σ8tσ7t3δ4δ1−δ3−δ42σ7t4,ϕ2=13σ8t2ϕ4β1σ8t−β3σ8t+β4σ7t2+ϕ5δ1σ8t−δ3σ8t+δ4σ7t2α44σ7t4+2σ7t2σ8t2+σ8t4.IV-IV:(59)Σ14=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosα4x+β4y+δ4z+σ4t+σ8tsinα4x+β4y+δ4z+σ4t,σ2t=−arctanσ5tσ6t+C1,σ8t=∫A2+A1−3σ7t3α4−3σ7tσ8t2α4dt,σ3t=∫−13A3+A4σ7tσ8tα4σ5t2+σ6t2σ7t2+σ8t2dt+C2,A1=−3α4σ7t2+σ8t2σ8t2ddtσ4t+σ7t2ddtσ4t+σ8tddtσ7t,A2=−3α42ϕ3σ7t2+σ8t22+ϕ4σ8t44β12−8β1β3+4β32−3β42+σ7tβ48σ8t3β1−8σ8t3β3−2σ8t2σ7tβ4−3σ7t3β4+ϕ5σ8t44δ12−8δ1δ3+4δ32−3δ42+σ7tδ48σ8t3δ1−8σ8t3δ3−2σ8t2σ7tδ4−3σ7t3δ4,A3=3σ8tα4σ7t2+σ8t2σ5t2+σ6t2σ8tddtσ4t−σ7tddtσ1t+ddtσ7t−σ7tσ5tddtσ5t+σ6tddtσ6t,A4=σ5t2+σ6t2σ8t43ϕ3α42−4β12ϕ4+8β1ϕ4β3−4ϕ4β32+3β42ϕ4−4δ12ϕ5+8δ1ϕ5δ3−4ϕ5δ32+3δ42ϕ5+σ7t43ϕ3α42+β42ϕ4+δ42ϕ5+2σ8t2σ7t23ϕ3α42+2β12ϕ4+2ϕ4β32−2β42ϕ4+2δ12ϕ5−2δ42ϕ5+2σ8t2σ7t23ϕ3α42+2β12ϕ4+2ϕ4β32−2β42ϕ4+2δ12ϕ5−4δ1ϕ5δ3+2ϕ5δ32−2δ42ϕ5−14σ8t3σ7tβ1ϕ4β4−β3ϕ4β4+δ1δ4ϕ5−δ3δ4ϕ5−2σ8tσ7t24σ8tβ1β3ϕ4−σ7tβ1β4ϕ4+σ7tβ3β4ϕ4−σ7tδ1δ4ϕ5,ϕ2=13σ8t2ϕ4β1σ8t−β3σ8t+β4σ7t2+ϕ5δ1σ8t−δ3σ8t+δ4σ7t2α44σ7t4+2σ7t2σ8t2+σ8t4.


### 4.2.5. Set V Solutions

(60)Σ15=12ϕ2ϕ1lngxx,g=eX1+σ1tσ5tcosα2x+α2β4yα4+δ2z+σ2t+σ6tsinα2x+α2β4yα4+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosα4x+β4y+δ4z+σ4t+σ8tsinα4x+β4y+δ4z+σ4t,X1=σ8tα2σ5t+α3σ6t−α4σ6tσ7txσ6tσ8t+σ8tα2β4σ5t+α4β3σ6t−α4β4σ6tσ7tyα4σ6tσ8t+12B1+B2+B3zα2α4σ6tσ8tα2σ5tσ8t−α4σ6tσ7t,B1=α2δ4+α4δ2σ5t2+2σ5tσ6tα4δ3+σ6t2α2δ4−α4δ2σ8t2α22,B2=−2σ7tσ5tσ8tσ6tα2α4α2δ4+α4δ2,B3=σ6t2α42σ7t2α2δ4+α4δ2−2σ7tσ8tα2δ3−σ8t2α2δ4−α4δ2,σ2t=∫−A1+A2σ6t2σ8t2α42α2δ4−α4δ22σ6t2+σ5t2dt+C1,A1=σ8t2σ6t2α42α2δ4−α4δ22ddtσ5tσ6t−ddtσ6tσ5t,A2=α2σ6t2+σ5t212σ7tσ6tα45δ22ϕ2σ7tσ6tα4−2σ5tσ8tα2+σ8t24α22α42ϕ23σ5t2α42δ22−σ6t2α22δ42+2σ6t2α2α4δ2δ4−σ6t2α42δ22+σ6t2α2δ4−α4δ22α42ϕ3+β42ϕ4,A3=−σ8t2σ6t2α4α2δ4−α4δ22σ7tddtσ8t−σ8tddtσ7t,A4=σ8t2+σ7t212α22α42δ42ϕ2σ7tσ6tα4−σ5tσ8tα22−σ8t2σ6t2α2δ4−α4δ224α44ϕ2−α42ϕ3−β42ϕ4,σ4t=∫−A3+A4σ6t2σ8t2α4α2δ4−α4δ22σ8t2+σ7t2dt,σ3t=∫−A5+A6σ6t3σ8t3α42α2δ4−α4δ22σ8t2+σ7t2σ6t2+σ5t2dt,A5=S1S2σ7tσ6tα4−σ5tσ8tα24σ7tσ6tα43ϕ2S3S4+4σ5t2σ8t2α22α42ϕ2S3+σ8t2σ6t2α2δ4−α4δ212α2α43ϕ2α2δ2−α4δ4+α42ϕ3+β42ϕ4α2δ4−α4δ2,A6=−σ8t3σ6t3α42α2δ4−α4δ22ddtσ1tS1S2+S2ddtσ5tσ5t+ddtσ6tσ6t−S1σ7tddtσ7t+σ8tddtσ8t,S1=σ6t2+σ5t2,S2=σ8t2+σ7t2,S3=α22δ42+α2α4δ2δ4+α42δ22,S4=σ7tσ6tα4−2σ5tσ8tα2,α1=σ5tσ8tα2+α3σ6tσ8t−σ7tσ6tα4σ6tσ8t,ϕ5=12σ5tσ8tα2−σ7tσ6tα42α42α22ϕ2α2δ4−α4δ22σ6t2σ8t2,δ1=1/2σ8t2α22+σ6t2α42σ7t2α2δ4+σ7t2α4δ2−σ8t2α2δ4+σ8t2α4δ2α2α4σ6tσ8tσ5tσ8tα2−σ7tσ6tα4,H1=σ5t2α2δ4+σ5t2α4δ2+σ6t2α2δ4−σ6t2α4δ2.


### 4.2.6. Set VI Solutions

(61)Σ16=12ϕ2ϕ1lngxx,g=eα3x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.


## 4.2.1. Set I Solutions

(38)σ7t=0,σ4t=C3,σ2t=−arctanσ5tσ6t+2δ2ϕ5β1δ2−β2δ1+β2δ3−β3δ2tα1β2−α3β2+C1,σ3t=∫Mtdt+C2,α2=α4=β4=δ4=0,ϕ4=−δ22ϕ5β22,Mt=−σ8tA5+A6−A7+A1+σ8tA2+A3−A4σ8tβ22σ5t2+σ6t2α1−α3,A1=σ8tddtσ1tβ22σ5t2+σ6t2α1−α3+σ8tβ22α1−α3ddtσ6tσ6t+σ5tddtσ5t,A2=σ6t2α14β22ϕ2+α34β22ϕ2+α12β22ϕ3+α32β22ϕ3−β12δ22ϕ5+β22δ12ϕ5+β22δ32ϕ5−β32δ22ϕ5,A3=σ5t2α14α34β24ϕ22+α32β22ϕ3+1,A4=ddtσ8tβ22σ5t2+σ6t2α1−α3,A5=2α1α3β22ϕ22α12−3α1α3+2α32σ5t2+σ6t2,A6=2σ6t2α1α3β22ϕ3−β1β3δ22ϕ5+β22δ1δ3ϕ5,A7=σ5t2α12β22ϕ3−2α1α3β22ϕ3−β12δ22ϕ5+2β1β3δ22ϕ5+β22δ12ϕ5−2β22δ1δ3ϕ5+β22δ32ϕ5−β32δ22ϕ5,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(39)Σ1=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+δ2z+σ2t+σ6tsinβ2y+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set I-I:(40)σ6t=0,σ8t=−tan−σ4t+C3σ7t,σ2t=2δ2ϕ5β1δ2−β2δ1+β2δ3−β3δ2tβ2α1−α3+C1,σ3t=∫Mtdt+C2,α2=α4=β4=δ4=0,ϕ4=−δ22ϕ5β22,Mt=−A1+A2σ7tσ5tβ22α1−α3+C2,A1=−σ5tσ7tβ22ϕ2α1−α34+β22ϕ3α1−α32−ϕ5β1δ2−β2δ1+β2δ3−β3δ2β1δ2+β2δ1−β2δ3−β3δ2,A2=β22α1−α3σ8tddtσ4tσ5t−σ5tσ7tddtσ1t+σ5tddtσ7t−σ7tddtσ5t,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(41)Σ2=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+δ2z+σ2t+σ6tsinβ2y+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set I-II:(42)σ8t=−tan−σ4t+C3σ7t,σ2t=−arctanσ5tσ6t+2δ2ϕ5β1δ2−β2δ1+β2δ3−β3δ2tα1β2−α3β2+C1,σ3t=∫Mtdt+C2,α2=α4=β4=δ4=0,ϕ4=−δ22ϕ5β22,Mt=−A1+A2+A3σ7tβ22σ5t2+σ6t2α1−α3+C2,A1=2σ7tσ5t2+σ6t22α13α3β22ϕ2−3α12α32β22ϕ2+2α1α33β22ϕ2+α1α3β22ϕ3−β1β3δ22ϕ5+β22δ1δ3ϕ5,A2=β22α1−α3σ5t2+σ6t2σ8tddtσ4t−σ7tddtσ1t+σ6t2ddtσ7t−σ6tddtσ6tσ7t+σ5t2ddtσ7t−σ5tσ7tddtσ5t,A3=−σ7tσ5t2+σ6t2α14β22ϕ2+α34β22ϕ2+α12β22ϕ3+α32β22ϕ3−β12δ22ϕ5+β22δ12ϕ5+β22δ32ϕ5−β32δ22ϕ5,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3,4, are free values. It concludes along with the bilinear equation; the exact solution will be as
(43)Σ3=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+δ2z+σ2t+σ6tsinβ2y+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.


## 4.2.2. Set II Solutions

(44)σ4t=C3,σ7t=0,σ2t=−arctanσ5tσ6t+C1,α2=β2=δ2=δ4=0,σ3t=∫A4+A1+A2+A3σ8tσ5t2+σ6t2α1−α3dt+C2,A1=σ8tσ5t2+σ6t2α14ϕ2+α34ϕ2+α12ϕ3+α32ϕ3+β12ϕ4+β32ϕ4+δ12ϕ5+ϕ5δ32,A2=−ddtσ8tσ5t2+σ6t2α1−α3,A3=σ8tα1−α3σ5t2ddtσ1t+ddtσ1tσ6t2+σ5tddtσ5t+σ6tddtσ6t,A4=−2σ8tσ5t2+σ6t22α13α3ϕ2−3α12α32ϕ2+2α1α33ϕ2+α1α3ϕ3+β1β3ϕ4+δ1ϕ5δ3,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3,4, are free values. It concludes along with the bilinear equation; the exact solution will be as
(45)Σ4=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set II-I:(46)σ2t=C1,σ5t=0,σ8t=−tan−σ4t+C3σ7t,α2=β2=δ2=δ4=0,σ3t=∫−A1+A2σ6tσ7tα1−α3dt+C2,A1=−σ6tσ7tϕ2α1−α34+ϕ3α1−α32+ϕ4β1−β32+ϕ5δ1−δ32,A2=−α1−α3ddtσ1tσ6tσ7t−σ8tσ6tddtσ4t−σ6tddtσ7t+ddtσ6tσ7t,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3, are free values. It concludes along with the bilinear equation; the exact solution will be as
(47)Σ5=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Set II-II:(48)σ2t=−arctanσ5tσ6t+C1,σ8t=−tan−σ4t+C3σ7t,α2=β2=δ2=δ4=0,σ3t=∫−A3+A1+A2σ7tσ5t2+σ6t2α1−α3dt+C2,A1=ddtσ7tσ5t2+σ6t2α1−α3,A2=−α1−α3σ5t2+σ6t2ddtσ1tσ7t−σ8tddtσ4t+σ5tddtσ5tσ7t+σ6tddtσ6tσ7t,A3=−σ7tσ5t2+σ6t2ϕ2α1−α34+ϕ3α1−α32+ϕ4β1−β32+ϕ5δ1−δ32,where σ1t,σ4t free functions of t and αl,βl,δl,l=1,2,3,4, are free values. It concludes along with the bilinear equation; the exact solution will be as
(49)Σ6=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.Figures7 and 8, respectively, show the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(50)C1=C2=1,C3=0.2,α1=δ1=0.2,β1=α3=0.1,β3=0.3,δ3=0.4,σ1t=σ5t=σ6t=σ7t=ϕ1=ϕ2=ϕ3=t,ϕ4=t+1,ϕ5=2t,z=1,y=1,C1=C2=1,C3=0.2,α1=δ1=0.2,β1=α3=0.1,β3=0.3,δ3=0.4,σ1t=sint,σ4t=sin2t,σ5t=sin3t,σ6t=sin4t,σ7t=sint,ϕ1=3,ϕ2=2,ϕ3=2,ϕ4=1,ϕ5=2,z=1,y=1,in equation (49).Figure 7
Plot of periodic wave solution (49) Σ6 (3D plot, density plot, and 2D plot t), respectively.Figure 8
Plot of periodic wave solution (49) Σ6 (3D plot, density plot, and 2D plot t), respectively.


## 4.2.3. Set III Solutions

III-I:(51)Σ7=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ6tsinβ2y+β2δ4zβ4+2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ42α1−α3+C1+eα3x+β3y+δ3z−∫A1+A2/σ8tσ6tβ42α1−α3dtσ8tsinβ4y+δ4z+2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ4α1−α3+C3,A1=−σ6tσ8tβ42ϕ2α1−α34+β42ϕ3α1−α32−ϕ5β1δ4−β3δ42−β4δ1−β4δ32,A2=−β42α1−α3σ6tσ8tddtσ1t−σ6tddtσ8t+σ8tddtσ6t.III-II:(52)Σ8=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosβ2y+β2δ4zβ4−arctanσ5tσ6t+A1+σ6tsinβ2y+β2δ4zβ4−arctanσ5tσ6t+A1+eα3x+β3y+δ3z−∫A2+A3/σ8tβ42σ5t2+σ6t2α1−α3dtσ8tsinβ4y+δ4z+2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ4α1−α3+C3,A1=2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ42α1−α3,A2=−β42α1−α3σ5t2+σ6t2σ8tddtσ1t−ddtσ8t+σ6tσ8tddtσ6t+σ8tσ5tddtσ5t,A3=−σ8tσ5t2+σ6t2β42ϕ2α1−α34+β42ϕ3α1−α32−ϕ5β1δ4−β3δ42−β4δ1−β4δ32.III-III:(53)Σ9=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ6tsinβ2y+β2δ4zβ4+2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3tβ4A0+C1+eα3x+β3y+δ3z−∫A2+A3/σ6tσ7tβ4A0dtσ7tcosβ4y+δ4z+σ4t−tanC3−∫A1A0dtσ7tsinβ4y+δ4z+σ4t,A0=β4α1−α3,A1=−2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3+A0ddtσ4t,A2=−β42α1−α3σ7tσ6tddtσ1t−ddtσ4tσ6tσ8t+σ7tddtσ6t−σ6tddtσ7t,A3=−σ6tσ7tβ42ϕ2α1−α34+σ7tβ42ϕ3α1−α32−ϕ5β1δ4−β3δ4−β4δ1+β4δ3σ7tβ1δ4−σ7tβ3δ4+σ7tβ4δ1−σ7tβ4δ3−2σ8tβ4δ4.III-IV:(54)Σ10=12ϕ2ϕ1lngxx,g=eα1x+β1y+δ1z+σ1tσ5tcosX1+σ6tsinX1+eα3x+β3y+δ3z−∫A3+A4/σ7tβ42σ5t2+σ6t2α1−α3dtσ7tcosX2−tanC3−∫A1β4α1−α3dtσ7tsinX2,X1=β2y+β2δ4zβ4−arctanσ5tσ6t+A2t,X2=β4y+δ4z+σ4t,A1=−2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3+β4α1−α3ddtσ4t,A2=2β2δ4ϕ5β1δ4−β3δ4−β4δ1+β4δ3β42α1−α3,A3=−σ5t2+σ6t2σ7tβ42ϕ2α1−α34+σ7tβ42ϕ3α1−α32−ϕ5β1δ4−β3δ4−β4δ1+β4δ3σ7tβ1δ4−σ7tβ3δ4+σ7tβ4δ1−σ7tβ4δ3−2σ8tβ4δ4,A4=−β42α1−α3σ5t2+σ6t2σ7tddtσ1t−ddtσ4tσ8t−ddtσ7t+ddtσ5tσ7tσ5t+σ7tσ6tddtσ6t.Figure9 shows the analysis of treatment of periodic wave solution with graphs of Σ with the below-selected parameters:
(55)C1=C2=1,C3=0.2,α1=δ1=0.2,β1=α3=0.1,β3=0.3,δ3=0.4,in Equation (54).Figure 9
Plot of periodic wave solution (54) Σ10 (3D plot, density plot, and 2D plot t), respectively.


## 4.2.4. Set IV Solutions

IV-I:(56)Σ11=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ6tsinC1+eα3x+β3y+δ3z+2tβ1ϕ4β4−β3ϕ4β4+δ1δ4ϕ5−δ3δ4ϕ5/α4+σ1t+lnσ6t/σ8tσ8t×sinα4x+β4y+δ4z−3ϕ3α42−ϕ44β12−8β1β3+4β32−3β42−ϕ54δ12−8δ1δ3+4δ32−3δ42t3α4,ϕ2=13ϕ4β1−β32+ϕ5δ1−δ32α44.IV-II:(57)Σ12=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ5tcosC1−arctanσ5tσ6t+σ6tsinC1−arctanσ5tσ6t+eα3x+β3y+δ3z+∫A2+A1/σ8tα4σ5t2+σ6t2dtσ8t×sinα4x+β4y+δ4z−3ϕ3α42−ϕ44β12−8β1β3+4β32−3β42−ϕ54δ12−8δ1δ3+4δ32−3δ42t3α4,A1=α4σ5t2+σ6t2σ8tddtσ1t−ddtσ8t+σ8tσ5tddtσ5t+σ6tddtσ6t,A2=2σ8tσ5t2+σ6t2β1ϕ4β4−β3ϕ4β4+δ1δ4ϕ5−δ3δ4ϕ5,ϕ2=ϕ4β1−β32+ϕ5δ1−δ323α44.IV-III:(58)Σ13=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosα4x+β4y+δ4z+σ4t+σ8tsinα4x+β4y+δ4z+σ4t,σ2t=C1,σ3t=∫−13A2+A3σ7tσ8tσ6tα4σ7t2+σ8t2dt+C2,σ8t=∫A1−3α4σ7t2+σ8t2σ8t2d/dtσ4t+σ7t2d/dtσ4t+σ8td/dtσ7t−3σ7t3α4−3σ7tσ8t2α4dt,A1=−σ8t43ϕ3α42−4β12ϕ4+8β1ϕ4β3−4ϕ4β32+3β42ϕ4−4δ12ϕ5+8δ1ϕ5δ3−4ϕ5δ32+3δ42ϕ5+2σ8t2σ7t4σ8tβ1β4ϕ4−4σ8tβ3β4ϕ4+4σ8tδ1δ4ϕ5−4σ8tδ3δ4ϕ5−3σ7tα42ϕ3−σ7tβ42ϕ4−σ7tδ42ϕ5−3σ7t4ϕ3α42+β42ϕ4+δ42ϕ5,A2=3σ8tα4σ7t2+σ8t2σ8tσ6tddtσ4t−σ7tddtσ1tσ6t−σ7tddtσ6t+σ6tddtσ7t,A3=σ6t3α42ϕ3σ7t2+σ8t22−ϕ4σ8t44β12−8β1β3+4β32−3β42+14σ8t3σ7tβ4β1−β3−4σ8t2σ7t2−β3+β1−β4−β3+β1+β4−2σ8tσ7t3β4β1−β3−β42σ7t4−ϕ5σ8t44δ12−8δ1δ3+4δ32−3δ42+14σ8t3σ7tδ4δ1−δ3−4σ8t2σ7t2−δ3+δ1−δ4−δ3+δ1+δ4−2σ8tσ7t3δ4δ1−δ3−δ42σ7t4,ϕ2=13σ8t2ϕ4β1σ8t−β3σ8t+β4σ7t2+ϕ5δ1σ8t−δ3σ8t+δ4σ7t2α44σ7t4+2σ7t2σ8t2+σ8t4.IV-IV:(59)Σ14=12ϕ2ϕ1lngxx,g=eσ8tα3−α4σ7tx/σ8t+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosα4x+β4y+δ4z+σ4t+σ8tsinα4x+β4y+δ4z+σ4t,σ2t=−arctanσ5tσ6t+C1,σ8t=∫A2+A1−3σ7t3α4−3σ7tσ8t2α4dt,σ3t=∫−13A3+A4σ7tσ8tα4σ5t2+σ6t2σ7t2+σ8t2dt+C2,A1=−3α4σ7t2+σ8t2σ8t2ddtσ4t+σ7t2ddtσ4t+σ8tddtσ7t,A2=−3α42ϕ3σ7t2+σ8t22+ϕ4σ8t44β12−8β1β3+4β32−3β42+σ7tβ48σ8t3β1−8σ8t3β3−2σ8t2σ7tβ4−3σ7t3β4+ϕ5σ8t44δ12−8δ1δ3+4δ32−3δ42+σ7tδ48σ8t3δ1−8σ8t3δ3−2σ8t2σ7tδ4−3σ7t3δ4,A3=3σ8tα4σ7t2+σ8t2σ5t2+σ6t2σ8tddtσ4t−σ7tddtσ1t+ddtσ7t−σ7tσ5tddtσ5t+σ6tddtσ6t,A4=σ5t2+σ6t2σ8t43ϕ3α42−4β12ϕ4+8β1ϕ4β3−4ϕ4β32+3β42ϕ4−4δ12ϕ5+8δ1ϕ5δ3−4ϕ5δ32+3δ42ϕ5+σ7t43ϕ3α42+β42ϕ4+δ42ϕ5+2σ8t2σ7t23ϕ3α42+2β12ϕ4+2ϕ4β32−2β42ϕ4+2δ12ϕ5−2δ42ϕ5+2σ8t2σ7t23ϕ3α42+2β12ϕ4+2ϕ4β32−2β42ϕ4+2δ12ϕ5−4δ1ϕ5δ3+2ϕ5δ32−2δ42ϕ5−14σ8t3σ7tβ1ϕ4β4−β3ϕ4β4+δ1δ4ϕ5−δ3δ4ϕ5−2σ8tσ7t24σ8tβ1β3ϕ4−σ7tβ1β4ϕ4+σ7tβ3β4ϕ4−σ7tδ1δ4ϕ5,ϕ2=13σ8t2ϕ4β1σ8t−β3σ8t+β4σ7t2+ϕ5δ1σ8t−δ3σ8t+δ4σ7t2α44σ7t4+2σ7t2σ8t2+σ8t4.


## 4.2.5. Set V Solutions

(60)Σ15=12ϕ2ϕ1lngxx,g=eX1+σ1tσ5tcosα2x+α2β4yα4+δ2z+σ2t+σ6tsinα2x+α2β4yα4+δ2z+σ2t+eα3x+β3y+δ3z+σ3tσ7tcosα4x+β4y+δ4z+σ4t+σ8tsinα4x+β4y+δ4z+σ4t,X1=σ8tα2σ5t+α3σ6t−α4σ6tσ7txσ6tσ8t+σ8tα2β4σ5t+α4β3σ6t−α4β4σ6tσ7tyα4σ6tσ8t+12B1+B2+B3zα2α4σ6tσ8tα2σ5tσ8t−α4σ6tσ7t,B1=α2δ4+α4δ2σ5t2+2σ5tσ6tα4δ3+σ6t2α2δ4−α4δ2σ8t2α22,B2=−2σ7tσ5tσ8tσ6tα2α4α2δ4+α4δ2,B3=σ6t2α42σ7t2α2δ4+α4δ2−2σ7tσ8tα2δ3−σ8t2α2δ4−α4δ2,σ2t=∫−A1+A2σ6t2σ8t2α42α2δ4−α4δ22σ6t2+σ5t2dt+C1,A1=σ8t2σ6t2α42α2δ4−α4δ22ddtσ5tσ6t−ddtσ6tσ5t,A2=α2σ6t2+σ5t212σ7tσ6tα45δ22ϕ2σ7tσ6tα4−2σ5tσ8tα2+σ8t24α22α42ϕ23σ5t2α42δ22−σ6t2α22δ42+2σ6t2α2α4δ2δ4−σ6t2α42δ22+σ6t2α2δ4−α4δ22α42ϕ3+β42ϕ4,A3=−σ8t2σ6t2α4α2δ4−α4δ22σ7tddtσ8t−σ8tddtσ7t,A4=σ8t2+σ7t212α22α42δ42ϕ2σ7tσ6tα4−σ5tσ8tα22−σ8t2σ6t2α2δ4−α4δ224α44ϕ2−α42ϕ3−β42ϕ4,σ4t=∫−A3+A4σ6t2σ8t2α4α2δ4−α4δ22σ8t2+σ7t2dt,σ3t=∫−A5+A6σ6t3σ8t3α42α2δ4−α4δ22σ8t2+σ7t2σ6t2+σ5t2dt,A5=S1S2σ7tσ6tα4−σ5tσ8tα24σ7tσ6tα43ϕ2S3S4+4σ5t2σ8t2α22α42ϕ2S3+σ8t2σ6t2α2δ4−α4δ212α2α43ϕ2α2δ2−α4δ4+α42ϕ3+β42ϕ4α2δ4−α4δ2,A6=−σ8t3σ6t3α42α2δ4−α4δ22ddtσ1tS1S2+S2ddtσ5tσ5t+ddtσ6tσ6t−S1σ7tddtσ7t+σ8tddtσ8t,S1=σ6t2+σ5t2,S2=σ8t2+σ7t2,S3=α22δ42+α2α4δ2δ4+α42δ22,S4=σ7tσ6tα4−2σ5tσ8tα2,α1=σ5tσ8tα2+α3σ6tσ8t−σ7tσ6tα4σ6tσ8t,ϕ5=12σ5tσ8tα2−σ7tσ6tα42α42α22ϕ2α2δ4−α4δ22σ6t2σ8t2,δ1=1/2σ8t2α22+σ6t2α42σ7t2α2δ4+σ7t2α4δ2−σ8t2α2δ4+σ8t2α4δ2α2α4σ6tσ8tσ5tσ8tα2−σ7tσ6tα4,H1=σ5t2α2δ4+σ5t2α4δ2+σ6t2α2δ4−σ6t2α4δ2.


## 4.2.6. Set VI Solutions

(61)Σ16=12ϕ2ϕ1lngxx,g=eα3x+β1y+δ1z+σ1tσ5tcosσ2t+σ6tsinσ2t+eα3x+β3y+δ3z+σ3tσ7tcosσ4t+σ8tsinσ4t.


## 5. Conclusion

This article investigated the combined damped sinusoidal oscillation solutions to the3+1-D variable-coefficient generalized nonlinear wave equation. The bilinear form of the equation has been described by means of Hirota derivatives. The governing equation is translated to nonlinear ODE using the Cole-Hopf algorithm. Two types of rational periodic solutions have been constructed, which are bright, singular, and periodic, damped sinusoidal oscillation solitons in terms of trigonometric and rational functions. The dynamic features of different types of traveling waves are analyzed in detail through numerical simulation. Meanwhile, the profiles of the surface for the deduced solutions have been depicted in 2D, density, and 3D for free parameters. From the acquired results, it can be concluded that the procedures followed in this analysis can be implemented in a simple and straightforward manner to create new exact solutions of many other nonlinear partial differential equations in terms of the Hirota operator. It is also a very easy-to-use mathematical tool to solve real-life problems in different areas of engineering and sciences.



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