# Lump and Interaction Solutions to the (3+1)-Dimensional Variable-Coefficient Nonlinear Wave Equation with Multidimensional Binary Bell Polynomials

Authors: Xuejun Zhou; Onur Alp Ilhan; Fangyuan Zhou; Sutarto Sutarto; Jalil Manafian; Mostafa Abotaleb
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we study the (3+1)-dimensional variable-coefficient nonlinear wave equation which is taken in soliton theory and generated by utilizing the Hirota bilinear technique. We obtain some new exact analytical solutions, containing interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles by the Maple symbolic package. Making use of Hirota’s bilinear scheme, we obtain its general soliton solutions in terms of bilinear form equation to the considered model which can be obtained by multidimensional binary Bell polynomials. Furthermore, we analyze typical dynamics of the high-order soliton solutions to show the regularity of solutions and also illustrate their behavior graphically.


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## Body

## 1. Introduction

It is known that there are a variety of useful and powerful tools to deal with the nonlocal equations, namely, the improvedtanϕ/2-expansion method [1], the homotopy perturbation method [2], Lie symmetry analysis [3], the Bäcklund transformation method [4], the sine-Gordon expansion approach [5], the (G′/G, 1/G), modified (G′/G2), and (1/G′)-expansion methods [6], the multiple Exp-function method [7–10], Hirota’s bilinear method including the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation [11], the generalized unstable space time fractional nonlinear Schrödinger equation [12], the inverse Cauchy problems [13], a generalized hyperelastic rod equation [14], the Kadomtsev-Petviashvili equation [15], the bKP equation [16], the generalized Burgers equation [16], the inverse scattering transformation method [17, 18], and the KP hierarchy reduction method [19]. Combining Hirota’s bilinear method with the KP reduction hierarchy method, very recently, Rao et al. [20] discussed the Kadomtsev-Petviashvili-based system and studied the fusion of lumps and line solitons into line solitons, fission of line solitons into lumps and line solitons, a mixture of fission and fusion processes of lumps and line solitons, and the inelastic collision of line rogue waves and line soliton. An improved Hirota bilinear method for the nonlocal complex MKdV equation was constructed in Ref. [21]. Sun et al. [22] investigated a generalized three-component Hirota-Satsuma coupled KdV equation describing the interactions of two long waves with different dispersion relations by applying Hirota bilinear operator theory. By employing the Hirota bilinear method, Ma [23] constructed the N-soliton solution for three integrable equations in (2+1)-dimensions including the (2+1)-dimensional KdV equation, the Kadomtsev-Petviashvili equation, and the (2+1)-dimensional Hirota-Satsuma-Ito equation and gave the asymptotic analysis of N-soliton solution. Also, by using the bilinear Bell polynomial approach, Cui [24] obtained some new exact solutions for the new extended (2+1)-dimensional Boussinesq equation which can be applied to describe the propagation of shallow water waves. And in Ref. [25], Cheng and coauthors obtained the velocity resonance mechanism and the two-, three-, and four-soliton molecules by utilizing the Hirota bilinear method for the combined (2N+1)th-order Lax’s KdV equation. In [26], the new solitary wave solutions for the (3+1)-dimensional extended Jimbo-Miwa equations were investigated. Moreover, Ma obtained lump solutions to a combined fourth-order nonlinear PDE in (2+1)-dimensions [27] and interaction solutions to the Hirota-Satsuma-Ito equation in (2+1)-dimensions [28].Consider the (3+1)-dimensional variable-coefficient nonlinear wave equation [29–31] which will be investigated below:
(1)ut+ϕ1tuux+ϕ2tuxxx+ϕ3tuxx+ϕ4tuyy+ϕ5tuzz=0,where u=ux,y,z,t is an unknown function and ϕjtj=1,⋯,5 are all optional amounts. And u is the wave amplitude, the variable coefficients ϕ1t,ϕ2t,ϕ3t,ϕ4t, and ϕ5t denote the bubble-liquid nonlinearity, the bubble-liquid dispersion, the bubble-liquid viscosity, the y-transverse-perturbation, and the z-transverse-perturbation, respectively, and they are all real functions of t. Equation (1) can be transformed to the following:
(i)
The well-known constant-coefficient Kadomtsev-Petviashvili equation withϕ1t=±6,ϕ2t=1,ϕ4t=3,ϕ3t=ϕ5t=0 as(2)ut±6uux+uxxxx+3uyy=0(ii)
The cylindrical KdV equation withϕ1t=±6,ϕ2t=1,ϕ3t=1/2t,ϕ4t=ϕ5t=0 as(3)ut±6uux+uxxx+12tux=0For equation (1), some solutions including the multisoliton, Bäcklund transformation, infinite conservation laws, lump solutions, and other soliton wave solutions have been investigated in Refs. [29, 30]. Also, Guo and Chen [31] studied equation (1) and obtained the multisoliton solutions and periodic solutions including X-periodic, Y-periodic, and 2-periodic wave solutions. The solitons, periodic, and travelling waves of a generalized (3+1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles were caught by Deng and Gao [29]. In [32], the first-order lump wave solution and second-order lump wave solution according to the corresponding two-soliton solution and four-soliton solution were presented. The multisoliton solutions and periodic solutions for the (3+1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles were reported by Guo and Chen [31]. Two different types of bright solutions for the generalized (3+1)-dimensional nonlinear wave equation by the traveling wave method were obtained by Guo and Chen [33]. The special N wave solutions by applying the linear superposition principle, the resonant multiple wave solutions, and complexiton solutions were investigated for the generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles in [34]. Ma and coauthors probed and analyzed N-soliton solutions and the Hirota conditions in (1+1)-dimensions [13] and (1+2)-dimensions [35].During the last years, the various analytical methods were developed to find the exact solutions by powerful scholars for interesting fields of research because of their wide number of applications in the engineering and manufacturing fields, nonlinear models, for example, nonlinear Schrodinger equation [36], the conformable nonlinear differential equation governing wave-propagation in low-pass electrical transmission lines [37], the (2+1)-dimensional coupled variant Boussinesq equations [38], the nonlinear directional couplers with metamaterials by including spatial-temporal fractional beta derivative evolution [39], a new (3+1)-dimensional Hirota bilinear equation [40], oblique resonant nonlinear waves with dual-power law nonlinearity [41], the coupled Schrödinger-Boussinesq system with the beta derivative [42], and the Hirota-Maccari system [43].The major aim of this paper is to obtain some novel exact analytical solutions, including interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the (3+1)-dimensional variable-coefficient (VC) nonlinear wave equation in liquid with gas bubbles through the method of the bilinear analysis.The rest of this article is organized as follows. In Section2, explanations of multidimensional binary Bell polynomials are given. Also, in Section 3, the bilinear form equation to the (3+1)-dimensional VC nonlinear wave equation is constructed. In Section 4, we obtain the interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions along with depicting 3D, density, and 2D graphs for the VC nonlinear wave equation. The conclusion is given in Section 5.


## 2. Multidimensional Binary Bell Polynomials

Based on Ref. [16, 44, 45], consider ξ=ξx1,x2,⋯,xn a C∞ function with multivariables; the polynomials of the following form
(4)Yn1x1,⋯,njxjξ≡Yn1,⋯,njξs1x1,⋯,sjxj=e−ξ∂x1n1⋯∂xjnjeξare called the multidimensional Bell polynomials:
(5)ξs1x1,⋯,sjxj=∂x1s1⋯∂xjsjξ,ξ0xi≡ξ,s1=0,⋯,n1;⋯;sj=0,⋯,nj,and we have
(6)Y1ξ=ξx,Y2ξ=ξ2x+ξx2,Y3ξ=ξ3x+3ξxξ2x+ξx3,⋯,ξ=ξx,t,Yx,tξ=ξx,t+ξxξt,Y2x,tξ=ξ2x,t+ξ2xξt+2ξx,tξx+ξx2ξt,⋯.The multidimensional binary Bell polynomials can be written as(7)Σn1x1,⋯,njxjα,β=Yn1,⋯,njξξs1x1,⋯,sjxj=αs1x1,⋯,sjxj,s1+s2+⋯+sj,isodd,βs1x1,⋯,sjxj,s1+s2+⋯+sj,iseven.We have the following conditions as(8)Σxα=αx,Σ2xα,β=β2x+αx2,Σx,tα,β=βx,t+αxαt,⋯.Proposition 1.
Supposeα=lnΘ/Δ,β=lnΘΔ,then the relations between binary Bell polynomials and HirotaD-operator can be written as below:(9)Σn1x1,⋯,njxjα,βα=lnΘ/Δ,β=lnΘΔ=ΘΔ−1Dx1n1⋯DxjnjΘΔ,with Hirota operator
(10)∏i=1jDxinig·η=∏i=1j∂∂xi−∂∂xi′niΘx1,⋯,xjΔx1′,⋯,xj′x1=x1′,⋯,xj=xj′.Proposition 2.
TakeΞγ=∑iδiPs1x1,⋯,sjxj=0andα=lnΘ/Δ,β=lnΘΔ,we have(11)∑iδ1iYn1x1,⋯,njxjα,β=0,∑iδ1iYs1x1,⋯,sjxjα,β=0,which need to satisfy
(12)Rγ′,γ=Rγ′−Rγ=Rβ+α−Rβ−α=0.
The generalized Bell polynomialsYn1x1,⋯,njxjξ are as
(13)ΘΔ−1Dx1n1⋯DxjnjΘΔ=Σn1x1,⋯,njxjα,βα=lnΘ/Δ,β=lnΘΔ=Σn1x1,⋯,njxjα,α+γα=lnΘ/Δ,γ=lnΘΔ=∑k1n1⋯∑kjnj∏i=1jnikiPk1x1,⋯,kjxjγYn1−k1x1,⋯,nj−kjxjα.
The Cole-Hopf transformation will be as(14)Yk1x1,⋯,kjxjα=lnφ=φn1x1,⋯,njxjφ,ΘΔ−1Dx1n1⋯DxjnjΘΔΔ=expγ/2,Θ/Δ=φ=φ−1∑k1n1⋯∑kjnj∏d=1jndkdPk1x1,⋯,kdxdγφn1−k1x1,⋯,nd−kdxd,with
(15)Ytα=φtφ,Y2xα,β=γ2x+φ2xφ,Y2x,yα,β=γ2xφyφ+2γx,yφxφ+φ2x,yφ.


## 3. Bilinear Form Equation to the (3+1)D VC Nonlinear Wave Equation

To find the linearizing representation, we consider the below:(16)u=cγxx+u0,γ=γx,y,z,t,c=ct.Inserting equation (16) into equation (1), one obtains
(17)Rγ=ddtct∂∂xγx,y,z,t+ct∂2∂x∂tγx,y,z,t+ϕ1tct2∂∂xγx,y,z,t2+ϕ1tct2γx,y,z,t∂2∂x2γx,y,z,t+ϕ2tct∂4∂x4γx,y,z,t+ϕ3tct∂2∂x2γx,y,z,t+ϕ4tct∂2∂y2γx,y,z,t+ϕ5tct∂2∂z2γx,y,z,t=0,with
(18)c=12.The new equationRγ is as
(19)Rγ=Px,t+ϕ2tP4x+3P2x2+ϕ3tP2x+ϕ4tP2y+ϕ5tP2z=0.Applying a change of dependent variable(20)γ=lng⟺u=12lngxx.Theorem 3.
With the following relations(21)γ=lng⟺u=12lngxx,into equation (1), the (3+1)D VC nonlinear wave equation can be linearized as the following bilinear equation:
(22)Rg=ggxt−gxgt+ϕ2tgg4x−4gxg3x+3gxx2+ϕ3tggxx−gx2,
ϕ4tggyy−gy2+ϕ5tggzz−gz2=1/2DxDt+ϕ2tDx4+ϕ3tDx2+ϕ4tDy2+ϕ5tDz2g·g=0, where g=gx,y,z,t and γ=γx,y,z,t.


## 4. Lump, Lump-Kink, and Other Wave Solutions

We would like to derive the general soliton solutions containing interaction between lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions.


### 4.1. Interaction between a Lump-Two Kink Soliton Solutions

In this section, we would like to present the general solutions of the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(23)g=τ12+τ22+expτ3+expτ4+expτ3+τ4+ε5t,τj=αjx+βjy+δjz+εjt,j=1,2,3,ε0>0.The valuesαi,βi,δi,εiti=1,2,3 are real constants to be computed. By substituting (23) into (22), we obtain a system containing 42 nonlinear equations. By solving the nonlinear system, the determined coefficients will be got as the below cases.Type I(24)ε1t=ε3t=ε4t=ε5t=0,ε2t=∫−α22ϕ2t+β22ϕ3tα2dt,α1=α3=α4=0,α2=α2,β1=β3=β4=0,β2=β2,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4.The solutions are given as follows:(25)u1=24α22δ12z2+α2x+β2y+δ2z+∫−α22ϕ2t+β22ϕ3t/α2dt2+eδ3z+eδ4z+eδ3z+δ4z−48α2x+β2y+δ2z+∫−α22ϕ2t+β22ϕ3t/α2dt2α22δ12z2+α2x+β2y+δ2z+∫−α22ϕ2t+β22ϕ3t/α2dt2+eδ3z+eδ4z+eδ3z+δ4z2.Ifτ12+τ22+expτ3+expτ4+expτ3+τ4+ε5t⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ1=1,δ2=2,δ3=1.5,δ4=3,α2=1,β2=3,ϕ2t=cost,ϕ3t=sint,x=1,y=1, then plots of equation (25) are plotted in Figure 1. And also, by selecting the parameters δ1=1,δ2=2,δ3=1.5,δ4=3,α2=1,β2=3,ϕ2t=1/4cos2t,ϕ3t=1/4sin1+2t,x=1,y=1, then plots of equation (25) are plotted in Figure 2.Figure 1
Plots of interaction lump with two solitons (25).Figure 2
Plots of interaction lump with two solitons (25).Type II(26)ε1t=∫−2δ1ϕ4tβ1δ2−β2δ1α2β1dt,ε2t=∫−α22β12ϕ2t+β12δ22ϕ4t−β22δ12ϕ4tβ12α2dt,ε4t=ε5t=0,ε3t=∫−2β3δ1ϕ4tβ1δ2−β2δ1β12α2dt,α1=α3=α4=0,α2=α2,β1=β1,β2=β2,β3=β3,β4=0,δ1=δ1,δ2=δ2,δ3=β3δ1β1,δ4=0,ϕ1t=0,ϕ3t=−δ12ϕ4tβ12.The solutions are given as follows:(27)u2=24α22τ12+τ22+2eβ3y+β3δ1z/β1+∫−2β3δ1ϕ4tβ1δ2−β2δ1/β12α2dt−48τ22α22τ12+τ22+2eβ3y+β3δ1z/β1+∫−2β3δ1ϕ4tβ1δ2−β2δ1/β12α2dt2,(28)τ1=β1y+δ1z+∫−2δ1ϕ4tβ1δ2−β2δ1α2β1dt,τ2=α2x+β2y+δ2z+∫−α22β12ϕ2t+β12δ22ϕ4t−β22δ12ϕ4tβ12α2dt.Ifτ12+τ22+expτ3+expτ4+expτ3+τ4+ε5t⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ1=1,δ2=2,α2=1,β1=2,β2=3,β3=4,ϕ2t=cost,ϕ4t=sint,x=1,y=1, then plots of equation (27) are plotted in Figure 3.Figure 3
Plots of interaction lump with two solitons (27).Type III(29)ε1t=∫2δ1ϕ4tβ2δ4−β4δ2α2β4dt,ε2t=∫−α22β42ϕ2t−β22δ42ϕ4t+β42δ22ϕ4tβ42α2dt,ε5t=0,ε3t=∫2δ3ϕ4tβ2δ4−β4δ2α2β4dt,ε4t=∫2δ4ϕ4tβ2δ4−β4δ2α2β4dt,α1=α3=α4=0,α2=α2,β1=β4δ1δ4,β2=β2,β3=δ3β4δ4,β4=β4,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ1t=0,ϕ3t=−δ42ϕ4tβ42.The solutions are given as follows:(30)u3=24α22τ12+τ22+F1+F2+F3−48τ22α22τ12+τ22+F1+F2+F32,(31)τ1=β4δ1yδ4+δ1z+∫2δ1ϕ4tβ2δ4−β4δ2α2β4dt,τ2=α2x+β2y+δ2z+∫−α22β42ϕ2t−β22δ42ϕ4t+β42δ22ϕ4tβ42α2dt,(32)F1=eδ3β4y/δ4+δ3z+∫2δ3ϕ4tβ2δ4−β4δ2/α2β4dt,F2=eβ4y+δ4z+2δ4tϕ4tβ2δ4−β4δ2/α2β4,(33)F3=eδ3β4y/δ4+δ3z+∫2δ3ϕ4tβ2δ4−β4δ2/α2β4dt+β4y+δ4z+2δ4tϕ4tβ2δ4−β4δ2/α2β4.By selecting the parametersδ1=1,δ2=2,δ3=3,δ4=4,α2=1,β2=3,β4=4,ϕ2t=cost,ϕ4t=sint,x=1,y=1, then plots of equation (30) are plotted in Figure 4.Figure 4
Plots of interaction lump with two solitons (30).Type IV(34)ε1t=−ε2tδ2δ1,ε3t=∫−α34ϕ1t+α32ϕ2t+β32ϕ3tα3dt,ε4t=0,ε5t=−2ε2tδ1ε2t−δ2ε1tδ1,α1=α2=α4=0,α3=α3,β1=β2=β4=0,β3=β3,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ4t=0.The solutions are given as follows:(35)u4=12α32F1+12α32F2δ1z−ε2tδ2/δ12+δ2z+ε2t2+F1+eδ4z+F2−2ε2tδ1ε2t−δ2ε1t/δ1,(36)12α3F1+α3F22δ1z−ε2tδ2/δ12+δ2z+ε2t2+F1+eδ4z+F2−2ε2tδ1ε2t−δ2ε1t/δ12,(37)F1=eα3x+β3y+δ3z+∫−α34ϕ1t+α32ϕ2t+β32ϕ3t/α3dt,F2=eα3x+β3y+δ3z+∫−α34ϕ1t+α32ϕ2t+β32ϕ3t/α3dt+δ4z.By selecting the parametersδ1=1,δ2=2,δ3=3,δ4=4,α2=1,β2=3,β4=4,ϕ2t=cost,ϕ4t=sint,x=1,y=1, then plots of equation (35) are plotted in Figure 5.Figure 5
Plots of interaction lump with two solitons (35).Type V(38)ε1t=−ε2tδ2δ1,ε3t=∫−α3α32ϕ1t+ϕ2tdt,ε4t=0,ε5t=∫−2d/dtε2tδ1ε2t−δ2ε1tδ1dt,α1=α2=α4=0,α3=α3,β1=β2=β4=0,β3=β3,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ3t=ϕ4t=0.The solutions are given as follows:(39)u5=12∂2∂x2lnF5,F5=δ1z+ε1t2+δ2z+ε2t2+eα3x+β3y+δ3z+ε3t+eβ4y+δ4z+ε4t+eα3x+β3y+δ3z+ε3t+β4y+δ4z+ε4t+ε5t.Type VI(40)ε3t=∫−α34ϕ1t+α32ϕ2t+δ32ϕ3tα3dt,ε2t=ε4t=0,ε5t=−ε1t2,α1=α2=α4=0,α3=α3,β1=0,β2=β2,β3=β3,β4=β4,δ1=δ2=δ4=0,δ3=δ3,ϕ3t=0.The solutions are given as follows:(41)u5=12∂2∂x2lnF6,F6=β2y+ε2t2+eα3x+β3y+δ3z+ε3t+eβ4y+ε4t+eα3x+β3y+δ3z+ε3t+β4y+ε4t.Type VII(42)ε3t=∫−α3α32ϕ1t+ϕ2tdt,ε1t=ε2t=ε4t=ε5t=0,α1=α2=α4=0,α3=α3,β1=0,β2=β2,β3=β3,β4=β4,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ3t=ϕ4t=0.The solutions are given as follows:(43)u7=12∂2∂x2lnF7,F7=z2δ12+β2y+δ2z2+eα3x+β3y+δ3z+∫−α3α32ϕ1t+ϕ2tdt+eβ4y+δ4z+eα3x+β3y+δ3z+∫−α3α32ϕ1t+ϕ2tdt+β4y+δ4z.Type VIII(44)ε2t=∫−2δ2ϕ4tβ2δ3−β3δ2α3β2dt,ε3t=∫−α34β22ϕ1t+α32β22ϕ2t+β22δ32ϕ4t−β32δ22ϕ4tβ22α3dt,ε4t=∫−2β2δ3−β3δ2δ4ϕ4tα3β2dt,ε5t=−ε12t,α1=α2=α4=0,α3=α3,β1=0,β2=β2,β3=β3,β4=β2δ4δ2,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ3t=−δ22ϕ4tβ22.The solutions are given as follows:(45)u8=12∂2∂x2lnF8,F8=ε1t2+β2y+δ2z+ε2t2+eα3x+β3y+δ3z+ε3t+eβ2δ4y/δ2+δ4z+ε4t+eα3x+β3y+δ3z+ε3t+β2δ4y/δ2+δ4z+ε4t+ε5t.Type IX(46)ε1t=−ε2tβ2β1,ε3t=∫−α3α32ϕ1t+ϕ2tdt,ε4t=0,ε5t=∫−2d/dtε2tβ1ε2t−β2ε1tβ1dt,α1=α2=α4=0,α3=α3,β1=β1,β2=β2,β3=β3,β4=β4,δ1=δ1,δ2=β2δ1β1,δ3=δ3,δ4=δ4,ϕ3t=ϕ4t=0.The solutions are given as follows:(47)u9=12∂2∂x2lnF9,F9=β1y+δ1z+ε1t2+β2y+β2δ1zβ1+ε2t2+eα3x+β3y+δ3z+ε3t+eβ4y+δ4z+ε4t+eα3x+β3y+δ3z+ε3t+β4y+δ4z+ε4t+ε5t.Type X(48)ε1t=∫−2β13δ1δ3ϕ4t−2β12β3δ12ϕ4t+2β1β22δ1δ3ϕ4t−2β22β3δ12ϕ4t+d/dtε2tα3β12β2α3β13dt,ε3t=∫−α34β12ϕ1t+α32β12ϕ2t+β12δ32ϕ4t−β32δ12ϕ4tα3β12dt,ε4t=∫−2β1δ3−β3δ1δ4ϕ4tα3β1dt,ε5t=∫−2ε2tβ14α3+∫S1+d/dtε2tα3β12β2dtβ2S2+d/dtε2tα3β12α32β16dt,S1=2δ1ϕ4tβ12+β22β1δ3−β3δ1,S2=2β2δ1ϕ4tβ1δ3−β3δ1,α1=α2=α4=0,β4=β1δ4δ1,δ1=δ1,δ2=β2δ1β1,δ3=δ3,δ4=δ4,ϕ3t=−δ12ϕ4tβ12.The solutions are given as follows:(49)u10=12∂2∂x2lnF10,F10=β1y+δ1z+ε1t2+β2y+β2δ1zβ1+ε2t2+eα3x+β3y+δ3z+ε3t+eβ1δ4y/δ1+δ4z+ε4t+eα3x+β3y+δ3z+ε3t+β1δ4y/δ1+δ4z+ε4t+ε5t.By selecting the parametersδ1=1,δ3=3,δ4=4,α3=1,β1=4,β2=2,β3=3,ϕ1t=sin2t,ϕ2t=cos2t,ϕ4t=cos3t,ε2t=expt,x=1,y=1, then plots of equation (49) are plotted in Figure 6. And also, by choosing the parameters δ1=1,δ3=3,δ4=4,α3=1,β1=4,β2=2,β3=3,ϕ1t=exptsint,ϕ2t=exptcos2t,ϕ4t=exptcos3t,ε2t=exp2t,x=1,y=1, then plots of equation (49) are plotted in Figure 7. Moreover, by choosing the parameters δ1=1,δ3=3,δ4=4,α3=1,β1=4,β2=2,β3=3,ϕ1t=1/1+sint,ϕ2t=cost/2+sint,ϕ4t=cost,ε2t=sin2t+tant,x=1,y=1, then plots of equation (49) are plotted in Figure 8.Figure 6
Plots of interaction lump with two solitons (49).Figure 7
Plots of interaction lump with two solitons (49).Figure 8
Plots of interaction lump with two solitons (49).


### 4.2. Interaction between Two Lump Solutions

Here, we offer interaction between two lump solutions containing combination of two functions for the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(50)g=τ14+τ22+τ32+ε4t,τj=αjx+βjy+δjz+εjt,j=1,2,3,ε4t>0.The valuesαi,βi,δi,εiti=1,2,3 are real constants to be computed. By appending (50) into (22), we receive to a system containing 42 nonlinear equations. To solve the nonlinear system, the determined coefficients will be got as the below cases.Type I(51)ε1t=C1−ε4t4,ε2t=∫−2β2β3ϕ3t+δ2δ3ϕ4tα3dt+C2,α1=α2=0,α3=α3,ε3t=∫−α32ϕ2t−ϕ3tβ22+β32ϕ3t−δ22ϕ4t+δ32ϕ4tα3dt+C3,β1=0,β2=β2,β3=β3,δ1=0,ϕ1t=−13r14+r4δ22ϕ4t+ϕ3tβ22α34.The solutions are given as follows:(52)u1=12∂2∂x2lnF1,F1=ε1t4+β2y+δ2z+ε2t2+α3x+β3y+δ3z+ε3t2+ε4t.Ifτ14+τ22+τ32+ε4t⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ2=2,δ3=3,α3=3,β2=2,β3=1,ϕ2t=t,ϕ3t=t2,ϕ4t=1+t,x=1,z=1, then plots of equation (52) are plotted in Figure 9. And also, by selecting the parameters δ2=2,δ3=3,α3=3,β2=2,β3=1,ϕ2t=cost,ϕ3t=sint,ϕ4t=cos2t,x=1,z=1, then plots of equation (52) are plotted in Figure 10. Moreover, by selecting the parameters δ2=2,δ3=3,α3=3,β2=2,β3=1,ϕ2t=t2sin2t,ϕ3t=tsin3t,ϕ4t=tcos3t,x=1,z=1, then plots of equation (52) are plotted in Figure 11.Figure 9
Plots of interaction with two lumps (52).Figure 10
Plots of interaction with two lumps (52).Figure 11
Plots of interaction with two lumps (52).Type II(53)ε1t=−−C1α23+∫Ftdtα24α2,α1=β1=δ1=ϕ1t=0,δ3=α3δ2α2,Ft=2ϕ2tα22α3+ϕ3tα3β22+ϕ4tα3δ22+ddtε3tα22ε3tα2−ε2tα3+ddtε4tα23,ε2t=∫−α22+α32δ22ϕ4t+ϕ3tβ22+ϕ2tα22+d/dtε3tα22α3α23dt+C2.The solutions are given as follows:(54)u2=12∂2∂x2lnF2,F2=ε1t4+α2x+β2y+δ2z+ε2t2+α3x+α3β2yα2+α3δ2zα2+ε3t2+ε4t.Type III(55)ε1t=∫−2ϕ4tδ1β1δ3−β3δ1α3β1dt,ε3t=∫−α32β12ϕ2t+β12δ32ϕ4t−β32δ12ϕ4tα3β12dt,ε4t=−ε2t2+C1,α1=α2=β2=δ2=ϕ1t=0,ϕ3t=−δ12ϕ4tβ12.The solutions are given as follows:(56)u3=12∂2∂x2lnF3,F3=β1y+δ1z+ε1t4+ε2t2+α3x+β3y+δ3z+ε3t2+ε4t.


### 4.3. Interaction between Two Lumps-Soliton Solutions

Here, we offer interaction between two lumps-soliton solutions containing combination of two functions for the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(57)g=τ14+τ22+τ32+expτ4+ε5t,τj=αjx+βjy+δjz+εjt,j=1,2,3,4,ε4t>0.The valuesαi,βi,δi,εiti=1,2,3,4 are real constants to be computed. By putting (57) into (22), we receive a system containing 119 nonlinear equations. To solve the nonlinear system, the determined coefficients will be got as the below cases.Type I(58)ε1t=A1t+A2tα24α2,A1t=2α32β22∫ε2tϕ3tdt−2α3∫ε3tϕ2tdtα23−2α3β22∫ε3tϕ3tdtα2,A2t=2α32∫ϕ2tε2tdtα22+2α3∫ε2tddtε3tdtα22−ε3t2α23−ε5tα23+C1α23,ε2t=∫−α22+α32ϕ2tα22+ϕ3tβ22+d/dtε3tα22α3α23dt+C2,ε4t=0,α1=α4=β1=β4=δ1=ϕ1t=ϕ4t=0,β3=α3β2α2,δ3=α3δ2α2.The solutions are given as follows:(59)u1=12∂2∂x2lnF1,F1=ε1t4+α2x+β2y+δ2z+ε2t2+α3x+α3β2yα2+α3δ2zα2+ε3t2+eδ4z+ε4t+ε5t.Ifτ14+τ22+τ32+expτ4+ε5t⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ2=1,δ4=2,α2=1,α3=2,β2=3,ϕ2t=cost,ϕ3t=sint,ε2t=cos2t,ε3t=sin2t,x=1,y=1, then plots of equation (59) are plotted in Figure 12. And also, by selecting the parameters δ2=1,δ4=2,α2=1,α3=2,β2=3,ϕ2t=cos1+t,ϕ3t=sin1+t,ε2t=cos1+3t,ε3t=sin1+3t,x=1,y=1, then plots of equation (52) are plotted in Figure 13.Figure 12
Plots of interaction with two lumps-soliton (59).Figure 13
Plots of interaction with two lumps-soliton (59).Type II(60)ε1t=A1t+A2t+A3tα42α234α4α2,A1t=2α32β42∫ε2tϕ3tdt−2α3∫ε3tϕ2tdtα42α2−2α3β42∫ε3tϕ3tdtα2,A2t=2α32∫ϕ2tε2tdtα42+2α32δ42∫ϕ4tε2tdt−2α3δ42∫ϕ4tε3tdtα2,A3t=2α3∫ε2tddtε3tdtα42−ε3t2α42α2−ε5tα42α2+C1α42α2,ε2t=∫−α22+α32α42ϕ2t+β42ϕ3t+δ42ϕ4t+d/dtε3tα3α42α42α2dt,ε4t=∫−α42ϕ2t+β42ϕ3t+δ42ϕ4tα4dt,α1=β1=δ1=ϕ1t=0,β2=α2β4α4,β3=α3β4α4,δ2=α2δ4α4,δ3=α3δ4α4.The solutions are given as follows:(61)u2=12∂2∂x2lnF2,(62)F2=ε1t4+α2x+α2β4yα4+α2δ4zα4+ε2t2+α3x+α3β4yα4+α3δ4zα4+ε3t2+eα4x+β4y+δ4z+ε4t+ε5t.By selecting the parametersδ4=2,α2=1,α3=2,α4=4,β4=3,ϕ2t=cost,ϕ3t=sint,ϕ4t=sin2t,ε2t=cos2t,ε3t=sin2t,ε4t=sin3t,C1=1,C2=2,x=1,y=1, then plots of equation (61) are plotted in Figure 14. And also, by selecting the parameters δ4=2,α2=1,α3=2,α4=4,β4=3,ϕ2t=cost,ϕ3t=sint,ϕ4t=sin2t,ε2t=t,ε3t=t2,ε4t=t3,C1=1,C2=2,x=1,y=1, then plots of equation (54) are plotted in Figure 15.Figure 14
Plots of interaction with two lumps-soliton (61).Figure 15
Plots of interaction with two lumps-soliton (61).


### 4.4. Lump-Periodic Solutions

In this section, we would like to present the general solutions of the including combination of two functions for the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(63)g=τ12+τ22+cosτ3+ε4t,τj=αjx+βjy+δjz+εjt,j=1,2,3,ε4t>0.The valuesαi,βi,δi,εiti=1,2,3 are real constants to be computed. By substituting (63) into (22), we obtain a system containing 24 nonlinear equations. To solve the nonlinear system, the determined coefficients will be got as the below cases.Type I(64)ε1t=−ε2tα2α1−α2α1∫α12ϕ2tα2+α2ϕ2tdt+C1,ε3t=0,ε4t=−α12+α22∫ϕ2tdt2α22+ε2t2α12−2ε2tα2α12∫ϕ2tα12+α22α2dtα2−C1α1−2∫−ϕ2tC1α1α22dtα12+C2,α3=ϕ1t=ϕ3t=0,β2=α2β1α1,δ2=α2δ1α1.The solutions are given as follows:(65)u1=12∂2∂x2lnF1,F1=α1x+β1y+δ1z+ε1t2+α2x+α2β1yα1+α2δ1zα1+ε2t2+cosβ3y+δ3z+ε3t+ε4t.From above, we can discover thatg is positive if ε4t>0, then in all space directions, u will be localized. Therefore, if τ12+τ22+cosτ3⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ1=1,δ3=3,α1=2,α2=4,β1=2,β3=3,ϕ2t=−2,ε2t=3,C1=1,C2=2,x=1,y=1, then plots of equation (65) are plotted in Figure 16.Figure 16
Plots of lump-periodic (65).Type II(66)ε1t=∫−α12+α22α12ϕ2+δ12ϕ4+d/dtε2tα12α2α13dt+C1,ε3t=0,ε4t=∫2ε1tα2−ε2tα1ϕ2tα12α2+ϕ4tα2δ12+d/dtε2tα12α13dt+C2,α3=δ3=ϕ1t=ϕ3t=0,β2=α2β1α1,δ2=α2δ1α1.The solutions are given as follows:(67)u2=12∂2∂x2lnF2,F2=α1x+β1y+δ1z+ε1t2+α2x+α2β1yα1+α2δ1zα1+ε2t2+cosβ3y+ε3t+ε4t.Type III(68)ε1t=∫−2δ1ϕ4tβ1δ2−β2δ1α2β1dt,ε2t=∫−α22β12ϕ2t+β12δ22ϕ4t−β22δ12ϕ4tα2β12dt,ε3t=∫−2δ3ϕ4tβ1δ2−β2δ1α2β1dt,ε4t=0,β3=β1δ3δ1,ϕ3t=−δ12ϕ4tβ12,α1=α3=ϕ1t=0,β2=α2β1α1,δ2=α2δ1α1.The solutions are given as follows:(69)u3=12∂2∂x2lnF3,F3=β1y+δ1z+ε1t2+α2x+β2y+δ2z+ε2t2+cosβ1δ3yδ1+δ3z+ε3t+ε4t.By selecting the parametersδ1=1,δ2=2,δ3=3,α2=4,β1=2,β2=3,ϕ2t=sint,ϕ4t=cost,x=1,y=1, then plots of equation (69) are plotted in Figure 17.Figure 17
Plots of lump-periodic (69).Type IV(70)ε2t=−tanα2∫ϕ2tε1tdt+C1ε1,ε4t=∫2ε1t2+ε2t2α2ϕ2tε2t−d/dtε1tε1tε1t2dt+C2,ε3t=0,α1=−ε2tα2ε1t,α3=β1=β2=δ1=δ2=ϕ1t=0,ϕ3t=−δ32ϕ4tβ32.The solutions are given as follows:(71)u4=12∂2∂x2lnF4,F4=−ε2tα2xε1t+ε1t2+α2x+ε2t2+cosβ3y+δ3z+ε3t+ε4t.Type V(72)ε2t=−tan1α2δ32∫α22δ32ϕ2t−β32δ22ϕ3tε1tdtε1t,ε3t=∫2β32δ2ϕ3tδ3α2dt,ε4t=∫2ε1t2+ε2t2ϕ2tε2tα22δ32−ϕ3tε2tβ32δ22−d/dtε1tα2δ32ε1tα2δ32ε1t2dt,α3=β1=β2=ϕ1t=0,α1=−α2ε2tε1t,δ1=−δ2ε2tε1t,ϕ4t=−β32ϕ3tδ32.The solutions are given as follows:(73)u5=12∂2∂x2lnF5,F5=−α2ε2txε1t−δ2ε2tzε1t+ε1t2+α2x+δ2z+ε2t2+cosβ3y+δ3z+ε3t+ε4t.Type VI(74)ε3t=0,ε4t=−ε1t2−ε2t2,α3=β1=β2=ϕ1t=ϕ3t=0,α1=−α2ε2tε1t,δ1=−δ2ε2tε1t,ϕ4t=−β32ϕ3tδ32,ϕ4t=−α2ϕ2tε1t2+ε2t2+d/dtε1tε1tε2t−d/dtε2tε1t2α2ε1t2+ε2t2δ22.The solutions are given as follows:(75)u6=12∂2∂x2lnF6,F6=−α2ε2txε1t−δ2ε2tzε1t+ε1t2+α2x+δ2z+ε2t2+cosβ3y−ε1t2−ε2t2.Type VII(76)ε1t=∫−ϕ2tα12+α22+d/dtε2tα2α1dt+C1,ε3t=0,ε4t=∫−2α2ϕ2tα1ε2t−α2ε1t+d/dtε2tα1ε2t−α2ε1tα1dt+C2,α3=β1=β2=β3=ϕ1t=ϕ4t=0,δ2=α2δ1α1.The solutions are given as follows:(77)u7=12∂2∂x2lnF7,F7=α1x+δ1z+ε1t2+α2x+α2δ1zα1+ε2t2+cosδ3z+ε3t+ε4t.By selecting the parametersδ1=2,α1=1,α2=2,β3=1,ϕ2t=t,ϕ4t=t2,ε1t=t,ε2t=1+t,ε3t=t2,x=1,y=1, then plots of equation (77) are plotted in Figure 18. And also, by selecting the parameters δ1=2,α1=1,α2=2,β3=1,ϕ2t=sint,ϕ4t=cost,ε1t=t,ε2t=sint,ε3t=cost,x=1,y=1, then plots of equation (77) are plotted in Figure 19. Moreover, by selecting the parameters δ1=2,α1=1,α2=2,β3=1,ϕ2t=1,ϕ4t=2,ε1t=t,ε2t=t,ε3t=t2,x=1,y=1, then plots of equation (77) are plotted in Figure 20. And finally, by selecting the parameters δ1=2,α1=1,α2=2,β3=1,ϕ2t=expt,ϕ4t=exp2t,ε1t=sinht,ε2t=cosht,ε3t=t2,x=1,y=1, then plots of equation (77) are plotted in Figure 21.Figure 18
Plots of lump-periodic (77).Figure 19
Plots of lump-periodic (77).Figure 20
Plots of lump-periodic (77).Figure 21
Plots of lump-periodic (77).


### 4.5. Lump-Three Kink Solutions

Here, we present lump-periodic solutions containing combination of two functions for the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(78)g=τ12+τ22+expτ3+expτ4+expτ5+ε6t,τj=αjx+βjy+δjz+εjt,j=1,2,3,4,5,ε6t>0.The valuesαi,βi,δi,εiti=1,⋯,5 are real constants to be computed. By appending (78) into (22), we receive a system containing 43 nonlinear equations. To solve the nonlinear system, the determined coefficients will be got as the below cases.Type I(79)ε1t=∫−2β5δ1ϕ3tβ2δ5−β5δ2δ52α2dt,ε2t=∫−α22δ52ϕ2t+β22δ52ϕ3t−β52δ22ϕ3tδ52α2dt,ε3t=∫−2β5δ3ϕ3tβ2δ5−β5δ2δ52α2dt,ε4t=∫−2β5δ4ϕ3tβ2δ5−β5δ2δ52α2dt,ε5t=∫−2β5ϕ3tβ2δ5−β5δ2α2δ5dt,ε6t=0,α1=α3=α4=α5=ϕ1t=0,β1=δ1β5δ5,β3=β5δ3δ5,β4=β5δ4δ5,ϕ4t=−β52ϕ3tδ52.The solutions are given as follows:(80)u1=12∂2∂x2lnF1,F1=δ1β5yδ5+δ1z+ε1t2+α2x+β2y+δ2z+ε2t2+eβ5δ3y/δ5+δ3z+ε3t+eβ5δ4y/δ5+δ4z+ε4t+eβ5y+δ5z+ε5t+ε6t.Type II(81)ε1t=∫−α12+α22α42ϕ2t+β42ϕ3t+δ42ϕ4t+d/dtε2tα2α42α1α42dt,ε3t=∫−α3α42ϕ2t+β42ϕ3t+δ42ϕ4tα42dt,ε4t=∫−α42ϕ2t+β42ϕ3t+δ42ϕ4tα4dt,ε5t=∫−α5α42ϕ2t+β42ϕ3t+δ42ϕ4tα42dt,ϕ1t=0,βi=αiβ4α4,δi=αiδ4α4,i=1,2,3,5,ε6t=∫−2α2α42ϕ2t+β42ϕ3t+δ42ϕ4tα1ε2t−α2ε1t+d/dtε2tα42α1ε2t−α2ε1tα1α42dt.The solutions are given as follows:(82)u2=12∂2∂x2lnF2,F2=α1x+α1β4yα4+α1δ4zα4+ε1t2+α2x+α2β4yα4+α2δ4zα4+ε2t2+eα3x+α3β4y/α4+α3δ4z/α4+ε3t+eα4x+β4y+δ4z+ε4t+eα5x+α5β4y/α4+α5δ4z/α4+ε5t+ε6t.Type III(83)ε1t=∫−ϕ2tα12+α22α1dt−ε2tα2α1,ε3t=∫−α3ϕ2tdt,ε4t=∫−α4ϕ2tdt,ε5t=∫−α5ϕ2tdt,ε6t=∫−2α1ε2t−α2ε1tα2ϕ2t+d/dtε2tα1dt,β2=α2β1α1,β3=β1δ3δ1,δ2=α2δ1α1,δ4=β4δ1β1,δ5=δ1β5β1,ϕ1t=0,ϕ4t=−β12ϕ3tδ12.The solutions are given as follows:(84)u3=12∂2∂x2lnF3,F3=α1x+β1y+δ1z+ε1t2+α2x+α2β1yα1+α2δ1zα1+ε2t2+eα3x+β1δ3y/δ1+δ3z+ε3t+eα4x+β4y+β4δ1z/β1+ε4t+eα5x+β5y+δ1β5z/β1+ε5t+ε6t.From above, we can discover thatg is positive if ε6t>0, then in all space directions u will be localized. Therefore, if τ12+τ22+expτ3+expτ4+expτ5+ε6t⟶∞, the lump-three kink solutions
(85)u⇒−12α32,α3>α4>α5,0,α4>α3>α3,0,α5>α3>α4,0,α4>α5>α3,at any t. By selecting the parameters α1=1,α2=2,α3=3,α4=4,α5=5,β1=1,β4=3,β5=4,δ1=1,δ3=1.5,ϕ2t=cost,ε1t=sin2t,ε2t=cos2t,x=1,y=1, then plots of equation (84) are plotted in Figure 22. And also, via selecting the parameters α1=1,α2=2,α3=3,α4=4,α5=5,β1=1,β4=3,β5=4,δ1=1,δ3=1.5,ϕ2t=1+t+t2,ε1t=1/21+t2,ε2t=1/31+t3,x=1,y=1, then plots of equation (84) are plotted in Figure 23.Figure 22
Plots of lump-three kink solutions (84).Figure 23
Plots of lump-three kink solutions (84).Type IV(86)ε1t=∫−α12+α22α12δ42ϕ2t−β42δ12ϕ3tα13δ42dt−ε2tα2α1,ε4t=∫2β42ϕ3tδ1α1δ4dt,ε3t=∫−α12α3δ42ϕ2t−β42δ1ϕ3t2α1δ3−α3δ1α12δ42dt,ε5t=∫−α12α5δ42ϕ2t−β42δ1ϕ3t2α1δ5−α5δ1α12δ42dt,ε6t=∫−2α1ε2t−α2ε1tϕ2tα12α2δ42−ϕ3tα2β42δ12+d/dtε2tα12δ42α13δ42dt,β3=α1δ3−α3δ1β4α1δ4,β5=β4α1δ5−α5δ1α1δ4,δ2=α2δ1α1,ϕ4t=−β42ϕ3tδ42,α4=β1=β2=ϕ1t=0.The solutions are given as follows:(87)u4=12∂2∂x2lnF4,F4=α1x+δ1z+ε1t2+α2x+α2δ1zα1+ε2t2+eα3x+α1δ3−α3δ1β4y/δ4α1+δ3z+ε3t+eβ4y+δ4z+ε4t+eα5x+β4α1δ5−α5δ1y/δ4α1+δ5z+ε5t+ε6t.By selecting the parametersα1=1,α2=2,α3=3,α5=5,β4=3,δ1=1,δ3=1.5,δ4=2,δ5=3,ϕ2t=t,ϕ3t=t2,ε2t=t4,x=1,y=1, then plots of equation (87) are plotted in Figure 24. And also, via selecting the parameters α1=1,α2=2,α3=3,α5=5,β4=3,δ1=1,δ3=1.5,δ4=2,δ5=3,ϕ2t=1,ϕ3t=2,ε2t=3,x=1,y=1, then plots of equation (87) are plotted in Figure 25. Moreover, via selecting the parameters α1=1,α2=2,α3=3,α5=5,β4=3,δ1=1,δ3=1.5,δ4=2,δ5=3,ϕ2t=sin3t,ϕ3t=cos3t,ε2t=sint+cost,x=1,y=1, then plots of equation (87) are plotted in Figure 26.Figure 24
Plots of lump-three kink solutions (87).Figure 25
Plots of lump-three kink solutions (87).Figure 26
Plots of lump-three kink solutions (87).


## 4.1. Interaction between a Lump-Two Kink Soliton Solutions

In this section, we would like to present the general solutions of the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(23)g=τ12+τ22+expτ3+expτ4+expτ3+τ4+ε5t,τj=αjx+βjy+δjz+εjt,j=1,2,3,ε0>0.The valuesαi,βi,δi,εiti=1,2,3 are real constants to be computed. By substituting (23) into (22), we obtain a system containing 42 nonlinear equations. By solving the nonlinear system, the determined coefficients will be got as the below cases.Type I(24)ε1t=ε3t=ε4t=ε5t=0,ε2t=∫−α22ϕ2t+β22ϕ3tα2dt,α1=α3=α4=0,α2=α2,β1=β3=β4=0,β2=β2,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4.The solutions are given as follows:(25)u1=24α22δ12z2+α2x+β2y+δ2z+∫−α22ϕ2t+β22ϕ3t/α2dt2+eδ3z+eδ4z+eδ3z+δ4z−48α2x+β2y+δ2z+∫−α22ϕ2t+β22ϕ3t/α2dt2α22δ12z2+α2x+β2y+δ2z+∫−α22ϕ2t+β22ϕ3t/α2dt2+eδ3z+eδ4z+eδ3z+δ4z2.Ifτ12+τ22+expτ3+expτ4+expτ3+τ4+ε5t⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ1=1,δ2=2,δ3=1.5,δ4=3,α2=1,β2=3,ϕ2t=cost,ϕ3t=sint,x=1,y=1, then plots of equation (25) are plotted in Figure 1. And also, by selecting the parameters δ1=1,δ2=2,δ3=1.5,δ4=3,α2=1,β2=3,ϕ2t=1/4cos2t,ϕ3t=1/4sin1+2t,x=1,y=1, then plots of equation (25) are plotted in Figure 2.Figure 1
Plots of interaction lump with two solitons (25).Figure 2
Plots of interaction lump with two solitons (25).Type II(26)ε1t=∫−2δ1ϕ4tβ1δ2−β2δ1α2β1dt,ε2t=∫−α22β12ϕ2t+β12δ22ϕ4t−β22δ12ϕ4tβ12α2dt,ε4t=ε5t=0,ε3t=∫−2β3δ1ϕ4tβ1δ2−β2δ1β12α2dt,α1=α3=α4=0,α2=α2,β1=β1,β2=β2,β3=β3,β4=0,δ1=δ1,δ2=δ2,δ3=β3δ1β1,δ4=0,ϕ1t=0,ϕ3t=−δ12ϕ4tβ12.The solutions are given as follows:(27)u2=24α22τ12+τ22+2eβ3y+β3δ1z/β1+∫−2β3δ1ϕ4tβ1δ2−β2δ1/β12α2dt−48τ22α22τ12+τ22+2eβ3y+β3δ1z/β1+∫−2β3δ1ϕ4tβ1δ2−β2δ1/β12α2dt2,(28)τ1=β1y+δ1z+∫−2δ1ϕ4tβ1δ2−β2δ1α2β1dt,τ2=α2x+β2y+δ2z+∫−α22β12ϕ2t+β12δ22ϕ4t−β22δ12ϕ4tβ12α2dt.Ifτ12+τ22+expτ3+expτ4+expτ3+τ4+ε5t⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ1=1,δ2=2,α2=1,β1=2,β2=3,β3=4,ϕ2t=cost,ϕ4t=sint,x=1,y=1, then plots of equation (27) are plotted in Figure 3.Figure 3
Plots of interaction lump with two solitons (27).Type III(29)ε1t=∫2δ1ϕ4tβ2δ4−β4δ2α2β4dt,ε2t=∫−α22β42ϕ2t−β22δ42ϕ4t+β42δ22ϕ4tβ42α2dt,ε5t=0,ε3t=∫2δ3ϕ4tβ2δ4−β4δ2α2β4dt,ε4t=∫2δ4ϕ4tβ2δ4−β4δ2α2β4dt,α1=α3=α4=0,α2=α2,β1=β4δ1δ4,β2=β2,β3=δ3β4δ4,β4=β4,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ1t=0,ϕ3t=−δ42ϕ4tβ42.The solutions are given as follows:(30)u3=24α22τ12+τ22+F1+F2+F3−48τ22α22τ12+τ22+F1+F2+F32,(31)τ1=β4δ1yδ4+δ1z+∫2δ1ϕ4tβ2δ4−β4δ2α2β4dt,τ2=α2x+β2y+δ2z+∫−α22β42ϕ2t−β22δ42ϕ4t+β42δ22ϕ4tβ42α2dt,(32)F1=eδ3β4y/δ4+δ3z+∫2δ3ϕ4tβ2δ4−β4δ2/α2β4dt,F2=eβ4y+δ4z+2δ4tϕ4tβ2δ4−β4δ2/α2β4,(33)F3=eδ3β4y/δ4+δ3z+∫2δ3ϕ4tβ2δ4−β4δ2/α2β4dt+β4y+δ4z+2δ4tϕ4tβ2δ4−β4δ2/α2β4.By selecting the parametersδ1=1,δ2=2,δ3=3,δ4=4,α2=1,β2=3,β4=4,ϕ2t=cost,ϕ4t=sint,x=1,y=1, then plots of equation (30) are plotted in Figure 4.Figure 4
Plots of interaction lump with two solitons (30).Type IV(34)ε1t=−ε2tδ2δ1,ε3t=∫−α34ϕ1t+α32ϕ2t+β32ϕ3tα3dt,ε4t=0,ε5t=−2ε2tδ1ε2t−δ2ε1tδ1,α1=α2=α4=0,α3=α3,β1=β2=β4=0,β3=β3,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ4t=0.The solutions are given as follows:(35)u4=12α32F1+12α32F2δ1z−ε2tδ2/δ12+δ2z+ε2t2+F1+eδ4z+F2−2ε2tδ1ε2t−δ2ε1t/δ1,(36)12α3F1+α3F22δ1z−ε2tδ2/δ12+δ2z+ε2t2+F1+eδ4z+F2−2ε2tδ1ε2t−δ2ε1t/δ12,(37)F1=eα3x+β3y+δ3z+∫−α34ϕ1t+α32ϕ2t+β32ϕ3t/α3dt,F2=eα3x+β3y+δ3z+∫−α34ϕ1t+α32ϕ2t+β32ϕ3t/α3dt+δ4z.By selecting the parametersδ1=1,δ2=2,δ3=3,δ4=4,α2=1,β2=3,β4=4,ϕ2t=cost,ϕ4t=sint,x=1,y=1, then plots of equation (35) are plotted in Figure 5.Figure 5
Plots of interaction lump with two solitons (35).Type V(38)ε1t=−ε2tδ2δ1,ε3t=∫−α3α32ϕ1t+ϕ2tdt,ε4t=0,ε5t=∫−2d/dtε2tδ1ε2t−δ2ε1tδ1dt,α1=α2=α4=0,α3=α3,β1=β2=β4=0,β3=β3,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ3t=ϕ4t=0.The solutions are given as follows:(39)u5=12∂2∂x2lnF5,F5=δ1z+ε1t2+δ2z+ε2t2+eα3x+β3y+δ3z+ε3t+eβ4y+δ4z+ε4t+eα3x+β3y+δ3z+ε3t+β4y+δ4z+ε4t+ε5t.Type VI(40)ε3t=∫−α34ϕ1t+α32ϕ2t+δ32ϕ3tα3dt,ε2t=ε4t=0,ε5t=−ε1t2,α1=α2=α4=0,α3=α3,β1=0,β2=β2,β3=β3,β4=β4,δ1=δ2=δ4=0,δ3=δ3,ϕ3t=0.The solutions are given as follows:(41)u5=12∂2∂x2lnF6,F6=β2y+ε2t2+eα3x+β3y+δ3z+ε3t+eβ4y+ε4t+eα3x+β3y+δ3z+ε3t+β4y+ε4t.Type VII(42)ε3t=∫−α3α32ϕ1t+ϕ2tdt,ε1t=ε2t=ε4t=ε5t=0,α1=α2=α4=0,α3=α3,β1=0,β2=β2,β3=β3,β4=β4,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ3t=ϕ4t=0.The solutions are given as follows:(43)u7=12∂2∂x2lnF7,F7=z2δ12+β2y+δ2z2+eα3x+β3y+δ3z+∫−α3α32ϕ1t+ϕ2tdt+eβ4y+δ4z+eα3x+β3y+δ3z+∫−α3α32ϕ1t+ϕ2tdt+β4y+δ4z.Type VIII(44)ε2t=∫−2δ2ϕ4tβ2δ3−β3δ2α3β2dt,ε3t=∫−α34β22ϕ1t+α32β22ϕ2t+β22δ32ϕ4t−β32δ22ϕ4tβ22α3dt,ε4t=∫−2β2δ3−β3δ2δ4ϕ4tα3β2dt,ε5t=−ε12t,α1=α2=α4=0,α3=α3,β1=0,β2=β2,β3=β3,β4=β2δ4δ2,δ1=δ1,δ2=δ2,δ3=δ3,δ4=δ4,ϕ3t=−δ22ϕ4tβ22.The solutions are given as follows:(45)u8=12∂2∂x2lnF8,F8=ε1t2+β2y+δ2z+ε2t2+eα3x+β3y+δ3z+ε3t+eβ2δ4y/δ2+δ4z+ε4t+eα3x+β3y+δ3z+ε3t+β2δ4y/δ2+δ4z+ε4t+ε5t.Type IX(46)ε1t=−ε2tβ2β1,ε3t=∫−α3α32ϕ1t+ϕ2tdt,ε4t=0,ε5t=∫−2d/dtε2tβ1ε2t−β2ε1tβ1dt,α1=α2=α4=0,α3=α3,β1=β1,β2=β2,β3=β3,β4=β4,δ1=δ1,δ2=β2δ1β1,δ3=δ3,δ4=δ4,ϕ3t=ϕ4t=0.The solutions are given as follows:(47)u9=12∂2∂x2lnF9,F9=β1y+δ1z+ε1t2+β2y+β2δ1zβ1+ε2t2+eα3x+β3y+δ3z+ε3t+eβ4y+δ4z+ε4t+eα3x+β3y+δ3z+ε3t+β4y+δ4z+ε4t+ε5t.Type X(48)ε1t=∫−2β13δ1δ3ϕ4t−2β12β3δ12ϕ4t+2β1β22δ1δ3ϕ4t−2β22β3δ12ϕ4t+d/dtε2tα3β12β2α3β13dt,ε3t=∫−α34β12ϕ1t+α32β12ϕ2t+β12δ32ϕ4t−β32δ12ϕ4tα3β12dt,ε4t=∫−2β1δ3−β3δ1δ4ϕ4tα3β1dt,ε5t=∫−2ε2tβ14α3+∫S1+d/dtε2tα3β12β2dtβ2S2+d/dtε2tα3β12α32β16dt,S1=2δ1ϕ4tβ12+β22β1δ3−β3δ1,S2=2β2δ1ϕ4tβ1δ3−β3δ1,α1=α2=α4=0,β4=β1δ4δ1,δ1=δ1,δ2=β2δ1β1,δ3=δ3,δ4=δ4,ϕ3t=−δ12ϕ4tβ12.The solutions are given as follows:(49)u10=12∂2∂x2lnF10,F10=β1y+δ1z+ε1t2+β2y+β2δ1zβ1+ε2t2+eα3x+β3y+δ3z+ε3t+eβ1δ4y/δ1+δ4z+ε4t+eα3x+β3y+δ3z+ε3t+β1δ4y/δ1+δ4z+ε4t+ε5t.By selecting the parametersδ1=1,δ3=3,δ4=4,α3=1,β1=4,β2=2,β3=3,ϕ1t=sin2t,ϕ2t=cos2t,ϕ4t=cos3t,ε2t=expt,x=1,y=1, then plots of equation (49) are plotted in Figure 6. And also, by choosing the parameters δ1=1,δ3=3,δ4=4,α3=1,β1=4,β2=2,β3=3,ϕ1t=exptsint,ϕ2t=exptcos2t,ϕ4t=exptcos3t,ε2t=exp2t,x=1,y=1, then plots of equation (49) are plotted in Figure 7. Moreover, by choosing the parameters δ1=1,δ3=3,δ4=4,α3=1,β1=4,β2=2,β3=3,ϕ1t=1/1+sint,ϕ2t=cost/2+sint,ϕ4t=cost,ε2t=sin2t+tant,x=1,y=1, then plots of equation (49) are plotted in Figure 8.Figure 6
Plots of interaction lump with two solitons (49).Figure 7
Plots of interaction lump with two solitons (49).Figure 8
Plots of interaction lump with two solitons (49).


## 4.2. Interaction between Two Lump Solutions

Here, we offer interaction between two lump solutions containing combination of two functions for the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(50)g=τ14+τ22+τ32+ε4t,τj=αjx+βjy+δjz+εjt,j=1,2,3,ε4t>0.The valuesαi,βi,δi,εiti=1,2,3 are real constants to be computed. By appending (50) into (22), we receive to a system containing 42 nonlinear equations. To solve the nonlinear system, the determined coefficients will be got as the below cases.Type I(51)ε1t=C1−ε4t4,ε2t=∫−2β2β3ϕ3t+δ2δ3ϕ4tα3dt+C2,α1=α2=0,α3=α3,ε3t=∫−α32ϕ2t−ϕ3tβ22+β32ϕ3t−δ22ϕ4t+δ32ϕ4tα3dt+C3,β1=0,β2=β2,β3=β3,δ1=0,ϕ1t=−13r14+r4δ22ϕ4t+ϕ3tβ22α34.The solutions are given as follows:(52)u1=12∂2∂x2lnF1,F1=ε1t4+β2y+δ2z+ε2t2+α3x+β3y+δ3z+ε3t2+ε4t.Ifτ14+τ22+τ32+ε4t⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ2=2,δ3=3,α3=3,β2=2,β3=1,ϕ2t=t,ϕ3t=t2,ϕ4t=1+t,x=1,z=1, then plots of equation (52) are plotted in Figure 9. And also, by selecting the parameters δ2=2,δ3=3,α3=3,β2=2,β3=1,ϕ2t=cost,ϕ3t=sint,ϕ4t=cos2t,x=1,z=1, then plots of equation (52) are plotted in Figure 10. Moreover, by selecting the parameters δ2=2,δ3=3,α3=3,β2=2,β3=1,ϕ2t=t2sin2t,ϕ3t=tsin3t,ϕ4t=tcos3t,x=1,z=1, then plots of equation (52) are plotted in Figure 11.Figure 9
Plots of interaction with two lumps (52).Figure 10
Plots of interaction with two lumps (52).Figure 11
Plots of interaction with two lumps (52).Type II(53)ε1t=−−C1α23+∫Ftdtα24α2,α1=β1=δ1=ϕ1t=0,δ3=α3δ2α2,Ft=2ϕ2tα22α3+ϕ3tα3β22+ϕ4tα3δ22+ddtε3tα22ε3tα2−ε2tα3+ddtε4tα23,ε2t=∫−α22+α32δ22ϕ4t+ϕ3tβ22+ϕ2tα22+d/dtε3tα22α3α23dt+C2.The solutions are given as follows:(54)u2=12∂2∂x2lnF2,F2=ε1t4+α2x+β2y+δ2z+ε2t2+α3x+α3β2yα2+α3δ2zα2+ε3t2+ε4t.Type III(55)ε1t=∫−2ϕ4tδ1β1δ3−β3δ1α3β1dt,ε3t=∫−α32β12ϕ2t+β12δ32ϕ4t−β32δ12ϕ4tα3β12dt,ε4t=−ε2t2+C1,α1=α2=β2=δ2=ϕ1t=0,ϕ3t=−δ12ϕ4tβ12.The solutions are given as follows:(56)u3=12∂2∂x2lnF3,F3=β1y+δ1z+ε1t4+ε2t2+α3x+β3y+δ3z+ε3t2+ε4t.


## 4.3. Interaction between Two Lumps-Soliton Solutions

Here, we offer interaction between two lumps-soliton solutions containing combination of two functions for the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(57)g=τ14+τ22+τ32+expτ4+ε5t,τj=αjx+βjy+δjz+εjt,j=1,2,3,4,ε4t>0.The valuesαi,βi,δi,εiti=1,2,3,4 are real constants to be computed. By putting (57) into (22), we receive a system containing 119 nonlinear equations. To solve the nonlinear system, the determined coefficients will be got as the below cases.Type I(58)ε1t=A1t+A2tα24α2,A1t=2α32β22∫ε2tϕ3tdt−2α3∫ε3tϕ2tdtα23−2α3β22∫ε3tϕ3tdtα2,A2t=2α32∫ϕ2tε2tdtα22+2α3∫ε2tddtε3tdtα22−ε3t2α23−ε5tα23+C1α23,ε2t=∫−α22+α32ϕ2tα22+ϕ3tβ22+d/dtε3tα22α3α23dt+C2,ε4t=0,α1=α4=β1=β4=δ1=ϕ1t=ϕ4t=0,β3=α3β2α2,δ3=α3δ2α2.The solutions are given as follows:(59)u1=12∂2∂x2lnF1,F1=ε1t4+α2x+β2y+δ2z+ε2t2+α3x+α3β2yα2+α3δ2zα2+ε3t2+eδ4z+ε4t+ε5t.Ifτ14+τ22+τ32+expτ4+ε5t⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ2=1,δ4=2,α2=1,α3=2,β2=3,ϕ2t=cost,ϕ3t=sint,ε2t=cos2t,ε3t=sin2t,x=1,y=1, then plots of equation (59) are plotted in Figure 12. And also, by selecting the parameters δ2=1,δ4=2,α2=1,α3=2,β2=3,ϕ2t=cos1+t,ϕ3t=sin1+t,ε2t=cos1+3t,ε3t=sin1+3t,x=1,y=1, then plots of equation (52) are plotted in Figure 13.Figure 12
Plots of interaction with two lumps-soliton (59).Figure 13
Plots of interaction with two lumps-soliton (59).Type II(60)ε1t=A1t+A2t+A3tα42α234α4α2,A1t=2α32β42∫ε2tϕ3tdt−2α3∫ε3tϕ2tdtα42α2−2α3β42∫ε3tϕ3tdtα2,A2t=2α32∫ϕ2tε2tdtα42+2α32δ42∫ϕ4tε2tdt−2α3δ42∫ϕ4tε3tdtα2,A3t=2α3∫ε2tddtε3tdtα42−ε3t2α42α2−ε5tα42α2+C1α42α2,ε2t=∫−α22+α32α42ϕ2t+β42ϕ3t+δ42ϕ4t+d/dtε3tα3α42α42α2dt,ε4t=∫−α42ϕ2t+β42ϕ3t+δ42ϕ4tα4dt,α1=β1=δ1=ϕ1t=0,β2=α2β4α4,β3=α3β4α4,δ2=α2δ4α4,δ3=α3δ4α4.The solutions are given as follows:(61)u2=12∂2∂x2lnF2,(62)F2=ε1t4+α2x+α2β4yα4+α2δ4zα4+ε2t2+α3x+α3β4yα4+α3δ4zα4+ε3t2+eα4x+β4y+δ4z+ε4t+ε5t.By selecting the parametersδ4=2,α2=1,α3=2,α4=4,β4=3,ϕ2t=cost,ϕ3t=sint,ϕ4t=sin2t,ε2t=cos2t,ε3t=sin2t,ε4t=sin3t,C1=1,C2=2,x=1,y=1, then plots of equation (61) are plotted in Figure 14. And also, by selecting the parameters δ4=2,α2=1,α3=2,α4=4,β4=3,ϕ2t=cost,ϕ3t=sint,ϕ4t=sin2t,ε2t=t,ε3t=t2,ε4t=t3,C1=1,C2=2,x=1,y=1, then plots of equation (54) are plotted in Figure 15.Figure 14
Plots of interaction with two lumps-soliton (61).Figure 15
Plots of interaction with two lumps-soliton (61).


## 4.4. Lump-Periodic Solutions

In this section, we would like to present the general solutions of the including combination of two functions for the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(63)g=τ12+τ22+cosτ3+ε4t,τj=αjx+βjy+δjz+εjt,j=1,2,3,ε4t>0.The valuesαi,βi,δi,εiti=1,2,3 are real constants to be computed. By substituting (63) into (22), we obtain a system containing 24 nonlinear equations. To solve the nonlinear system, the determined coefficients will be got as the below cases.Type I(64)ε1t=−ε2tα2α1−α2α1∫α12ϕ2tα2+α2ϕ2tdt+C1,ε3t=0,ε4t=−α12+α22∫ϕ2tdt2α22+ε2t2α12−2ε2tα2α12∫ϕ2tα12+α22α2dtα2−C1α1−2∫−ϕ2tC1α1α22dtα12+C2,α3=ϕ1t=ϕ3t=0,β2=α2β1α1,δ2=α2δ1α1.The solutions are given as follows:(65)u1=12∂2∂x2lnF1,F1=α1x+β1y+δ1z+ε1t2+α2x+α2β1yα1+α2δ1zα1+ε2t2+cosβ3y+δ3z+ε3t+ε4t.From above, we can discover thatg is positive if ε4t>0, then in all space directions, u will be localized. Therefore, if τ12+τ22+cosτ3⟶∞, the lump solutions u⟶0 at any t. By selecting the parameters δ1=1,δ3=3,α1=2,α2=4,β1=2,β3=3,ϕ2t=−2,ε2t=3,C1=1,C2=2,x=1,y=1, then plots of equation (65) are plotted in Figure 16.Figure 16
Plots of lump-periodic (65).Type II(66)ε1t=∫−α12+α22α12ϕ2+δ12ϕ4+d/dtε2tα12α2α13dt+C1,ε3t=0,ε4t=∫2ε1tα2−ε2tα1ϕ2tα12α2+ϕ4tα2δ12+d/dtε2tα12α13dt+C2,α3=δ3=ϕ1t=ϕ3t=0,β2=α2β1α1,δ2=α2δ1α1.The solutions are given as follows:(67)u2=12∂2∂x2lnF2,F2=α1x+β1y+δ1z+ε1t2+α2x+α2β1yα1+α2δ1zα1+ε2t2+cosβ3y+ε3t+ε4t.Type III(68)ε1t=∫−2δ1ϕ4tβ1δ2−β2δ1α2β1dt,ε2t=∫−α22β12ϕ2t+β12δ22ϕ4t−β22δ12ϕ4tα2β12dt,ε3t=∫−2δ3ϕ4tβ1δ2−β2δ1α2β1dt,ε4t=0,β3=β1δ3δ1,ϕ3t=−δ12ϕ4tβ12,α1=α3=ϕ1t=0,β2=α2β1α1,δ2=α2δ1α1.The solutions are given as follows:(69)u3=12∂2∂x2lnF3,F3=β1y+δ1z+ε1t2+α2x+β2y+δ2z+ε2t2+cosβ1δ3yδ1+δ3z+ε3t+ε4t.By selecting the parametersδ1=1,δ2=2,δ3=3,α2=4,β1=2,β2=3,ϕ2t=sint,ϕ4t=cost,x=1,y=1, then plots of equation (69) are plotted in Figure 17.Figure 17
Plots of lump-periodic (69).Type IV(70)ε2t=−tanα2∫ϕ2tε1tdt+C1ε1,ε4t=∫2ε1t2+ε2t2α2ϕ2tε2t−d/dtε1tε1tε1t2dt+C2,ε3t=0,α1=−ε2tα2ε1t,α3=β1=β2=δ1=δ2=ϕ1t=0,ϕ3t=−δ32ϕ4tβ32.The solutions are given as follows:(71)u4=12∂2∂x2lnF4,F4=−ε2tα2xε1t+ε1t2+α2x+ε2t2+cosβ3y+δ3z+ε3t+ε4t.Type V(72)ε2t=−tan1α2δ32∫α22δ32ϕ2t−β32δ22ϕ3tε1tdtε1t,ε3t=∫2β32δ2ϕ3tδ3α2dt,ε4t=∫2ε1t2+ε2t2ϕ2tε2tα22δ32−ϕ3tε2tβ32δ22−d/dtε1tα2δ32ε1tα2δ32ε1t2dt,α3=β1=β2=ϕ1t=0,α1=−α2ε2tε1t,δ1=−δ2ε2tε1t,ϕ4t=−β32ϕ3tδ32.The solutions are given as follows:(73)u5=12∂2∂x2lnF5,F5=−α2ε2txε1t−δ2ε2tzε1t+ε1t2+α2x+δ2z+ε2t2+cosβ3y+δ3z+ε3t+ε4t.Type VI(74)ε3t=0,ε4t=−ε1t2−ε2t2,α3=β1=β2=ϕ1t=ϕ3t=0,α1=−α2ε2tε1t,δ1=−δ2ε2tε1t,ϕ4t=−β32ϕ3tδ32,ϕ4t=−α2ϕ2tε1t2+ε2t2+d/dtε1tε1tε2t−d/dtε2tε1t2α2ε1t2+ε2t2δ22.The solutions are given as follows:(75)u6=12∂2∂x2lnF6,F6=−α2ε2txε1t−δ2ε2tzε1t+ε1t2+α2x+δ2z+ε2t2+cosβ3y−ε1t2−ε2t2.Type VII(76)ε1t=∫−ϕ2tα12+α22+d/dtε2tα2α1dt+C1,ε3t=0,ε4t=∫−2α2ϕ2tα1ε2t−α2ε1t+d/dtε2tα1ε2t−α2ε1tα1dt+C2,α3=β1=β2=β3=ϕ1t=ϕ4t=0,δ2=α2δ1α1.The solutions are given as follows:(77)u7=12∂2∂x2lnF7,F7=α1x+δ1z+ε1t2+α2x+α2δ1zα1+ε2t2+cosδ3z+ε3t+ε4t.By selecting the parametersδ1=2,α1=1,α2=2,β3=1,ϕ2t=t,ϕ4t=t2,ε1t=t,ε2t=1+t,ε3t=t2,x=1,y=1, then plots of equation (77) are plotted in Figure 18. And also, by selecting the parameters δ1=2,α1=1,α2=2,β3=1,ϕ2t=sint,ϕ4t=cost,ε1t=t,ε2t=sint,ε3t=cost,x=1,y=1, then plots of equation (77) are plotted in Figure 19. Moreover, by selecting the parameters δ1=2,α1=1,α2=2,β3=1,ϕ2t=1,ϕ4t=2,ε1t=t,ε2t=t,ε3t=t2,x=1,y=1, then plots of equation (77) are plotted in Figure 20. And finally, by selecting the parameters δ1=2,α1=1,α2=2,β3=1,ϕ2t=expt,ϕ4t=exp2t,ε1t=sinht,ε2t=cosht,ε3t=t2,x=1,y=1, then plots of equation (77) are plotted in Figure 21.Figure 18
Plots of lump-periodic (77).Figure 19
Plots of lump-periodic (77).Figure 20
Plots of lump-periodic (77).Figure 21
Plots of lump-periodic (77).


## 4.5. Lump-Three Kink Solutions

Here, we present lump-periodic solutions containing combination of two functions for the (3+1)-dimensional variable-coefficient nonlinear wave equation through utilizing the bilinear method as the below frame:
(78)g=τ12+τ22+expτ3+expτ4+expτ5+ε6t,τj=αjx+βjy+δjz+εjt,j=1,2,3,4,5,ε6t>0.The valuesαi,βi,δi,εiti=1,⋯,5 are real constants to be computed. By appending (78) into (22), we receive a system containing 43 nonlinear equations. To solve the nonlinear system, the determined coefficients will be got as the below cases.Type I(79)ε1t=∫−2β5δ1ϕ3tβ2δ5−β5δ2δ52α2dt,ε2t=∫−α22δ52ϕ2t+β22δ52ϕ3t−β52δ22ϕ3tδ52α2dt,ε3t=∫−2β5δ3ϕ3tβ2δ5−β5δ2δ52α2dt,ε4t=∫−2β5δ4ϕ3tβ2δ5−β5δ2δ52α2dt,ε5t=∫−2β5ϕ3tβ2δ5−β5δ2α2δ5dt,ε6t=0,α1=α3=α4=α5=ϕ1t=0,β1=δ1β5δ5,β3=β5δ3δ5,β4=β5δ4δ5,ϕ4t=−β52ϕ3tδ52.The solutions are given as follows:(80)u1=12∂2∂x2lnF1,F1=δ1β5yδ5+δ1z+ε1t2+α2x+β2y+δ2z+ε2t2+eβ5δ3y/δ5+δ3z+ε3t+eβ5δ4y/δ5+δ4z+ε4t+eβ5y+δ5z+ε5t+ε6t.Type II(81)ε1t=∫−α12+α22α42ϕ2t+β42ϕ3t+δ42ϕ4t+d/dtε2tα2α42α1α42dt,ε3t=∫−α3α42ϕ2t+β42ϕ3t+δ42ϕ4tα42dt,ε4t=∫−α42ϕ2t+β42ϕ3t+δ42ϕ4tα4dt,ε5t=∫−α5α42ϕ2t+β42ϕ3t+δ42ϕ4tα42dt,ϕ1t=0,βi=αiβ4α4,δi=αiδ4α4,i=1,2,3,5,ε6t=∫−2α2α42ϕ2t+β42ϕ3t+δ42ϕ4tα1ε2t−α2ε1t+d/dtε2tα42α1ε2t−α2ε1tα1α42dt.The solutions are given as follows:(82)u2=12∂2∂x2lnF2,F2=α1x+α1β4yα4+α1δ4zα4+ε1t2+α2x+α2β4yα4+α2δ4zα4+ε2t2+eα3x+α3β4y/α4+α3δ4z/α4+ε3t+eα4x+β4y+δ4z+ε4t+eα5x+α5β4y/α4+α5δ4z/α4+ε5t+ε6t.Type III(83)ε1t=∫−ϕ2tα12+α22α1dt−ε2tα2α1,ε3t=∫−α3ϕ2tdt,ε4t=∫−α4ϕ2tdt,ε5t=∫−α5ϕ2tdt,ε6t=∫−2α1ε2t−α2ε1tα2ϕ2t+d/dtε2tα1dt,β2=α2β1α1,β3=β1δ3δ1,δ2=α2δ1α1,δ4=β4δ1β1,δ5=δ1β5β1,ϕ1t=0,ϕ4t=−β12ϕ3tδ12.The solutions are given as follows:(84)u3=12∂2∂x2lnF3,F3=α1x+β1y+δ1z+ε1t2+α2x+α2β1yα1+α2δ1zα1+ε2t2+eα3x+β1δ3y/δ1+δ3z+ε3t+eα4x+β4y+β4δ1z/β1+ε4t+eα5x+β5y+δ1β5z/β1+ε5t+ε6t.From above, we can discover thatg is positive if ε6t>0, then in all space directions u will be localized. Therefore, if τ12+τ22+expτ3+expτ4+expτ5+ε6t⟶∞, the lump-three kink solutions
(85)u⇒−12α32,α3>α4>α5,0,α4>α3>α3,0,α5>α3>α4,0,α4>α5>α3,at any t. By selecting the parameters α1=1,α2=2,α3=3,α4=4,α5=5,β1=1,β4=3,β5=4,δ1=1,δ3=1.5,ϕ2t=cost,ε1t=sin2t,ε2t=cos2t,x=1,y=1, then plots of equation (84) are plotted in Figure 22. And also, via selecting the parameters α1=1,α2=2,α3=3,α4=4,α5=5,β1=1,β4=3,β5=4,δ1=1,δ3=1.5,ϕ2t=1+t+t2,ε1t=1/21+t2,ε2t=1/31+t3,x=1,y=1, then plots of equation (84) are plotted in Figure 23.Figure 22
Plots of lump-three kink solutions (84).Figure 23
Plots of lump-three kink solutions (84).Type IV(86)ε1t=∫−α12+α22α12δ42ϕ2t−β42δ12ϕ3tα13δ42dt−ε2tα2α1,ε4t=∫2β42ϕ3tδ1α1δ4dt,ε3t=∫−α12α3δ42ϕ2t−β42δ1ϕ3t2α1δ3−α3δ1α12δ42dt,ε5t=∫−α12α5δ42ϕ2t−β42δ1ϕ3t2α1δ5−α5δ1α12δ42dt,ε6t=∫−2α1ε2t−α2ε1tϕ2tα12α2δ42−ϕ3tα2β42δ12+d/dtε2tα12δ42α13δ42dt,β3=α1δ3−α3δ1β4α1δ4,β5=β4α1δ5−α5δ1α1δ4,δ2=α2δ1α1,ϕ4t=−β42ϕ3tδ42,α4=β1=β2=ϕ1t=0.The solutions are given as follows:(87)u4=12∂2∂x2lnF4,F4=α1x+δ1z+ε1t2+α2x+α2δ1zα1+ε2t2+eα3x+α1δ3−α3δ1β4y/δ4α1+δ3z+ε3t+eβ4y+δ4z+ε4t+eα5x+β4α1δ5−α5δ1y/δ4α1+δ5z+ε5t+ε6t.By selecting the parametersα1=1,α2=2,α3=3,α5=5,β4=3,δ1=1,δ3=1.5,δ4=2,δ5=3,ϕ2t=t,ϕ3t=t2,ε2t=t4,x=1,y=1, then plots of equation (87) are plotted in Figure 24. And also, via selecting the parameters α1=1,α2=2,α3=3,α5=5,β4=3,δ1=1,δ3=1.5,δ4=2,δ5=3,ϕ2t=1,ϕ3t=2,ε2t=3,x=1,y=1, then plots of equation (87) are plotted in Figure 25. Moreover, via selecting the parameters α1=1,α2=2,α3=3,α5=5,β4=3,δ1=1,δ3=1.5,δ4=2,δ5=3,ϕ2t=sin3t,ϕ3t=cos3t,ε2t=sint+cost,x=1,y=1, then plots of equation (87) are plotted in Figure 26.Figure 24
Plots of lump-three kink solutions (87).Figure 25
Plots of lump-three kink solutions (87).Figure 26
Plots of lump-three kink solutions (87).


## 5. Conclusion

In this paper, under the multidimensional binary Bell polynomials, we derive that the lump-soliton and its interaction solutions of the (3+1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles has been successfully obtained. And then, ten classes for interaction between a lump-two kink solitons, three classes for interaction between two lumps, two classes for interaction between two lumps-soliton, seven classes for lump-periodic, and four classes for lump-three kink solutions were successfully constructed by employing the bilinear scheme. With the combination of the binary Bell polynomials and Hirota’s bilinear method, some concrete solutions are expressed in terms of determined functions. We illustrate the regularity of solutions with some parameter constrains on constant and rational and periodic backgrounds. We also consider the classical dynamical behaviors and the elasticity of the collisions of soliton solutions and their interactions when plotting them. Finally, we conclude two dynamical behaviors of general lump-periodic and multikink soliton solutions.



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