# Investigating One-, Two-, and Triple-Wave Solutions via Multiple Exp-Function Method Arising in Engineering Sciences

Authors: Pengbo Wan; Jalil Manafian; Hajar Farhan Ismael; Sizar Abid Mohammed
Journal: Advances in Mathematical Physics
 (2020)

---

## Abstract

The multiple Exp-function method is employed for searching the multiple soliton solutions for the new extended (3+1)-dimensional Jimbo-Miwa-like (JM) equation, the extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (eCBS) equation, the generalization of the (2+1)-dimensional Bogoyavlensky-Konopelchenko (BK) equation, and a variable-coefficient extension of the DJKM (vDJKM) equation, which contain one-soliton-, two-soliton-, and triple-soliton-kind solutions. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.


---

## Body

## 1. Introduction

Many nonlinear physical phenomena arise in various fields of engineering and science such as fluid dynamics, nuclear reactor dynamics, hydrodynamics, molecular biology, quantum mechanics, plasma physics, optical fibres, and solid state physics. To describe these complex physical phenomena, nonlinear differential equations play a significant role. Therefore, obtaining the solutions of these nonlinear equations is a topic of great interest in the study of many fields of science. To better understand the working of a physical problem, a mathematical model was brought into the picture in the form of nonlinear PDEs. The solutions of partial differential equations give the detailed summary about the nature of phenomena involved. Many numerical and analytical methods have been derived to deal with these kinds of scientific problems. We need to adopt an effective and powerful method to investigate such type of mathematical model which gives the solutions upholding physical reality. In most of the analytic techniques, linearization of the system is the main topic to focus on, and also, it is assumed that the nonlinearities are relatively insignificant. Sometimes, these assumptions make a strong effect on the solutions in respect to the real physics of the phenomena involved. Thus, finding the solutions of nonlinear ODEs and PDEs is still a significant problem. For this, we need new techniques to develop approximate and exact solutions. Several analytical and numerical techniques have been formulated for tackling these types of nonlinear models, including the Exp-function method [1, 2], the homotopy analysis method [3], the homotopy perturbation method [4], the (G′/G)-expansion method [5], the improved tanϕ/2-expansion method ([6–8]), the Hirota bilinear method ([9–13]), the He variational principle [14, 15], the binary Darboux transformation [16], the Lie group analysis [17, 18], and the Bäcklund transformation method [19]. Moreover, many powerful methods have been used to investigate the new properties of mathematical models which are symbolizing serious real world problems ([20–24]). This study is aimed at investigating the following four forms of nonlinear PDEs:
(i)
Extended (3+1)-dimensional Jimbo-Miwa-like (JM) equations ([25–34]) is given as
(1)Ψxxxy+3ΨxxΨy+3ΨxΨxy+2Ψyt−3Ψxz+Ψyz+Ψzz=0,and
(2)Ψxxxy+3ΨxxΨy+3ΨxΨxy−3Ψxz+2Ψxt+Ψyt+Ψzt=0,explains some (3+1)-dimensional waves in physics [25, 26], in which Ψx,y,z,t is a function including spaces x,y,z and time t. In addition, this equation cannot pass certain conventional integrability tests [27]. Qi et al. [32] studied the extended (3+1)-dimensional Jimbo-Miwa-like equation via the Hirota method and obtained the solitary-wave solutions and novel exact soliton solutions.(ii)
The extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (eCBS) equation [35] is given as
(3)Ψxt+Ψxxxy+6ΨxxΨy+6ΨxΨxy+δ1Ψxy+δ2Ψxx+δ3Ψxxxx+12ΨxΨxx=0,where δ1,δ2, and δ3 are arbitrary values. The eCBS equation will be changed to a (2+1)-dimensional CBS equation when δ1=δ2=δ3=0 [36]. Ren et al. [35] investigated the extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff-like equation by using the generalized bilinear operators based on a prime number p=3.(iii)
The generalization of the (2+1)-dimensional Bogoyavlensky-Konopelchenko (BK) equation [37] is given as
(4)Ψxt+αΨxxxx+6ΨxΨxx+βΨxxxy+3ΨxxΨy+3ΨxΨxy+δ1Ψxx+δ2Ψxy+δ3Ψyy=0.The (3+1)-dimensional VC B-type KP equation is extended from the KP equation and can explain some notable (3+1)-dimensional waves in fluid dynamics [37]. Also, Zhang and Pang [38] found the lump and lump-type solutions through the Hirota bilinear form for the variable-coefficient B-type Kadomtsev-Petviashvili equation.(iv)
The variable-coefficient extension of the DJKM (vDJKM) equation ([39–48]) is given as
(5)Ψxxxxy+4ΨxΨxxy+2ΨyΨxxx+6ΨxxΨxy−αΨyyy−2βgtΨxxt+htΨxxy=0,where the wave amplitude ux,y,t is a function of variables and x,y,t, gt, and ht are functions of t. The vD JKM equation (6) reduces to the DJKM equation when gt=1 and ht=0 as follows:
(6)Ψxxxxy+4ΨxΨxxy+2ΨyΨxxx+6ΨxxΨxy−αΨyyy−2βΨxxt=0.The soliton solutions to a few (3+1)-dimensional generalized nonlinear integrable equations have been constructed. The adopted approach is the multiple Exp-function method, which was presented earlier in the valuable work by Ma et al. [49]. Recently, special kinds of reductions of soliton solutions to rational functions that are being actively studied are lump solutions to nonlinear partial differential equations by Ma and Zhou [50] and their interactions with solitons to the Hirota-Satsuma-Ito equation in (2+1)-dimensions by Ma [51], even for linear PDEs by the same author [52]. Cordero et al. constructed the stability analysis of the fourth-order iterative method for finding multiple roots of nonlinear equations [53]. Gao et al. obtained the optical soliton solutions of the cubic-quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equations with the parabolic law [54]. The truncated Painlevé expansion was employed to derive a Bäcklund transformation of a (2+1)-dimensional nonlinear system by Zhao and Han [55]. Also, Gao et al. studied on the conformable (2+1)-dimensional Ablowitz-KaupNewell-Segur equation in order to show the existence of complex combined dark-bright soliton solutions [56]. The same authors presented the nonlinear Zoomeron equation by using the newly extended direct algebraic technique [57]. The instability modulation for the (2+1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr’s media by utilizing the modified auxiliary expansion method have been studied in [58]. In [59], the authors found new complex solitons to the perturbed nonlinear Schrödinger model with the help of an analytical method. In this paper, we will study the multiple Exp-function method for determining the multiple soliton solutions. The multiple Exp-function method used by some of the powerful authors for various nonlinear equations include the nonlinear evolution equations [60], the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation [61], the generalized (1+1)-dimensional and (2+1)-dimensional Ito equations [62], a new generalization of the associated Camassa-Holm equation [63], the (3+1)-dimensional generalized KP and BKP equations [64], and the new (2+1)-dimensional Korteweg-de Vries equation [65]. In [65], Liu et al. utilized the multiple Exp-function method for the most well-known equation, namely, the Korteweg-de Vries (KdV) equation, and gained one-soliton-, two-soliton-, and three-soliton-type solutions with interpretations for the obtained soliton solutions. Moreover, they analyzed and illustrated the propagation and interaction of some soliton solutions by selecting appropriate values. Also, the elasticity and being unchanged for some solutions when interacting among three solitons and after the collision were investigated.The rest of this paper is structured as follows: the multiple Exp-function scheme is summarized in Section2. In Sections 3–7, the extended JM equations, the extended CBS equation, the generalized BK equation, and the vDJKM equation, respectively, will be investigated to find one-soliton, two-soliton, and triple-soliton solutions. In the last section, the conclusions are given.


## 2. Multiple Exp-Function Method

This section elucidates a systematic explanation of the multiple Exp-function method [60–64] so that it can be further applied to the nonlinear PDEs in order to furnish its exact solutions:Step 1.
The following NLPDE are as follows:(7)Nx,y,t,Ψ,Ψx,Ψy,Ψz,Ψt,Ψxx,Ψtt,⋯=0.
We commence a sequence of novel variablesξi=ξix,y,z,t, 1≤i≤n, by solvable PDEs, for example, the linear ones:
(8)ξi,x=αiξi,ξi,y=βiξi,ξi,z=γiξi,ξi,t=δiξi,1≤i≤n,where αi,βi,γi,1≤i≤n are the angular wave numbers and δi,1≤i≤n are the wave frequencies. It must be pointed out that this is frequently the initiating step for constructing the exact solutions to nonlinear partial differential equations; moreover, solving such linear equations redounds to the exponential function solutions,
(9)ξi=ϖieθi,θi=αix+βiy+γiz−δit,1≤i≤n,in which ϖi,1≤i≤n, are undetermined values.Step 2.
Supposing the solution of equation (7) happens to be of the following form in terms of new variables ξi,1≤i≤n:
(10)Ψx,y,z,t=Δξ1,ξ2,⋯,ξnΩξ1,ξ2,⋯,ξn,Δ=∑r,s=1n∑i,j=0MΔrs,ijξriξsj,Ω=∑r,s=1n∑i,j=0NΩrs,ijξriξsj,in which Δrs,ij and Ωrs,ij are values to be settled. Plugging equation (10) into equation (7) and ordering the numerator of the rational function to zero, we can gain a series of the nonlinear algebraic equations about the variables αi,βi,γi,δi,Δrs,ij and Ωrs,ij. Solving the solutions for these nonlinear algebraic equations and inserting these solutions into equation (10), the multiple soliton solutions for equation (7) can be achieved as follows:
(11)Ψx,y,z,t=Δϖ1eα1x+β1y+γ1x−δ1t,⋯,ϖneαnx+βny+γnx−δntΩϖ1eα1x+β1y+γ1x−δ1t,⋯,ϖneαnx+βny+γnx−δnt,and also we have
(12)Δt=∑i=1nΔξiξi,t,Ωt=∑i=1nΩξiξi,t,Δx=∑i=1nΔξiξi,xΩx=∑i=1nΩξiξi,xΔy=∑i=1nΔξiξi,y,Ωy=∑i=1nΩξiξi,y,Δ=Δξ1,ξ2,⋯,ξn,Ω=Ωξ1,ξ2,⋯,ξn,Δξi=dΔdξi,Ωξi=dΩdξi,ξi,t=dξidt,ξi,x=dξidx,ξi,y=dξidy,Ψt=Ω∑i=1nΔξiξi,t−Δ∑i=1nΩξiξi,tΩ2,Ψx=Ω∑i=1nΔξiξi,x−Δ∑i=1nΩξiξi,xΩ2,Ψy=Ω∑i=1nΔξiξi,y−Δ∑i=1nΩξiξi,yΩ2,Ψz=Ω∑i=1nΔξiξi,z−Δ∑i=1nΩξiξi,zΩ2.


## 3. Multiple Soliton Solutions for the Extended (3+1) JM Equation (1)

### 3.1. Set I: One-Wave Solution

We commence with a one-wave function based on the statement in Step2 in the previous section; we suppose that equation (1) has the rational function of the one-wave solution as shown in the following form:
(13)Ψx,y,z,t=Δ1Ω1,Ω1=1+ρ1eα1x+β1y+γ1z−δ1t,Δ1=σ1+ρ1σ2eα1x+β1y+γ1z−δ1t,in which σ1 and σ2 are unfound constants. Plugging (13) into equation (1), we gain to the following case:
(14)ρ1=ρ1,σ1=σ1,σ2=ρ22α1ρ1+σ1ρ1,δ1=α13β1−3α1γ1−3β1γ1−3γ122β1.Therefore, the resulting one-wave solution reads as(15)Ψx,y,z,t=σ1+ρ22α1ρ1+σ1eα1x+β1y+γ1z−α13β1−3α1γ1−3β1γ1−3γ12/2β1t1+ρ1eα1x+β1y+γ1z−α13β1−3α1γ1−3β1γ1−3γ12/2β1t.By selecting the suitable values of parameters, the graphic presentation of the periodic wave solution is presented in Figure1 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 1
The one-wave solution (15) at α1=1,β1=1.5,σ2=1.5,ρ1=2.2,γ1=2,ρ2=3,y=10,andz=2.


### 3.2. Set II: Two-Wave Solutions

We commence with the two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (1) has the rational function of two-wave solutions as shown in the following form:
(16)Ψx,y,z,t=Δ2Ω2,(17)Ω2=1+σ1eα1x+β1y+γ1z−δ1t+σ2eα2x+β2y+γ2z−δ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1+γ2z−δ1+δ2t,Δ2=ρ1eα1x+β1y+γ1z−δ1t+ρ2eα2x+β2y+γ2z−δ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y+γ1+γ2z−δ1+δ2t.Plugging (16) along with (17) into equation (1), we get to the following case:
(18)σ1=σ1,σ2=σ2,δ1=12α13,δ2=12α23,ρ1=2α1σ1,ρ2=2α2σ2,ρ12=β1−β2α1−α22α1α2β1+β2,σ12=β1−β2α1−α2β1+β2α1+α2.Therefore, the resulting two-wave solution reads as(19)Ψ2x,y,z,t=2α1σ1eΛ1+2α2σ2eΛ2+2σ1σ2β1−β2α1−α2/β1+β2eΛ1+Λ21+σ1eΛ1+σ2eΛ2+σ1σ2β1−β2α1−α2/β1+β2α1+α2eΛ1+Λ2,in which Λi=αix+βiy+γiz−1/2αi3t,i=1,2. By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 2 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 2
The two-wave solution (19) at α1=1,α2=1.5,β1=1.5,β2=2,σ1=1.1,σ2=2,γ1=1.5,γ2=2,y=10,andz=2.


### 3.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (1) has the rational function of triple-wave solutions as shown in the following form:
(20)Ψx,y,z,t=Δ3Ω3,(21)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy+γix−δit,i=1,2,3. Plugging (20) along with (21) into equation (1), we obtain the following case:
(22)σi=σi,δi=12αi3,ρi=2αiσi,i=1,2,3,ρ12=β1−β2α1−α22α1α2β1+β2,σ12=β1−β2α1−α2β1+β2α1+α2,ρ13=β1−β3α1−α32α1α3β1+β3,σ13=β1−β3α1−α3β1+β3α1+α3,ρ23=β2−β3α2−α32α2α3β2+β3,σ23=β2−β3α2−α3β2+β3α2+α3.Therefore, the resulting three-wave solution reads as(23)Ψ3x,y,z,t=Δ3Ω3,in which Δ3 and Ω3 with their relations are given in (21) and (22). By selecting suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 3 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 3
The three-wave solution (23) at α1=1,α2=1.5,α3=2,σ1=1.1,σ2=2,σ3=2.7,β1=2.5,β2=2,β3=2.6,γ1=1.5,γ2=2,γ3=3,y=10,andz=2.


## 3.1. Set I: One-Wave Solution

We commence with a one-wave function based on the statement in Step2 in the previous section; we suppose that equation (1) has the rational function of the one-wave solution as shown in the following form:
(13)Ψx,y,z,t=Δ1Ω1,Ω1=1+ρ1eα1x+β1y+γ1z−δ1t,Δ1=σ1+ρ1σ2eα1x+β1y+γ1z−δ1t,in which σ1 and σ2 are unfound constants. Plugging (13) into equation (1), we gain to the following case:
(14)ρ1=ρ1,σ1=σ1,σ2=ρ22α1ρ1+σ1ρ1,δ1=α13β1−3α1γ1−3β1γ1−3γ122β1.Therefore, the resulting one-wave solution reads as(15)Ψx,y,z,t=σ1+ρ22α1ρ1+σ1eα1x+β1y+γ1z−α13β1−3α1γ1−3β1γ1−3γ12/2β1t1+ρ1eα1x+β1y+γ1z−α13β1−3α1γ1−3β1γ1−3γ12/2β1t.By selecting the suitable values of parameters, the graphic presentation of the periodic wave solution is presented in Figure1 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 1
The one-wave solution (15) at α1=1,β1=1.5,σ2=1.5,ρ1=2.2,γ1=2,ρ2=3,y=10,andz=2.


## 3.2. Set II: Two-Wave Solutions

We commence with the two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (1) has the rational function of two-wave solutions as shown in the following form:
(16)Ψx,y,z,t=Δ2Ω2,(17)Ω2=1+σ1eα1x+β1y+γ1z−δ1t+σ2eα2x+β2y+γ2z−δ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1+γ2z−δ1+δ2t,Δ2=ρ1eα1x+β1y+γ1z−δ1t+ρ2eα2x+β2y+γ2z−δ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y+γ1+γ2z−δ1+δ2t.Plugging (16) along with (17) into equation (1), we get to the following case:
(18)σ1=σ1,σ2=σ2,δ1=12α13,δ2=12α23,ρ1=2α1σ1,ρ2=2α2σ2,ρ12=β1−β2α1−α22α1α2β1+β2,σ12=β1−β2α1−α2β1+β2α1+α2.Therefore, the resulting two-wave solution reads as(19)Ψ2x,y,z,t=2α1σ1eΛ1+2α2σ2eΛ2+2σ1σ2β1−β2α1−α2/β1+β2eΛ1+Λ21+σ1eΛ1+σ2eΛ2+σ1σ2β1−β2α1−α2/β1+β2α1+α2eΛ1+Λ2,in which Λi=αix+βiy+γiz−1/2αi3t,i=1,2. By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 2 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 2
The two-wave solution (19) at α1=1,α2=1.5,β1=1.5,β2=2,σ1=1.1,σ2=2,γ1=1.5,γ2=2,y=10,andz=2.


## 3.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (1) has the rational function of triple-wave solutions as shown in the following form:
(20)Ψx,y,z,t=Δ3Ω3,(21)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy+γix−δit,i=1,2,3. Plugging (20) along with (21) into equation (1), we obtain the following case:
(22)σi=σi,δi=12αi3,ρi=2αiσi,i=1,2,3,ρ12=β1−β2α1−α22α1α2β1+β2,σ12=β1−β2α1−α2β1+β2α1+α2,ρ13=β1−β3α1−α32α1α3β1+β3,σ13=β1−β3α1−α3β1+β3α1+α3,ρ23=β2−β3α2−α32α2α3β2+β3,σ23=β2−β3α2−α3β2+β3α2+α3.Therefore, the resulting three-wave solution reads as(23)Ψ3x,y,z,t=Δ3Ω3,in which Δ3 and Ω3 with their relations are given in (21) and (22). By selecting suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 3 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 3
The three-wave solution (23) at α1=1,α2=1.5,α3=2,σ1=1.1,σ2=2,σ3=2.7,β1=2.5,β2=2,β3=2.6,γ1=1.5,γ2=2,γ3=3,y=10,andz=2.


## 4. Multiple Soliton Solutions for the Extended (3+1) JM Equation (2)

### 4.1. Set I: One-Wave Solution

We commence with the one-wave function based on the statement in Step2 in the previous section; we suppose that equation (2) has the rational function of the one-wave solution as shown in the following form:
(24)Ψx,y,z,t=Δ1Ω1,Ω1=ρ1+ρ2eα1x+β1y+γ1z−δ1t,Δ1=σ1+σ2eα1x+β1y+γ1z−δ1t,in which ρ1,ρ2,σ1, and σ2 are unfound constants. Plugging (24) into equation (1), we gain the following case:
(25)ρ1=ρ1,ρ2=ρ2,σ1=σ1,σ2=ρ22α1ρ1+σ1ρ1,δ1=α1α12β1−3γ12α1+2β1+2γ1.Therefore, the resulting one-wave solution reads as(26)Ψx,y,z,t=σ1+ρ22α1ρ1+σ1eα1x+β1y+γ1z−α1α12β1−3γ1/2α1+2β1+2γ1tρ1+ρ2eα1x+β1y+γ1z−α1α12β1−3γ1/2α1+2β1+2γ1t.By choosing the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure4 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 4
The one-wave solution (26) at α1=1,β1=1.5,σ1=1.4,ρ1=2.2,ρ2=3,γ1=2,y=1,andz=2.


### 4.2. Set II: Two-Wave Solutions

We commence with two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (2) has the rational function of two-wave solutions as shown in the following form:
(27)Ψx,y,z,t=Δ2Ω2,(28)Ω2=1+σ1eα1x+β1y+γ1z−δ1t+σ2eα2x+β2y+γ2z−δ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1+γ2z−δ1+δ2t,Δ2=ρ1eα1x+β1y+γ1z−δ1t+ρ2eα2x+β2y+γ2z−δ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y+γ1+γ2z−δ1+δ2t.Plugging (27) along with (28) into equation (2), we get to the following case:
(29)σi=σi,δi=αi3βi2αi+2βi,ρi=2αiσi,i=1,2,ρ12=Σ122α1α2Γ12,σ12=Y12Φ12.(30)Σ12=α1α2α12β2+2α1α2β1−2α1α2β2−α22β1+3β12β2−3β1β22α12−α22+α22β12α1+α23α12−3α1α2+α22+2α1α2β1β2α1+α2α12−3α1α2+α22+α12β22α1+α2α12−3α1α2+3α22,Γ12=α1α2α1+α2α12β2+2α1α2β1+2α1α2β2+α22β1+3β12β2+3β1β22+α22β123α12+3α1α2+α22+2α1α2β1β2α12+3α1α2+α22+α12β22α12+3α1α2+3α22,Y12=α1α2α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1+3β12β2−3β1β22+α22β123α12−3α1α2+α22+2α1α2β1β2α12−3α1α2+α22+α12β22α12−3α1α2+3α22,Φ12=α1α2α1+α2α12β2+2α1α2β1+2α1α2β2+α22β1+3β12β2+3β1β22+α22β123α12+3α1α2+α22+2α1α2β1β2α12+3α1α2+α22+α12β22α12+3α1α2+3α22.Therefore, the resulting two-wave solution reads as(31)Ψ2x,y,z,t=2α1σ1eΛ1+2α2σ2eΛ2+2σ1σ2Σ12/Γ12eΛ1+Λ21+σ1eΛ1+σ2eΛ2+σ1σ2Y12/Φ12eΛ1+Λ2,in which Λi=αix+βiy+γiz−αi3βi/2αi+2βit,i=1,2. By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 5 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 5
The two-wave solution (31) at α1=1,α2=1.5,β1=1.5,β2=2,σ1=1.1,σ2=2,γ1=1.5,γ2=2,y=10,andz=2.


### 4.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (2) has the rational function of triple-wave solutions as shown in the following form:
(32)Ψx,y,z,t=Δ3Ω3,(33)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy+γix−δit,i=1,2,3. Plugging (32) along with (33) into equation (2), we obtain the following case:
(34)σi=σi,δi=αi3βi2αi+2βi,ρi=2αiσi,i=1,2,3,ρ12=Σ122α1α2Γ12,σ12=Y12Φ12.(35)ρ13=Σ132α1α3Γ13,σ13=Y13Φ13,ρ23=Σ232α2α3Γ23,σ23=Y23Φ23,Σ13=substituteα2=α3,β2=β3,atΣ12,Σ23=substituteα1=α2,β1=β2,atΣ13,Γ13=substituteα2=α3,β2=β3,atΓ12,Γ23=substituteα1=α2,β1=β2,atΓ13,Y13=substituteα2=α3,β2=β3,atY12,Y23=substituteα1=α2,β1=β2,atY13,Φ13=substituteα2=α3,β2=β3,atΦ12,Φ23=substituteα1=α2,β1=β2,atΦ13.Therefore, the resulting three-wave solution reads as(36)Ψ3x,y,z,t=Δ3Ω3,in which Δ3 and Ω3 with their relations are given in (33) and (34). By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 6 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 6
The three-wave solution (36) at α1=1,α2=1.5,α3=2,σ1=1.1,σ2=2,σ3=2.7,β1=2.5,β2=2,β3=2.6,γ1=1.5,γ2=2,γ3=3,y=10,andz=2.


## 4.1. Set I: One-Wave Solution

We commence with the one-wave function based on the statement in Step2 in the previous section; we suppose that equation (2) has the rational function of the one-wave solution as shown in the following form:
(24)Ψx,y,z,t=Δ1Ω1,Ω1=ρ1+ρ2eα1x+β1y+γ1z−δ1t,Δ1=σ1+σ2eα1x+β1y+γ1z−δ1t,in which ρ1,ρ2,σ1, and σ2 are unfound constants. Plugging (24) into equation (1), we gain the following case:
(25)ρ1=ρ1,ρ2=ρ2,σ1=σ1,σ2=ρ22α1ρ1+σ1ρ1,δ1=α1α12β1−3γ12α1+2β1+2γ1.Therefore, the resulting one-wave solution reads as(26)Ψx,y,z,t=σ1+ρ22α1ρ1+σ1eα1x+β1y+γ1z−α1α12β1−3γ1/2α1+2β1+2γ1tρ1+ρ2eα1x+β1y+γ1z−α1α12β1−3γ1/2α1+2β1+2γ1t.By choosing the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure4 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 4
The one-wave solution (26) at α1=1,β1=1.5,σ1=1.4,ρ1=2.2,ρ2=3,γ1=2,y=1,andz=2.


## 4.2. Set II: Two-Wave Solutions

We commence with two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (2) has the rational function of two-wave solutions as shown in the following form:
(27)Ψx,y,z,t=Δ2Ω2,(28)Ω2=1+σ1eα1x+β1y+γ1z−δ1t+σ2eα2x+β2y+γ2z−δ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1+γ2z−δ1+δ2t,Δ2=ρ1eα1x+β1y+γ1z−δ1t+ρ2eα2x+β2y+γ2z−δ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y+γ1+γ2z−δ1+δ2t.Plugging (27) along with (28) into equation (2), we get to the following case:
(29)σi=σi,δi=αi3βi2αi+2βi,ρi=2αiσi,i=1,2,ρ12=Σ122α1α2Γ12,σ12=Y12Φ12.(30)Σ12=α1α2α12β2+2α1α2β1−2α1α2β2−α22β1+3β12β2−3β1β22α12−α22+α22β12α1+α23α12−3α1α2+α22+2α1α2β1β2α1+α2α12−3α1α2+α22+α12β22α1+α2α12−3α1α2+3α22,Γ12=α1α2α1+α2α12β2+2α1α2β1+2α1α2β2+α22β1+3β12β2+3β1β22+α22β123α12+3α1α2+α22+2α1α2β1β2α12+3α1α2+α22+α12β22α12+3α1α2+3α22,Y12=α1α2α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1+3β12β2−3β1β22+α22β123α12−3α1α2+α22+2α1α2β1β2α12−3α1α2+α22+α12β22α12−3α1α2+3α22,Φ12=α1α2α1+α2α12β2+2α1α2β1+2α1α2β2+α22β1+3β12β2+3β1β22+α22β123α12+3α1α2+α22+2α1α2β1β2α12+3α1α2+α22+α12β22α12+3α1α2+3α22.Therefore, the resulting two-wave solution reads as(31)Ψ2x,y,z,t=2α1σ1eΛ1+2α2σ2eΛ2+2σ1σ2Σ12/Γ12eΛ1+Λ21+σ1eΛ1+σ2eΛ2+σ1σ2Y12/Φ12eΛ1+Λ2,in which Λi=αix+βiy+γiz−αi3βi/2αi+2βit,i=1,2. By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 5 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 5
The two-wave solution (31) at α1=1,α2=1.5,β1=1.5,β2=2,σ1=1.1,σ2=2,γ1=1.5,γ2=2,y=10,andz=2.


## 4.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (2) has the rational function of triple-wave solutions as shown in the following form:
(32)Ψx,y,z,t=Δ3Ω3,(33)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy+γix−δit,i=1,2,3. Plugging (32) along with (33) into equation (2), we obtain the following case:
(34)σi=σi,δi=αi3βi2αi+2βi,ρi=2αiσi,i=1,2,3,ρ12=Σ122α1α2Γ12,σ12=Y12Φ12.(35)ρ13=Σ132α1α3Γ13,σ13=Y13Φ13,ρ23=Σ232α2α3Γ23,σ23=Y23Φ23,Σ13=substituteα2=α3,β2=β3,atΣ12,Σ23=substituteα1=α2,β1=β2,atΣ13,Γ13=substituteα2=α3,β2=β3,atΓ12,Γ23=substituteα1=α2,β1=β2,atΓ13,Y13=substituteα2=α3,β2=β3,atY12,Y23=substituteα1=α2,β1=β2,atY13,Φ13=substituteα2=α3,β2=β3,atΦ12,Φ23=substituteα1=α2,β1=β2,atΦ13.Therefore, the resulting three-wave solution reads as(36)Ψ3x,y,z,t=Δ3Ω3,in which Δ3 and Ω3 with their relations are given in (33) and (34). By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 6 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 6
The three-wave solution (36) at α1=1,α2=1.5,α3=2,σ1=1.1,σ2=2,σ3=2.7,β1=2.5,β2=2,β3=2.6,γ1=1.5,γ2=2,γ3=3,y=10,andz=2.


## 5. Multiple Soliton Solutions for the (2+1) eCBS Equation

### 5.1. Set I: One-Wave Solution

We start with one-wave function based on the statement in Step2 in the previous section; we suppose that equation (3) has the rational function of the one-wave solution as shown in the following form:
(37)Ψx,y,t=Δ1Ω1,Ω1=ρ1+ρ2eα1x+β1y−λ1t,Δ1=σ1+σ2eα1x+β1y−λ1t,in which σ1 and σ2 are unfound constants. Substituting (37) into equation (3), we get to the following case:
(38)ρ1=ρ1,ρ2=ρ2,σ1=σ1,σ2=ρ2α1ρ1+σ1ρ1,λ1=α13δ3+α12β1+α1δ2+β1δ1.Therefore, the resulting one-wave solution reads as(39)Ψx,y,t=σ1+ρ2α1ρ1+σ1/ρ1eα1x+β1y−α13δ3+α12β1+α1δ2+β1δ1tρ1+ρ2eα1x+β1y−α13δ3+α12β1+α1δ2+β1δ1t.By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure7 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 7
The one-wave solution (39) at α1=1,σ1=2,ρ1=2.2,ρ2=3,ρ1=2,β1=1.5,δ1=1.5,δ2=2.5,δ3=0.5,andy=1.


### 5.2. Set II: Two-Wave Solutions

We begin with two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (3) has the rational function of two-wave solutions as shown in the following form:
(40)Ψx,y,z,t=Δ2Ω2,(41)Ω2=1+σ1eα1x+β1y−λ1t+σ2eα2x+β2y−λ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1−λ1+λ2t,Δ2=ρ1eα1x+β1y−λ1t+ρ2eα2x+β2y−λ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y−λ1+λ2t.Inserting (40) along with (41) into equation (3), we obtain the following case:
(42)σi=σi,λi=αi3δ3+αi2βi+αiδ2+βiδ1,ρi=2αiσi,i=1,2,ρ12=3α1α2δ3α1−α22+α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1α2α13α1α2δ3α1+α2+2α1α2β1+β2+α12β2+α22β1.(43)σ12=3α1α2δ3α1−α22+α1−α2α12β2+2α1α2β1−2α1α2β2−α22β13α1α2δ3α1+α22+α1+α2α12β2+2α1α2β1+2α1α2β2+α22β1.Therefore, the resulting two-wave solution reads as(44)Ψ2x,y,t=σ1eΛ1+σ2eΛ2+σ1σ2σ12eΛ1+Λ2ρ1eΛ1+ρ2eΛ2+4α1α2σ1σ2ρ12eΛ1+Λ2,in which Λi=αix+βiy−αi3δ3+αi2βi+αiδ2+βiδ1t,i=1,2. By choosing the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 8 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 8
The two-wave solution (44) at α1=1,α2=1.6,σ2=1.5,ρ=1.5,ρ1=2,ρ2=3,β1=2.5,β2=3.4,γ1=0.75,γ2=1.5,at=sint,bt=cost,y=1,andz=2.


### 5.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (3) has the rational function of triple-wave solutions as shown in the following form:
(45)Ψx,y,z,t=Δ3Ω3,(46)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy−λit,i=1,2,3. Plugging (45) along with (46) into equation (3), we obtain the following case:
(47)σi=σi,λi=αi3δ3+αi2βi+αiδ2+βiδ1,ρi=2αiσi,i=1,2,3,ρ12=3α1α2δ3α1−α22+α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1α2α13α1α2δ3α1+α2+2α1α2β1+β2+α12β2+α22β1,σ12=3α1α2δ3α1−α22+α1−α2α12β2+2α1α2β1−2α1α2β2−α22β13α1α2δ3α1+α22+α1+α2α12β2+2α1α2β1+2α1α2β2+α22β1,ρ13=substituteα2=α3,β2=β3,atρ12,σ13=substituteα2=α3,β2=β3,atσ12,ρ23=substituteα1=α2,β1=β2,atρ13,σ23=substituteα1=α2,β1=β2,atσ13.Therefore, the resulting three-wave solution reads as(48)Ψ3x,y,z,t=Δ3Ω3.In (48), Δ3 and Ω3 with their relations are given in (46) and (47). By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 9 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 9
The three-wave solution (48) at α1=1,α2=2.3,α3=3.6,ρ1=2,ρ2=2.3,ρ3=4.2,σ1=0.5,σ2=1.6,σ3=2.3,δ1=1,δ2=0.5,δ3=2.2,β1=1.5,β2=3.3,andβ3=5.4.


## 5.1. Set I: One-Wave Solution

We start with one-wave function based on the statement in Step2 in the previous section; we suppose that equation (3) has the rational function of the one-wave solution as shown in the following form:
(37)Ψx,y,t=Δ1Ω1,Ω1=ρ1+ρ2eα1x+β1y−λ1t,Δ1=σ1+σ2eα1x+β1y−λ1t,in which σ1 and σ2 are unfound constants. Substituting (37) into equation (3), we get to the following case:
(38)ρ1=ρ1,ρ2=ρ2,σ1=σ1,σ2=ρ2α1ρ1+σ1ρ1,λ1=α13δ3+α12β1+α1δ2+β1δ1.Therefore, the resulting one-wave solution reads as(39)Ψx,y,t=σ1+ρ2α1ρ1+σ1/ρ1eα1x+β1y−α13δ3+α12β1+α1δ2+β1δ1tρ1+ρ2eα1x+β1y−α13δ3+α12β1+α1δ2+β1δ1t.By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure7 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 7
The one-wave solution (39) at α1=1,σ1=2,ρ1=2.2,ρ2=3,ρ1=2,β1=1.5,δ1=1.5,δ2=2.5,δ3=0.5,andy=1.


## 5.2. Set II: Two-Wave Solutions

We begin with two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (3) has the rational function of two-wave solutions as shown in the following form:
(40)Ψx,y,z,t=Δ2Ω2,(41)Ω2=1+σ1eα1x+β1y−λ1t+σ2eα2x+β2y−λ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1−λ1+λ2t,Δ2=ρ1eα1x+β1y−λ1t+ρ2eα2x+β2y−λ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y−λ1+λ2t.Inserting (40) along with (41) into equation (3), we obtain the following case:
(42)σi=σi,λi=αi3δ3+αi2βi+αiδ2+βiδ1,ρi=2αiσi,i=1,2,ρ12=3α1α2δ3α1−α22+α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1α2α13α1α2δ3α1+α2+2α1α2β1+β2+α12β2+α22β1.(43)σ12=3α1α2δ3α1−α22+α1−α2α12β2+2α1α2β1−2α1α2β2−α22β13α1α2δ3α1+α22+α1+α2α12β2+2α1α2β1+2α1α2β2+α22β1.Therefore, the resulting two-wave solution reads as(44)Ψ2x,y,t=σ1eΛ1+σ2eΛ2+σ1σ2σ12eΛ1+Λ2ρ1eΛ1+ρ2eΛ2+4α1α2σ1σ2ρ12eΛ1+Λ2,in which Λi=αix+βiy−αi3δ3+αi2βi+αiδ2+βiδ1t,i=1,2. By choosing the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 8 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 8
The two-wave solution (44) at α1=1,α2=1.6,σ2=1.5,ρ=1.5,ρ1=2,ρ2=3,β1=2.5,β2=3.4,γ1=0.75,γ2=1.5,at=sint,bt=cost,y=1,andz=2.


## 5.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (3) has the rational function of triple-wave solutions as shown in the following form:
(45)Ψx,y,z,t=Δ3Ω3,(46)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy−λit,i=1,2,3. Plugging (45) along with (46) into equation (3), we obtain the following case:
(47)σi=σi,λi=αi3δ3+αi2βi+αiδ2+βiδ1,ρi=2αiσi,i=1,2,3,ρ12=3α1α2δ3α1−α22+α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1α2α13α1α2δ3α1+α2+2α1α2β1+β2+α12β2+α22β1,σ12=3α1α2δ3α1−α22+α1−α2α12β2+2α1α2β1−2α1α2β2−α22β13α1α2δ3α1+α22+α1+α2α12β2+2α1α2β1+2α1α2β2+α22β1,ρ13=substituteα2=α3,β2=β3,atρ12,σ13=substituteα2=α3,β2=β3,atσ12,ρ23=substituteα1=α2,β1=β2,atρ13,σ23=substituteα1=α2,β1=β2,atσ13.Therefore, the resulting three-wave solution reads as(48)Ψ3x,y,z,t=Δ3Ω3.In (48), Δ3 and Ω3 with their relations are given in (46) and (47). By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 9 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−1, x=0, and x=1.Figure 9
The three-wave solution (48) at α1=1,α2=2.3,α3=3.6,ρ1=2,ρ2=2.3,ρ3=4.2,σ1=0.5,σ2=1.6,σ3=2.3,δ1=1,δ2=0.5,δ3=2.2,β1=1.5,β2=3.3,andβ3=5.4.


## 6. Multiple Soliton Solutions for the Generalized (2+1) BK Equation

### 6.1. Set I: One-Wave Solution

We start with one-wave function based on the statement in Step2 in the previous section; we suppose that equation (4) has the rational function of the one-wave solution as shown in the following form:
(49)Ψx,y,t=Δ1Ω1,Ω1=ρ1+ρ2eα1x+β1y−λ1t,Δ1=σ1+σ2eα1x+β1y−λ1t,in which σ1 and σ2 are unfound constants. Putting (49) into equation (4), we get to the following case:
(50)ρ1=ρ1,ρ2=ρ2,σ1=σ1,σ2=ρ22α1ρ1+σ1ρ1,λ1=αα14+βα13β1+α12δ1+α1β1δ2+β12δ3α1.Therefore, the resulting one-wave solution reads as(51)Ψx,y,t=σ1+ρ22α1ρ1+σ1/ρ1eα1x+β1y−αα14+βα13β1+α12δ1+α1β1δ2+β12δ3/α1tρ1+ρ2eα1x+β1y−αα14+βα13β1+α12δ1+α1β1δ2+β12δ3/α1t.By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure10 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 10
The one-wave solution (51) at α=0.5,β=0.1,α1=1,β1=1.5,σ1=2,rho1=2.2,rho2=3,δ1=1,δ2=1.5,δ3=2,δ1=1,δ2=1.5,δ3=2,andy=1.


### 6.2. Set II: Two-Wave Solutions

We begin with two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (4) has the rational function of two-wave solutions as shown in the following form:
(52)Ψx,y,z,t=Δ2Ω2,(53)Ω2=1+σ1eα1x+β1y−λ1t+σ2eα2x+β2y−λ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1−λ1+λ2t,Δ2=ρ1eα1x+β1y−λ1t+ρ2eα2x+β2y−λ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y−λ1+λ2t.Inserting (52) along with (53) into equation (4), we obtain the following case:
(54)σi=σi,λi=ααi4+βαi3βi+αi2δ1+αiβiδ2+βi2δ3αi,ρi=2αiσi,i=1,2,ρ12=3αα12α22α2+α1α1−α22+α2+α1βα1α2α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1−δ3α1β2−α2β122α1α2α1α2α2+α13αα1α2α2+α1+βα12β2+2α1α2β1+2α1α2β2+α22β1−δ3α1β2−α2β12,σ12=α1α2−α2+α13αα1α2−α2+α1+βα12β2+2α1α2β1−2α1α2β2−α22β1−δ3α1β2−α2β12α1α2α2+α13αα1α2α2+α1+βα12β2+2α1α2β1+2α1α2β2+α22β1−δ3α1β2−α2β12.Therefore, the resulting two-wave solution reads as(55)Ψ2x,y,t=σ1eΛ1+σ2eΛ2+σ1σ2σ12eΛ1+Λ2ρ1eΛ1+ρ2eΛ2+4α1α2σ1σ2ρ12eΛ1+Λ2,in which Λi=αix+βiy−ααi4+βαi3βi+αi2δ1+αiβiδ2+βi2δ3/αit,i=1,2. By choosing suitable values of parameters, the analytical treatment of the periodic wave solution is presented in Figure 11 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 11
The two-wave solution (55) at α=0.5,β=0.1,α1=1,α2=1.6,rho1=2,rho2=2.3,σ1=1.3,σ2=2,β1=0.5,β2=1.5,δ1=0.1,δ2=0.5,δ3=0.2,andy=1.


### 6.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (4) has the rational function of triple-wave solutions as shown in the following form:
(56)Ψx,y,z,t=Δ3Ω3,(57)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy−λit,i=1,2,3. Plugging (56) along with (57) into equation (4), we obtain the following case:
(58)σi=σi,λi=ααi4+βαi3βi+αi2δ1+αiβiδ2+βi2δ3αi,ρi=2αiσi,i=1,2,3,ρ12=3αα12α22α2+α1α1−α22+α2+α1βα1α2α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1−δ3α1β2−α2β122α1α2α1α2α2+α13αα1α2α2+α1+βα12β2+2α1α2β1+2α1α2β2+α22β1−δ3α1β2−α2β12,σ12=α1α2−α2+α13αα1α2−α2+α1+βα12β2+2α1α2β1−2α1α2β2−α22β1−δ3α1β2−α2β12α1α2α2+α13αα1α2α2+α1+βα12β2+2α1α2β1+2α1α2β2+α22β1−δ3α1β2−α2β12,ρ13=substituteα2=α3,β2=β3,atρ12,σ13=substituteα2=α3,β2=β3,atσ12,ρ23=substituteα1=α2,β1=β2,atρ13,σ23=substituteα1=α2,β1=β2,atσ13.Therefore, the resulting three-wave solution reads as(59)Ψ3x,y,z,t=Δ3Ω3,in which Δ3 and Ω3 with their relations are given in (57) and (58). By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 12 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 12
The three-wave solution (59) at alpha=0.5,beta=0.1,α1=1,α2=2.3,α3=3.6,ρ1=2,ρ2=2.3,ρ3=4.2,σ1=0.5,σ2=1.6,σ3=2.3,δ1=1,δ2=0.5,δ3=2.2,β1=1.5,β2=3.3,andβ3=5.4.


## 6.1. Set I: One-Wave Solution

We start with one-wave function based on the statement in Step2 in the previous section; we suppose that equation (4) has the rational function of the one-wave solution as shown in the following form:
(49)Ψx,y,t=Δ1Ω1,Ω1=ρ1+ρ2eα1x+β1y−λ1t,Δ1=σ1+σ2eα1x+β1y−λ1t,in which σ1 and σ2 are unfound constants. Putting (49) into equation (4), we get to the following case:
(50)ρ1=ρ1,ρ2=ρ2,σ1=σ1,σ2=ρ22α1ρ1+σ1ρ1,λ1=αα14+βα13β1+α12δ1+α1β1δ2+β12δ3α1.Therefore, the resulting one-wave solution reads as(51)Ψx,y,t=σ1+ρ22α1ρ1+σ1/ρ1eα1x+β1y−αα14+βα13β1+α12δ1+α1β1δ2+β12δ3/α1tρ1+ρ2eα1x+β1y−αα14+βα13β1+α12δ1+α1β1δ2+β12δ3/α1t.By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure10 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 10
The one-wave solution (51) at α=0.5,β=0.1,α1=1,β1=1.5,σ1=2,rho1=2.2,rho2=3,δ1=1,δ2=1.5,δ3=2,δ1=1,δ2=1.5,δ3=2,andy=1.


## 6.2. Set II: Two-Wave Solutions

We begin with two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (4) has the rational function of two-wave solutions as shown in the following form:
(52)Ψx,y,z,t=Δ2Ω2,(53)Ω2=1+σ1eα1x+β1y−λ1t+σ2eα2x+β2y−λ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1−λ1+λ2t,Δ2=ρ1eα1x+β1y−λ1t+ρ2eα2x+β2y−λ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y−λ1+λ2t.Inserting (52) along with (53) into equation (4), we obtain the following case:
(54)σi=σi,λi=ααi4+βαi3βi+αi2δ1+αiβiδ2+βi2δ3αi,ρi=2αiσi,i=1,2,ρ12=3αα12α22α2+α1α1−α22+α2+α1βα1α2α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1−δ3α1β2−α2β122α1α2α1α2α2+α13αα1α2α2+α1+βα12β2+2α1α2β1+2α1α2β2+α22β1−δ3α1β2−α2β12,σ12=α1α2−α2+α13αα1α2−α2+α1+βα12β2+2α1α2β1−2α1α2β2−α22β1−δ3α1β2−α2β12α1α2α2+α13αα1α2α2+α1+βα12β2+2α1α2β1+2α1α2β2+α22β1−δ3α1β2−α2β12.Therefore, the resulting two-wave solution reads as(55)Ψ2x,y,t=σ1eΛ1+σ2eΛ2+σ1σ2σ12eΛ1+Λ2ρ1eΛ1+ρ2eΛ2+4α1α2σ1σ2ρ12eΛ1+Λ2,in which Λi=αix+βiy−ααi4+βαi3βi+αi2δ1+αiβiδ2+βi2δ3/αit,i=1,2. By choosing suitable values of parameters, the analytical treatment of the periodic wave solution is presented in Figure 11 including the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 11
The two-wave solution (55) at α=0.5,β=0.1,α1=1,α2=1.6,rho1=2,rho2=2.3,σ1=1.3,σ2=2,β1=0.5,β2=1.5,δ1=0.1,δ2=0.5,δ3=0.2,andy=1.


## 6.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (4) has the rational function of triple-wave solutions as shown in the following form:
(56)Ψx,y,z,t=Δ3Ω3,(57)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy−λit,i=1,2,3. Plugging (56) along with (57) into equation (4), we obtain the following case:
(58)σi=σi,λi=ααi4+βαi3βi+αi2δ1+αiβiδ2+βi2δ3αi,ρi=2αiσi,i=1,2,3,ρ12=3αα12α22α2+α1α1−α22+α2+α1βα1α2α1−α2α12β2+2α1α2β1−2α1α2β2−α22β1−δ3α1β2−α2β122α1α2α1α2α2+α13αα1α2α2+α1+βα12β2+2α1α2β1+2α1α2β2+α22β1−δ3α1β2−α2β12,σ12=α1α2−α2+α13αα1α2−α2+α1+βα12β2+2α1α2β1−2α1α2β2−α22β1−δ3α1β2−α2β12α1α2α2+α13αα1α2α2+α1+βα12β2+2α1α2β1+2α1α2β2+α22β1−δ3α1β2−α2β12,ρ13=substituteα2=α3,β2=β3,atρ12,σ13=substituteα2=α3,β2=β3,atσ12,ρ23=substituteα1=α2,β1=β2,atρ13,σ23=substituteα1=α2,β1=β2,atσ13.Therefore, the resulting three-wave solution reads as(59)Ψ3x,y,z,t=Δ3Ω3,in which Δ3 and Ω3 with their relations are given in (57) and (58). By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 12 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 12
The three-wave solution (59) at alpha=0.5,beta=0.1,α1=1,α2=2.3,α3=3.6,ρ1=2,ρ2=2.3,ρ3=4.2,σ1=0.5,σ2=1.6,σ3=2.3,δ1=1,δ2=0.5,δ3=2.2,β1=1.5,β2=3.3,andβ3=5.4.


## 7. Multiple Soliton Solutions for the (2+1) VC DJKM Equation

### 7.1. Set I: One-Wave Solution

We start with one-wave function based on the statement in Step2 in the previous section; we suppose that equation (6) has the rational function of one-wave solutions as shown in the following form:
(60)Ψx,y,t=Δ1Ω1,Ω1=ρ1+ρ2eα1x+β1y−λ1t,Δ1=σ1+σ2eα1x+β1y−λ1t,in which ρ1,ρ2,σ1 and σ2 are unfound constants. Inserting (60) into equation (6), we gain the following case:
(61)ρ1=ρ1,ρ2=ρ2,σ1=−ρ12α1ρ2−σ2ρ2,σ2=σ2,λ1=β1−α14+αβ12−htα122βgtα12.Therefore, the resulting one-wave solution reads as(62)Ψx,y,t=−ρ12α1ρ2−σ2/ρ2+σ2eα1x+β1y−β1−α14+αβ12−htα12/2βgtα12tρ1+ρ2eα1x+β1y−β1−α14+αβ12−htα12/2βgtα12t.By choosing the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure13 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−5, x=0, and x=5.Figure 13
The one-wave solution (62) at α=0.5,β=0.2,α1=1,β1=1.5,σ2=2,rho1=2.2,ρ2=3,gt=sint,ht=cost,andy=1.


### 7.2. Set II: Two-Wave Solutions

We begin with two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (6) has the rational function of two-wave solutions as shown in the following form:
(63)Ψx,y,z,t=Δ2Ω2,(64)Ω2=1+σ1eα1x+β1y−λ1t+σ2eα2x+β2y−λ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1−λ1+λ2t,Δ2=ρ1eα1x+β1y−λ1t+ρ2eα2x+β2y−λ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y−λ1+λ2t.Inserting (63) along with (64) into equation (6), we obtain the following case:
(65)σi=σi,λi=βi−αi4+αβi2−htαi22βgtαi2,ρi=2αiσi,i=1,2.(66)ρ12=12α12α22α1+α2−α2+α12+αα1+α2α1β2−α2β12α2α1α12α22α1+α22+αα1β2−α2β12,σ12=α12α22−α2+α12+αα1β2−α2β12α12α22α1+α22+αα1β2−α2β12.Therefore, the resulting two-wave solution reads as(67)Ψ2x,y,t=σ1eΛ1+σ2eΛ2+σ1σ2σ12eΛ1+Λ2ρ1eΛ1+ρ2eΛ2+4α1α2σ1σ2ρ12eΛ1+Λ2,in which Λi=αix+βiy−βi−αi4+αβi2−htαi2/2βgtαi2t,i=1,2. By choosing the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 14 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 14
The two-wave solution (67) at α=0.5,β=0.2,α1=1,α2=1.5,β1=1.5,β2=2,σ1=1.1,σ2=2,ρ1=2.2,ρ2=3,gt=sint,ht=cost,andy=1.


### 7.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (6) has the rational function of triple-wave solutions as shown in the following form:
(68)Ψx,y,z,t=Δ3Ω3,(69)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy−λit,i=1,2,3. Plugging (68) along with (69) into equation (6), we obtain the following case:
(70)σi=σi,λi=βi−αi4+αβi2−htαi22βgtαi2,ρi=2αiσi,i=1,2,3,ρ12=12α12α22α1+α2−α2+α12+αα1+α2α1β2−α2β12α2α1α12α22α1+α22+αα1β2−α2β12,σ12=α12α22−α2+α12+αα1β2−α2β12α12α22α1+α22+αα1β2−α2β12,ρ13=substituteα2=α3,β2=β3,atρ12,σ13=substituteα2=α3,β2=β3,atσ12,ρ23=substituteα1=α2,β1=β2,atρ13,σ23=substituteα1=α2,β1=β2,atσ13.Therefore, the resulting three-wave solution reads as(71)Ψ3x,y,z,t=Δ3Ω3.In (71), Δ3 and Ω3 with their relations are given in (69) and (70). By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 15 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 15
The three-wave solution (71) at α=0.5,β=0.2,α1=1,α2=1.5,α3=2,β1=1.5,β2=2,β3=3.2,σ1=1.1,σ2=2,σ3=1.5,ρ1=2.2,ρ2=3,gt=sint,ht=cost,andy=1.This paper finds many novel one-soliton-, two-soliton-, and triple-soliton-type solutions to governing models. With the help of some computations, surfaces of results reported have been observed in Figures1–15. These figures are dependent on the family conditions which are of importance physically. It has been investigated that all figures plotted have been symbolized for the four types of the nonlinear PDEs containing the extended (3+1)-dimensional Jimbo-Miwa-like equation, the extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation, the generalization of the (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, and a variable-coefficient extension of the DJKM equation. These mathematical properties come from exponential function properties. In this sense, from the mathematical and physical points of view, these results play an important role in explaining wave propagation of nonlinear phenomena. Hence, we consider that surfaces plotted in this paper have proven such physical meaning of the obtained solutions.


## 7.1. Set I: One-Wave Solution

We start with one-wave function based on the statement in Step2 in the previous section; we suppose that equation (6) has the rational function of one-wave solutions as shown in the following form:
(60)Ψx,y,t=Δ1Ω1,Ω1=ρ1+ρ2eα1x+β1y−λ1t,Δ1=σ1+σ2eα1x+β1y−λ1t,in which ρ1,ρ2,σ1 and σ2 are unfound constants. Inserting (60) into equation (6), we gain the following case:
(61)ρ1=ρ1,ρ2=ρ2,σ1=−ρ12α1ρ2−σ2ρ2,σ2=σ2,λ1=β1−α14+αβ12−htα122βgtα12.Therefore, the resulting one-wave solution reads as(62)Ψx,y,t=−ρ12α1ρ2−σ2/ρ2+σ2eα1x+β1y−β1−α14+αβ12−htα12/2βgtα12tρ1+ρ2eα1x+β1y−β1−α14+αβ12−htα12/2βgtα12t.By choosing the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure13 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−5, x=0, and x=5.Figure 13
The one-wave solution (62) at α=0.5,β=0.2,α1=1,β1=1.5,σ2=2,rho1=2.2,ρ2=3,gt=sint,ht=cost,andy=1.


## 7.2. Set II: Two-Wave Solutions

We begin with two-wave functions based on the statement in Step2 in the previous section; we suppose that equation (6) has the rational function of two-wave solutions as shown in the following form:
(63)Ψx,y,z,t=Δ2Ω2,(64)Ω2=1+σ1eα1x+β1y−λ1t+σ2eα2x+β2y−λ2t+σ1σ2σ12eα1+α2x+β1+β2y+γ1−λ1+λ2t,Δ2=ρ1eα1x+β1y−λ1t+ρ2eα2x+β2y−λ2t+ρ1ρ2ρ12eα1+α2x+β1+β2y−λ1+λ2t.Inserting (63) along with (64) into equation (6), we obtain the following case:
(65)σi=σi,λi=βi−αi4+αβi2−htαi22βgtαi2,ρi=2αiσi,i=1,2.(66)ρ12=12α12α22α1+α2−α2+α12+αα1+α2α1β2−α2β12α2α1α12α22α1+α22+αα1β2−α2β12,σ12=α12α22−α2+α12+αα1β2−α2β12α12α22α1+α22+αα1β2−α2β12.Therefore, the resulting two-wave solution reads as(67)Ψ2x,y,t=σ1eΛ1+σ2eΛ2+σ1σ2σ12eΛ1+Λ2ρ1eΛ1+ρ2eΛ2+4α1α2σ1σ2ρ12eΛ1+Λ2,in which Λi=αix+βiy−βi−αi4+αβi2−htαi2/2βgtαi2t,i=1,2. By choosing the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 14 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 14
The two-wave solution (67) at α=0.5,β=0.2,α1=1,α2=1.5,β1=1.5,β2=2,σ1=1.1,σ2=2,ρ1=2.2,ρ2=3,gt=sint,ht=cost,andy=1.


## 7.3. Set III: Triple-Wave Solutions

We commence with three-wave functions based on the statement in Step2 in the previous section; we suppose that equation (6) has the rational function of triple-wave solutions as shown in the following form:
(68)Ψx,y,z,t=Δ3Ω3,(69)Ω3=1+ρ1eΛ1+ρ2eΛ2+ρ3eΛ3+ρ1ρ2ρ12eΛ1+Λ2+ρ1ρ3ρ13eΛ1+Λ3+ρ2ρ3ρ23eΛ2+Λ3+ρ1ρ2ρ3ρ12ρ13ρ23eΛ1+Λ2+Λ3,Δ3=σ1eΛ1+σ2eΛ2+σ3eΛ3+σ1σ2σ12eΛ1+Λ2+σ1σ3σ13eΛ1+Λ3+σ2σ3σ23eΛ2+Λ3+σ1σ2σ3σ12σ13σ23eΛ1+Λ2+Λ3,in which Λi=αix+βiy−λit,i=1,2,3. Plugging (68) along with (69) into equation (6), we obtain the following case:
(70)σi=σi,λi=βi−αi4+αβi2−htαi22βgtαi2,ρi=2αiσi,i=1,2,3,ρ12=12α12α22α1+α2−α2+α12+αα1+α2α1β2−α2β12α2α1α12α22α1+α22+αα1β2−α2β12,σ12=α12α22−α2+α12+αα1β2−α2β12α12α22α1+α22+αα1β2−α2β12,ρ13=substituteα2=α3,β2=β3,atρ12,σ13=substituteα2=α3,β2=β3,atσ12,ρ23=substituteα1=α2,β1=β2,atρ13,σ23=substituteα1=α2,β1=β2,atσ13.Therefore, the resulting three-wave solution reads as(71)Ψ3x,y,z,t=Δ3Ω3.In (71), Δ3 and Ω3 with their relations are given in (69) and (70). By selecting the suitable values of parameters, the graphic representation of the periodic wave solution is presented in Figure 15 containing the 3D plot, the contour plot, the density plot, and the 2D plot when three spaces arise at spaces x=−10, x=0, and x=10.Figure 15
The three-wave solution (71) at α=0.5,β=0.2,α1=1,α2=1.5,α3=2,β1=1.5,β2=2,β3=3.2,σ1=1.1,σ2=2,σ3=1.5,ρ1=2.2,ρ2=3,gt=sint,ht=cost,andy=1.This paper finds many novel one-soliton-, two-soliton-, and triple-soliton-type solutions to governing models. With the help of some computations, surfaces of results reported have been observed in Figures1–15. These figures are dependent on the family conditions which are of importance physically. It has been investigated that all figures plotted have been symbolized for the four types of the nonlinear PDEs containing the extended (3+1)-dimensional Jimbo-Miwa-like equation, the extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation, the generalization of the (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, and a variable-coefficient extension of the DJKM equation. These mathematical properties come from exponential function properties. In this sense, from the mathematical and physical points of view, these results play an important role in explaining wave propagation of nonlinear phenomena. Hence, we consider that surfaces plotted in this paper have proven such physical meaning of the obtained solutions.


## 8. Conclusion

In this article, we obtained the multiple soliton solutions of the novel extended (3+1)-dimensional Jimbo-Miwa-like equation, the extended (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff equation, the generalization of the (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, and a variable-coefficient extension of the DJKM equation via operating the multiple Exp-function method, including one-soliton-, two-soliton-, and triple-soliton-type solutions. It is quite visible that this novel scheme has plenty of family of solutions containing rational exponential functions by selecting particular parameters. Thus, this paper provides a lot of encouragement for future research in soliton topics. The behaviours of the solutions for the known nonlinear equations obtained by the multiple Exp-function method by choosing the suitable values are cited in Figures 1–15. Moreover, the numerical simulations include gaining the coefficientsρi,σi,andλi and have been carried out to show that the projected algorithm is applicable and efficient. The analytical study has been conducted for the solutions Ψx,y,t obtained by employing the aforementioned method, in which obtained solutions were as one-soliton, two-soliton, and triple-soliton. The results are beneficial to the study of wave propagation. All computations in this paper have been made quickly with the aid of Maple.



---

*Source: 8018064-2020-06-08.xml*