# Bäcklund Transformation and Exact Solutions to a Generalized (3 + 1)-Dimensional Nonlinear Evolution Equation

Authors: Yali Shen; Ying Yang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this article, a generalized (3 + 1)-dimensional nonlinear evolution equation (NLEE), which can be obtained by a multivariate polynomial, is investigated. Based on the Hirota bilinear method, theN-soliton solution and bilinear Bäcklund transformation (BBT) with explicit formulas are successfully constructed. By using BBT, two traveling wave solutions and a mixed solution of the generalized (3 + 1)-dimensional NLEE are obtained. Furthermore, the lump and the interaction solutions for the equation are constructed. Finally, the dynamic properties of the lump and the interaction solutions are described graphically.


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## Body

## 1. Introduction

With recent advancements in nonlinear science, the application of nonlinear evolution equations (NLEEs) has gained more prominence amongst mathematicians and physicists. The use of NLEEs is important because NLEEs can be used to describe many interesting nonlinear dynamic behaviors in nonlinear fields like optical fibers, plasma physics, atmospheric science, biologic nerve propagation, and marine science. Although the use of NLEEs is significant, the task of finding an explicit solution for the NLEEs has always been difficult and tedious. However, this task is also an important and meaningful exercise.Many effective methods for solving NLEEs have been proposed by previous researchers [1–9]. For instance, Zhang et al. obtained the exact solutions for generalized nonlinear diffusion equations by using the separation of variables method [4]. He and Wu also used the Exp-function method to solve the nonlinear wave equations [5]. Xu et al. presented analytic solutions of the discrete PT-symmetric nonlocal nonlinear Schrödinger equation by Darboux transformation [9]. Amongst the proposed methods [1–9], the Hirota bilinear method is one of the most direct and effective methods for obtaining the exact solution of the NLEEs.The most critical step in using the Hirota bilinear method is to transform the original nonlinear equation into a bilinear equation by introducing appropriate variable transformations. After obtaining the bilinear equation, the multisoliton solution of the original nonlinear equation can then be constructed. Interestingly, Ma et al. showed that multivariable polynomials can also produce Hirota bilinear equations [10]. The necessary criterion for the existence of a linear subspace for the exponential traveling wave solutions of the Hirota bilinear equation was given by Ma et al. [10]. Generally, the Hirota bilinear equations derived from multivariable polynomials have linear subspace solutions.By considering the multivariable polynomial given by [10](1)Px,y,z,t=c1x4+c2xt+c3y2+c4yz+c5z2,where cii=1,…,5 are arbitrary constants whose corresponding Hirota bilinear equation is(2)c1Dx4+c2DxDt+c3Dy2+c4DyDz+c5Dz2f⋅f=0,where f=fx,y,z,t and D is the Hirota bilinear operator [2].(3)DxmDtna⋅b=∂∂x1−∂∂x2m∂∂t1−∂∂t2nax1,t1bx2,t2|x1=x2t1=t2.If the coefficientsci are chosen as c1=c2=1 and c3=c4=c5=0, then (2) can be reduced to the bilinear form of the Korteweg–de Vries (KdV) equation. Furthermore, if the coefficients ci are set as c1=c2=1, c3=c5=3, and c4=0, then (2) can be reduced to the bilinear form of (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation.By introducing the variable transformation given by(4)u=2lnfx,(2) is then mapped to(5)c1uxxxx+c2uxt+c3uyy+c4uyz+c5uzz+6c1uxuxx=0,where f=fx,y,z,t and u=ux,y,z,t. Based on our literature review, (5) is a generalized (3 + 1)-dimensional NLEE which has not been presented by previous authors. Recently, many generalized NLEEs have attracted more attention as a result of their widespread application in many physical fields [11–14]. By investigating different kinds of generalized NLEEs, more general properties of the equations can be obtained.This article is divided into sections and it focuses on the generalized (3 + 1)-dimensional NLEE represented by equation (5). In Section 2, the bilinear Bäcklund transformation (BBT) of equation (5) is constructed based on its Hirota bilinear equation. Armed with the BBT, the traveling wave solution and a mixed solution of (5) are obtained. In Section 3, the one-soliton, the two-soliton, and the three-soliton solutions are obtained, and then the N-soliton solution is derived. In Section 4, the lump and the interaction solutions of (5) are given. The solutions also included some free parameters based on the bilinear form of (5). However, their dynamical behaviors are shown under some selected values of parameters. The article is then concluded with a summary and the implications of this research.


## 2. Bilinear Bäcklund Transformation and Its Application

Bäcklund transformation is important in the study of soliton systems. Some bilinear identities are given as follows [2]:(6)Dx4a⋅ab2−Dx4b⋅ba2=2DxDx3a⋅b⋅ab+3Dx2a⋅b⋅Dxb⋅a,(7)DxDta⋅ab2−DxDtb⋅ba2=2DxDta⋅b⋅ba,(8)Dx2a⋅ab2−Dx2b⋅ba2=2DxDxa⋅b⋅ba,(9)DtDxa⋅b⋅ba=DxDta⋅b⋅ba,(10)Dxa⋅a=0,where a=ax,t and b=bx,t.


### 2.1. Bilinear Bäcklund Transformation

The BBT for (2) is then constructed and summarized as follows.Theorem 1.
Iff satisfies (2) as well as the following relations:(11)B1f^⋅f=c1Dx3+λ1Dy+λ2Dz+c2Dt+λ3f^⋅f=0,(12)B2f^⋅f=3c1Dx2+λ4Dxf^⋅f=0,(13)B3f^⋅f=c3Dy+λ5Dz+c4Dz−λ1Dx+λ6f^⋅f=0,(14)B4f^⋅f=c5Dz−λ2Dx−λ5Dy+λ7f^⋅f=0,then f^ also satisfies (2) with (11)–(14), where λi  i=1,…,7 are arbitrary constants.Proof.
For the following function,(15)P=c1Dx4+c2DxDt+c3Dy2+c4DyDz+c5Dz2f^⋅f^,we show that f^ satisfying (11)–(14) can make P=0.(16)Pff=c1Dx4+c2DxDt+c3Dy2+c4DyDz+c5Dz2f^⋅f^ff−c1Dx4+c2DxDt+c3Dy2+c4DyDz+c5Dz2f⋅ff^f^=c1Dx4f^⋅f^f2−Dx4f⋅ff^2+c2DxDtf^⋅f^f2−DxDtf⋅ff^2+c3Dy2f^⋅f^f2−Dy2f⋅ff^2+c4DyDzf^⋅f^f2−DyDzf⋅ff^2+c5Dz2f^⋅f^f2−Dz2f⋅ff^26−8=2c1DxDx3f^⋅f⋅ff^+3Dx2f^⋅f⋅Dxf⋅f^+2c2DxDtf^⋅f⋅ff^+2c3DyDyf^⋅f⋅ff^+2c4DyDzf^⋅f⋅ff^+2c5DzDzf^⋅f⋅ff^=2Dxc1Dx3f^⋅f+λ1Dyf^⋅f+λ2Dzf^⋅f+λ3f^⋅f⋅ff^+3c1Dx2f^⋅f+λ4Dxf^⋅f⋅Dxf⋅f^+2Dxc2Dtf^⋅f⋅ff^+2Dyc3Dyf^⋅f+λ5Dzf^⋅f+λ6f^⋅f⋅ff^+2Dyc4Dzf^⋅f−λ1Dxf^⋅f⋅ff^+2Dzc5Dzf^⋅f−λ2Dxf^⋅f−λ5Dyf^⋅f+λ7f^⋅f⋅ff^11−14=2DxB1f^⋅f⋅ff^+2DxB2f^⋅f⋅Dxf⋅f^+2DyB3f^⋅f⋅ff^+2DzB4f^⋅f⋅ff^≡0.
Based on (9) and (10) respectively, it is clear that the coefficients of λi  i=1,2,5 and λi  i=3,4,6,7 are zeros. Hence, the proof of Theorem 1 is complete.


### 2.2. Application of BBT

The valuef=1 is the solution of (2) and the corresponding original variable is u=lnfx=0. Due to Dxna⋅1=∂n/∂xna,n≥1, BBT (11)–(14) associated with f=1 is reduced to the following equations:(17)c1f^xxx+λ1f^y+λ2f^z+c2f^t+λ3f^=0,3c1f^xx+λ4f^x=0,c3f^y+λ5f^z+c4f^z−λ1f^x+λ6f^=0,c5f^z−λ2f^x−λ5f^y+λ7f^=0.Different types of solutions to (5) can be obtained by substituting different forms of f^ into (17).


#### 2.2.1. Traveling Wave Solution

First consider the following form:(18)f^=1+εekx+ly+mz+ωt,where ε, k, l, m, and ω are undetermined constants. By inserting (18) into (17) and assuming λ3=λ6=λ7=0, these parameters can be determined by calculation as follows:(19)λ2=c3c5m+λ52m+c4λ5m−λ1λ5kc3k,λ4=−3c1k,l=λ1k−λ5m−c4mc3,ω=2λ1λ5mk−c1c3k4+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2c2c3k.Therefore,(20)u=2kεekx+λ1k−λ5m−c4m/c3y+mz+2λ1λ5mk−c1c3k4+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2/c2c3kt1+εekx+λ1k−λ5m−c4m/c3y+mz+2λ1λ5mk−c1c3k4+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2/c2c3kt.This solves (5), where ε, m, λ5, λ7, and cii=1,4,5 are arbitrary constants and kc2c3≠0. Solution (20) is a one-stripe soliton solution of (5).By taking a simple polynomial solution,(21)f^=kx+ly+mz+ωt,where k, l, m, and ω are constants to be determined. Similarly, by substituting (21) into (17) and supposing λi=0i=3,6,7, a direct computation gets(22)λ2=c3c5m+λ52m+c4λ5m−λ5λ1kc3k,l=λ1k−λ5m−c4mc3,λ4=0,ω=2λ1λ5mk+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2c2c3k,with kc2c3≠0 and m, λ1, and λ5 are arbitrary constants.Hence,(23)u=2kkx+ly+mz+ωtis a rational solution to (5), and parameters k, l, m, and ω satisfy (22).


#### 2.2.2. Mixed Solution

f^ is now taken as a combination of a polynomial function and cosine function:(24)f^=k1x+l1y+m1z+ω1t+ε1coshk2x+l2y+m2z+ω2t,where ε1, ki, li, mi, and ωii=1,2 are constants to be determined. By inserting (24) into (17), these parameters can be determined by calculation as follows:(25)λ2=2c5λ1c4±c42−4c3c5,λ5=−2c3c5c4±c42−4c3c5,l1=2λ1k1−m1c4±c42−4c3c52c3,l2=−m2c4±c42−4c3c52c3,ω1=λ1−λ1k1±m1c42−4c3c5c2c3,ω2=±λ1m2c42−4c3c5c2c3,k2=λ3=λ4=λ6=λ7=0,where ε1, λ1, k1, m1, and m2 are arbitrary constants and c2c3≠0 and c42≥4c3c5.Thus,(26)u=k1+ε1k2sinhk2x+l2y+m2z+ω2tk1x+l1y+m1z+ω1t+ε1coshk2x+l2y+m2z+ω2t,is a mixed form solution to (5), and ki, li, mi, and ωii=1,2 satisfy (25).


## 2.1. Bilinear Bäcklund Transformation

The BBT for (2) is then constructed and summarized as follows.Theorem 1.
Iff satisfies (2) as well as the following relations:(11)B1f^⋅f=c1Dx3+λ1Dy+λ2Dz+c2Dt+λ3f^⋅f=0,(12)B2f^⋅f=3c1Dx2+λ4Dxf^⋅f=0,(13)B3f^⋅f=c3Dy+λ5Dz+c4Dz−λ1Dx+λ6f^⋅f=0,(14)B4f^⋅f=c5Dz−λ2Dx−λ5Dy+λ7f^⋅f=0,then f^ also satisfies (2) with (11)–(14), where λi  i=1,…,7 are arbitrary constants.Proof.
For the following function,(15)P=c1Dx4+c2DxDt+c3Dy2+c4DyDz+c5Dz2f^⋅f^,we show that f^ satisfying (11)–(14) can make P=0.(16)Pff=c1Dx4+c2DxDt+c3Dy2+c4DyDz+c5Dz2f^⋅f^ff−c1Dx4+c2DxDt+c3Dy2+c4DyDz+c5Dz2f⋅ff^f^=c1Dx4f^⋅f^f2−Dx4f⋅ff^2+c2DxDtf^⋅f^f2−DxDtf⋅ff^2+c3Dy2f^⋅f^f2−Dy2f⋅ff^2+c4DyDzf^⋅f^f2−DyDzf⋅ff^2+c5Dz2f^⋅f^f2−Dz2f⋅ff^26−8=2c1DxDx3f^⋅f⋅ff^+3Dx2f^⋅f⋅Dxf⋅f^+2c2DxDtf^⋅f⋅ff^+2c3DyDyf^⋅f⋅ff^+2c4DyDzf^⋅f⋅ff^+2c5DzDzf^⋅f⋅ff^=2Dxc1Dx3f^⋅f+λ1Dyf^⋅f+λ2Dzf^⋅f+λ3f^⋅f⋅ff^+3c1Dx2f^⋅f+λ4Dxf^⋅f⋅Dxf⋅f^+2Dxc2Dtf^⋅f⋅ff^+2Dyc3Dyf^⋅f+λ5Dzf^⋅f+λ6f^⋅f⋅ff^+2Dyc4Dzf^⋅f−λ1Dxf^⋅f⋅ff^+2Dzc5Dzf^⋅f−λ2Dxf^⋅f−λ5Dyf^⋅f+λ7f^⋅f⋅ff^11−14=2DxB1f^⋅f⋅ff^+2DxB2f^⋅f⋅Dxf⋅f^+2DyB3f^⋅f⋅ff^+2DzB4f^⋅f⋅ff^≡0.
Based on (9) and (10) respectively, it is clear that the coefficients of λi  i=1,2,5 and λi  i=3,4,6,7 are zeros. Hence, the proof of Theorem 1 is complete.


## 2.2. Application of BBT

The valuef=1 is the solution of (2) and the corresponding original variable is u=lnfx=0. Due to Dxna⋅1=∂n/∂xna,n≥1, BBT (11)–(14) associated with f=1 is reduced to the following equations:(17)c1f^xxx+λ1f^y+λ2f^z+c2f^t+λ3f^=0,3c1f^xx+λ4f^x=0,c3f^y+λ5f^z+c4f^z−λ1f^x+λ6f^=0,c5f^z−λ2f^x−λ5f^y+λ7f^=0.Different types of solutions to (5) can be obtained by substituting different forms of f^ into (17).


### 2.2.1. Traveling Wave Solution

First consider the following form:(18)f^=1+εekx+ly+mz+ωt,where ε, k, l, m, and ω are undetermined constants. By inserting (18) into (17) and assuming λ3=λ6=λ7=0, these parameters can be determined by calculation as follows:(19)λ2=c3c5m+λ52m+c4λ5m−λ1λ5kc3k,λ4=−3c1k,l=λ1k−λ5m−c4mc3,ω=2λ1λ5mk−c1c3k4+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2c2c3k.Therefore,(20)u=2kεekx+λ1k−λ5m−c4m/c3y+mz+2λ1λ5mk−c1c3k4+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2/c2c3kt1+εekx+λ1k−λ5m−c4m/c3y+mz+2λ1λ5mk−c1c3k4+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2/c2c3kt.This solves (5), where ε, m, λ5, λ7, and cii=1,4,5 are arbitrary constants and kc2c3≠0. Solution (20) is a one-stripe soliton solution of (5).By taking a simple polynomial solution,(21)f^=kx+ly+mz+ωt,where k, l, m, and ω are constants to be determined. Similarly, by substituting (21) into (17) and supposing λi=0i=3,6,7, a direct computation gets(22)λ2=c3c5m+λ52m+c4λ5m−λ5λ1kc3k,l=λ1k−λ5m−c4mc3,λ4=0,ω=2λ1λ5mk+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2c2c3k,with kc2c3≠0 and m, λ1, and λ5 are arbitrary constants.Hence,(23)u=2kkx+ly+mz+ωtis a rational solution to (5), and parameters k, l, m, and ω satisfy (22).


### 2.2.2. Mixed Solution

f^ is now taken as a combination of a polynomial function and cosine function:(24)f^=k1x+l1y+m1z+ω1t+ε1coshk2x+l2y+m2z+ω2t,where ε1, ki, li, mi, and ωii=1,2 are constants to be determined. By inserting (24) into (17), these parameters can be determined by calculation as follows:(25)λ2=2c5λ1c4±c42−4c3c5,λ5=−2c3c5c4±c42−4c3c5,l1=2λ1k1−m1c4±c42−4c3c52c3,l2=−m2c4±c42−4c3c52c3,ω1=λ1−λ1k1±m1c42−4c3c5c2c3,ω2=±λ1m2c42−4c3c5c2c3,k2=λ3=λ4=λ6=λ7=0,where ε1, λ1, k1, m1, and m2 are arbitrary constants and c2c3≠0 and c42≥4c3c5.Thus,(26)u=k1+ε1k2sinhk2x+l2y+m2z+ω2tk1x+l1y+m1z+ω1t+ε1coshk2x+l2y+m2z+ω2t,is a mixed form solution to (5), and ki, li, mi, and ωii=1,2 satisfy (25).


## 2.2.1. Traveling Wave Solution

First consider the following form:(18)f^=1+εekx+ly+mz+ωt,where ε, k, l, m, and ω are undetermined constants. By inserting (18) into (17) and assuming λ3=λ6=λ7=0, these parameters can be determined by calculation as follows:(19)λ2=c3c5m+λ52m+c4λ5m−λ1λ5kc3k,λ4=−3c1k,l=λ1k−λ5m−c4mc3,ω=2λ1λ5mk−c1c3k4+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2c2c3k.Therefore,(20)u=2kεekx+λ1k−λ5m−c4m/c3y+mz+2λ1λ5mk−c1c3k4+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2/c2c3kt1+εekx+λ1k−λ5m−c4m/c3y+mz+2λ1λ5mk−c1c3k4+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2/c2c3kt.This solves (5), where ε, m, λ5, λ7, and cii=1,4,5 are arbitrary constants and kc2c3≠0. Solution (20) is a one-stripe soliton solution of (5).By taking a simple polynomial solution,(21)f^=kx+ly+mz+ωt,where k, l, m, and ω are constants to be determined. Similarly, by substituting (21) into (17) and supposing λi=0i=3,6,7, a direct computation gets(22)λ2=c3c5m+λ52m+c4λ5m−λ5λ1kc3k,l=λ1k−λ5m−c4mc3,λ4=0,ω=2λ1λ5mk+c4λ1mk−λ12k2−c3c5m2−λ52m2−c4λ5m2c2c3k,with kc2c3≠0 and m, λ1, and λ5 are arbitrary constants.Hence,(23)u=2kkx+ly+mz+ωtis a rational solution to (5), and parameters k, l, m, and ω satisfy (22).


## 2.2.2. Mixed Solution

f^ is now taken as a combination of a polynomial function and cosine function:(24)f^=k1x+l1y+m1z+ω1t+ε1coshk2x+l2y+m2z+ω2t,where ε1, ki, li, mi, and ωii=1,2 are constants to be determined. By inserting (24) into (17), these parameters can be determined by calculation as follows:(25)λ2=2c5λ1c4±c42−4c3c5,λ5=−2c3c5c4±c42−4c3c5,l1=2λ1k1−m1c4±c42−4c3c52c3,l2=−m2c4±c42−4c3c52c3,ω1=λ1−λ1k1±m1c42−4c3c5c2c3,ω2=±λ1m2c42−4c3c5c2c3,k2=λ3=λ4=λ6=λ7=0,where ε1, λ1, k1, m1, and m2 are arbitrary constants and c2c3≠0 and c42≥4c3c5.Thus,(26)u=k1+ε1k2sinhk2x+l2y+m2z+ω2tk1x+l1y+m1z+ω1t+ε1coshk2x+l2y+m2z+ω2t,is a mixed form solution to (5), and ki, li, mi, and ωii=1,2 satisfy (25).


## 3. Soliton Solutions

It is known that many interesting solutions, including breathers, lump, peakon, and rogue solutions, are all special reductions ofN-soliton solutions in different situations. This section shows solutions with a few solitons to (5). The standard perturbation method is used to find possible soliton-like solutions. By using a tedious but straightforward calculation, the one-soliton, two-soliton, three-soliton, and N-soliton solutions can be obtained.


### 3.1. One-Soliton Solution

To obtain the one-soliton solution, we assume thatf in (2) is expressed as(27)f=1+eη1,with(28)η1=k1x+p1y+q1z+ω1t+l1,where k1, p1, q1, l1, and ω1 are arbitrary constants. By substituting (27) into (2) and making the coefficients of all exponential terms zero, the dispersion relation is obtained as follows:(29)ω1=−c1k14+c4p1q1+c3p12+c5q12c2k1.Therefore, by substituting (27)–(29) into (4), the one-soliton solution of (5) can be obtained.


### 3.2. Two-Soliton Solution

To obtain the two-soliton solution, letf in (2) be expressed as(30)f=1+eη1+eη2+α12eη1+η2,with(31)ηi=kix+piy+qiz+ωit+li,i=1,2,where ki, pi, qi, li, and ωii=1,2 are arbitrary constants. By substituting (30) into (2) and making the coefficients of all exponential terms zero and after some reduction, the followings are obtained:(32)ωi=−c1ki4+c4piqi+c3pi2+c5qi2c2k1,i=1,2,α12=3k12k22k1−k22c1−k1p2−k2p12c3−k1q2−k2q1k1p2−k2p1c4−k1q2−k2q12c53k12k22k1+k22c1−k1p2−k2p12c3−k1q2−k2q1k1p2−k2p1c4−k1q2−k2q12c5.By substituting (30)–(32) into (4), the two-soliton solution of (5) can be obtained.


### 3.3. Three-Soliton Solution

To obtain the three-soliton solution, it is assumed thatf in (2) is expressed as(33)f=1+eη1+eη2+eη3+α12eη1+η2+α13eη1+η3+α23eη2+η3+α123eη1+η2+η3,with(34)ηi=kix+piy+qiz+ωit+li,i=1,2,3,where ki, pi, qi, li, and ωii=1,2,3 are arbitrary constants. Similar to the above method, but through a more complex process and then reduction, the following results are finally obtained:(35)ωi=−c1ki4+c4piqi+c3pi2+c5qi2c2k1,i=1,2,3,αij=3ki2kj2ki−kj2c1−kipj−kjpi2c3−kiqj−kjqikipj−kjpic4−kiqj−kjqi2c53ki2kj2ki+kj2c1−kipj−kjpi2c3−kiqj−kjqikipj−kjpic4−kiqj−kjqi2c5,α123=α12α13α23,1≤i<j≤3.By substituting (33)–(35) into (4), the three-soliton solution of (5) can be obtained. When the three-soliton are substituted into (4) for verification, it is shown that (35) needs to satisfy the constraint condition c42=4c3c5.Figure1 shows the three-soliton solution of (5) through selecting the appropriate parameters. As it is shown in Figure 1, the wave shapes are kink solitons. Both one and two solitons are also kink solitons, whose graphs are omitted here.Figure 1
The three-soliton solution for equation (5) by choosing suitable parameters: k1=2,k2=2,k3=3,p1=1,p2=2,p3=1,q1=1,q2=1,q3=1,l1=0,l2=0,l3=0,c1=1,c2=5,c3=1,c4=4,andc5=4. (a) x=0,y=0; (b) x=0,t=0; (c) z=0,t=0.
(a)(b)(c)


### 3.4.N-Soliton Solution

(5) admits multiple-soliton solution (4) with f of the following form:(36)f=∑μ=0,1exp∑i=1Nμiηi+∑1≤i<j≤NNμiμjlnαij,where(37)ηi=kix+piy+qiz+ωit+li,ωi=−c1ki4+c4piqi+c3pi2+c5qi2c2k1,i=1,2,…,N,αij=3ki2kj2ki−kj2c1−kipj−kjpi2c3−kiqj−kjqikipj−kjpic4−kiqj−kjqi2c53ki2kj2ki+kj2c1−kipj−kjpi2c3−kiqj−kjqikipj−kjpic4−kiqj−kjqi2c5,1≤i<j≤N,with ki, pi, qi, and li being arbitrary constants, c42=4c3c5, and ∑μ=0,1 is the summation that over all possible combinations of μi,μj=0,1i,j=1,2,…,N. By substituting (36) and (37) into (4), the N-soliton solution of (5) can be obtained.


## 3.1. One-Soliton Solution

To obtain the one-soliton solution, we assume thatf in (2) is expressed as(27)f=1+eη1,with(28)η1=k1x+p1y+q1z+ω1t+l1,where k1, p1, q1, l1, and ω1 are arbitrary constants. By substituting (27) into (2) and making the coefficients of all exponential terms zero, the dispersion relation is obtained as follows:(29)ω1=−c1k14+c4p1q1+c3p12+c5q12c2k1.Therefore, by substituting (27)–(29) into (4), the one-soliton solution of (5) can be obtained.


## 3.2. Two-Soliton Solution

To obtain the two-soliton solution, letf in (2) be expressed as(30)f=1+eη1+eη2+α12eη1+η2,with(31)ηi=kix+piy+qiz+ωit+li,i=1,2,where ki, pi, qi, li, and ωii=1,2 are arbitrary constants. By substituting (30) into (2) and making the coefficients of all exponential terms zero and after some reduction, the followings are obtained:(32)ωi=−c1ki4+c4piqi+c3pi2+c5qi2c2k1,i=1,2,α12=3k12k22k1−k22c1−k1p2−k2p12c3−k1q2−k2q1k1p2−k2p1c4−k1q2−k2q12c53k12k22k1+k22c1−k1p2−k2p12c3−k1q2−k2q1k1p2−k2p1c4−k1q2−k2q12c5.By substituting (30)–(32) into (4), the two-soliton solution of (5) can be obtained.


## 3.3. Three-Soliton Solution

To obtain the three-soliton solution, it is assumed thatf in (2) is expressed as(33)f=1+eη1+eη2+eη3+α12eη1+η2+α13eη1+η3+α23eη2+η3+α123eη1+η2+η3,with(34)ηi=kix+piy+qiz+ωit+li,i=1,2,3,where ki, pi, qi, li, and ωii=1,2,3 are arbitrary constants. Similar to the above method, but through a more complex process and then reduction, the following results are finally obtained:(35)ωi=−c1ki4+c4piqi+c3pi2+c5qi2c2k1,i=1,2,3,αij=3ki2kj2ki−kj2c1−kipj−kjpi2c3−kiqj−kjqikipj−kjpic4−kiqj−kjqi2c53ki2kj2ki+kj2c1−kipj−kjpi2c3−kiqj−kjqikipj−kjpic4−kiqj−kjqi2c5,α123=α12α13α23,1≤i<j≤3.By substituting (33)–(35) into (4), the three-soliton solution of (5) can be obtained. When the three-soliton are substituted into (4) for verification, it is shown that (35) needs to satisfy the constraint condition c42=4c3c5.Figure1 shows the three-soliton solution of (5) through selecting the appropriate parameters. As it is shown in Figure 1, the wave shapes are kink solitons. Both one and two solitons are also kink solitons, whose graphs are omitted here.Figure 1
The three-soliton solution for equation (5) by choosing suitable parameters: k1=2,k2=2,k3=3,p1=1,p2=2,p3=1,q1=1,q2=1,q3=1,l1=0,l2=0,l3=0,c1=1,c2=5,c3=1,c4=4,andc5=4. (a) x=0,y=0; (b) x=0,t=0; (c) z=0,t=0.
(a)(b)(c)


## 3.4.N-Soliton Solution

(5) admits multiple-soliton solution (4) with f of the following form:(36)f=∑μ=0,1exp∑i=1Nμiηi+∑1≤i<j≤NNμiμjlnαij,where(37)ηi=kix+piy+qiz+ωit+li,ωi=−c1ki4+c4piqi+c3pi2+c5qi2c2k1,i=1,2,…,N,αij=3ki2kj2ki−kj2c1−kipj−kjpi2c3−kiqj−kjqikipj−kjpic4−kiqj−kjqi2c53ki2kj2ki+kj2c1−kipj−kjpi2c3−kiqj−kjqikipj−kjpic4−kiqj−kjqi2c5,1≤i<j≤N,with ki, pi, qi, and li being arbitrary constants, c42=4c3c5, and ∑μ=0,1 is the summation that over all possible combinations of μi,μj=0,1i,j=1,2,…,N. By substituting (36) and (37) into (4), the N-soliton solution of (5) can be obtained.


## 4. Lump and Interaction Solutions

### 4.1. Lump Solutions

Many studies of the literature have given positive quadratic functions to get lump solutions [15–17]. Based on these studies of the literature, we have the following:(38)f=m1x+m2y+m3z+m4t2+n1x+n2y+n3z+n4t2+f0,where mi, nii=1,…,4 and f0 are real parameters to be determined.By substitutingf into (2) along with a direct but tedious calculation, 12 sets of solutions are obtained. By removing the redundant parameter solutions, the following three cases are finally obtained.Case 1.
(39)m3=±n12−3c1c5f0,m4=∓c4n2+2c5n3m3c2n1,n4=−c5n32+c4n3n2+c3n22f0+3c1n14c2n1f0,m1=m2=0,where c2n1≠0, c1c5f0<0, and n2, n3 are arbitrary constants.
By substituting (38) and (39) into (4), the lump solution of (5) is as follows:(40)u=4n1n1x+n2y+n3z+n4tm3z+m4t2+n1x+n2y+n3z+n4t2+f0,where mi and nii=1,…,4 satisfy (39).Case 2.
(41)m4=∓m3n2c4+2n2m2c3+2n3m3c5+n3m2c4c2n1,n1=∓−c5m32+c4m3m2+c3m22f03c11/4,n4=±c3m22−n22+c4m2m3−n2n3+c5m32−n32c2n1,m1=0,where c2≠0, c5m32+c4m3m2+c3m22f0c1<0, and n2, n3 are arbitrary constants.
By substituting (38) and (41) into (4), the lump solution of (5) is as follows:(42)u=4n1n1x+n2y+n3z+n4tm2y+m3z+m4t2+n1x+n2y+n3z+n4t2+f0,where miandnii=1,…,4 satisfy (41). The first set parameter solution in (41) is then chosen. Figure 2 shows the spatial structures of the lump solution and the contour plots for (42) by selecting the appropriate parameter values.Figure 2
Spatial structures of lump solution (42) with parameters: c1=−1andc2=c3=c4=c5=m2=m3=n2=n3=f0=1. (a) y=0,z=0; (b) x=0,z=0; (c) z=0,t=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)Case 3.
(43)m2=Θ1±Θ22c3n1f0,m4=−c4n3Θ1±Θ2±2c3n2Θ1±Θ2+Θ32c2c3n12f0,n4=−3c1m12+n122+f0c5n32+c4n2n3+c3n22f0c2n1,where(44)Θ1=c4n3m1−c4n1m3+2c3n2m1f0,Θ2=−12c1c3c5f0n12+m123−c5f02n3m1−n1m324c3c5−c42,Θ3=−3c1c5m1n12+m122+c4c52n2n1m3−n3m1f0+c522c5m3n3n1−c5n32m1−n22m1c3f0,where c2c3n1f0≠0, Θ2≥0, and m1, m3, n2, and n3 are arbitrary constants.
By substituting (38) and (43) into (4), the lump solution of (5) is obtained as follows:(45)u=4m1m1x+m2y+m3z+m4t+4n1n1x+n2y+n3z+n4tm1x+m2y+m3z+m4t2+n1x+n2y+n3z+n4t2+f0,where miandnii=1,…,4 satisfy (43).
It is noteworthy to state that the spatial structures of the lump solutions in Case1 and Case 3 are like those in Case 2. From the above three cases, the obtained lump waves have the following asymptotic behavior: limx⟶±∞u=0, limt⟶∞u=0, limy⟶±∞u=0, and limz⟶±∞u=0.


### 4.2. Interaction Solution between Solitary and Lump Waves

To construct interaction solutions, the following form is taken:(46)f=aξ12+bξ22+kgξ3+c,with(47)ξ1=m1x+m2y+m3z+m4t,ξ2=n1x+n2y+n3z+n4t,ξ3=k1x+k2y+k3z+k4t,where a, b, k, c, mi, ni, and kii=1,…,4 are arbitrary parameters and g is an arbitrary function. By choosing different functions for g, abundant interaction solutions can be obtained [18–21].This section mainly considers the interaction between the lump and the soliton solutions and then discusses their corresponding dynamical behaviors. The following values are chosen fora=1 and b=c=0 in (46) to obtain(48)f=m1x+m2y+m3z+m4t2+kcoshk1x+k2y+k3z+k4t.By substituting (48) into (2) and combining direct but tedious calculations, 31 sets of solutions are obtained for the above undetermined parameters. To ensure the generality of parametric solutions, the following four cases are obtained.Case 4.
(49)k4=−k22c3+c4k2k3+c5k32c2k1,m4=−c4k2+2c5k3±k2c42−4c3c52m14c2c5k12,m3=±c4k2+2c5k3±k2c42−4c3c5m12c5k1,c1=m2=0,where c2c5k1≠0, c42≥4c3c5, and k, k2, k3, and m1 are arbitrary constants.
By substituting (48) and (49) into (4), the interaction solution of (5) is obtained as follows:(50)u=4m1m1x+m3z+m4t+2kk1sinhk1x+k2y+k3z+k4tm1x+m3z+m4t2+kcoshk1x+k2y+k3z+k4t,where miandkii=1,…,4 satisfy (49) and the first set parameter solution in (49) is chosen. Figure 3 shows the spatial structures of the interaction solution and the contour plots for (50) by selecting the appropriate parameter values. Interestingly, it was found that Figure 3(a) is consistent with the result of the interaction between the kink soliton and the lump waves (see Figures 1 and 2).Figure 3
Spatial structures of interaction solution (50) with parameters: c2=1,c3=1,c4=4,c5=4,andm1=k1=k2=k3=k4=1. (a) z=0,t=0; (b) y=0,z=0; (c) x=0,y=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)Case 5.
(51)k1=k2m1m2,k3=k2m3m2,k4=−k2c3m22+c4m3m2+c5m32c2m1m2,m4=−c3m22+c4m3m2+c5m32c2m1,c1=0,where c2m1m2≠0, and k, k2, m3, and cii=3,4,5 are arbitrary constants.
By substituting (48) and (51) into (4), the interaction solution of (5) is as follows:(52)u=4m1m1x+m2y+m3z+m4t+2kk1sinhk1x+k2y+k3z+k4tm1x+m2y+m3z+m4t2+kcoshk1x+k2y+k3z+k4t,where miandkii=1,…,4 satisfy (51). Figure 4 shows the spatial structures of the interaction solution and the contour plots for (52) by selecting the appropriate parameter values.Figure 4
Spatial structures of interaction solution (52) with parameters: c2=c3=c4=c5=k1=k2=1,m1=1,m2=2,andm3=3. (a) y=0,z=0; (b) x=0,z=0; (c) z=0,t=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)Case 6.
(53)k4=−c3k22+c4k3k2+c5k32c2k1,m3=m2−c4±c42−4c3c52c5,m4=m2c42−4c3c5k2c42−4c3c5∓c4k2+2c5k32c2c5k1,c1=m1=0,where c2c5k1≠0, c42≥4c3c5, and k, k2, and k3 are arbitrary constants.
By substituting (48) and (53) into (4), the interaction solution of (5) is obtained as follows:(54)u=2kk1sinhk1x+k2y+k3z+k4tm2y+m3z+m4t2+kcoshk1x+k2y+k3z+k4t,where mi,kii=1,…,4 satisfy (53). The first set parameter solution in (53) is then chosen. Figure 5 shows the spatial structures of the interaction solution and the contour plots for (54) by selecting the appropriate parameter values.Figure 5
Spatial structures of interaction solution (54) with parameters: c2=c3=c5=k1=k2=k3=k=1,c4=3,andm2=2. (a) y=0,z=0; (b) x=0,z=0; (c) z=0,t=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)Case 7.
(55)k3=−c4k2m1±c42−4c3c52k1m2−k2m12c5m1,k4=m2−c42+4c3c5k1m2−k2m1c2c5m12,m3=m2−c4±c42−4c3c52c5,c1=m4=0,where c2c5m1≠0, c42≥4c3c5, and k, k2, m1, and m2 are arbitrary constants.
By substituting (48) and (55) into (4), the interaction solution of (5) is obtained as follows:(56)u=4m1m1x+m2y+m3z+2kk1sinhk1x+k2y+k3z+k4tm1x+m2y+m3z2+kcoshk1x+k2y+k3z+k4t,where mi,kii=1,…,4 satisfies (55). The first set parameter solution in (55) is then chosen. Figure 6 shows the spatial structures of the interaction solution and the contour plots for (56) by selecting the appropriate parameter values.Figure 6
Spatial structures of interaction solution (56) with parameters: c2=c3=c5=k1=k2=k3=k=m1=1,c4=3,andm2=2. (a) y=0,z=0; (b) x=0,z=0; (c) y=0,t=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)


## 4.1. Lump Solutions

Many studies of the literature have given positive quadratic functions to get lump solutions [15–17]. Based on these studies of the literature, we have the following:(38)f=m1x+m2y+m3z+m4t2+n1x+n2y+n3z+n4t2+f0,where mi, nii=1,…,4 and f0 are real parameters to be determined.By substitutingf into (2) along with a direct but tedious calculation, 12 sets of solutions are obtained. By removing the redundant parameter solutions, the following three cases are finally obtained.Case 1.
(39)m3=±n12−3c1c5f0,m4=∓c4n2+2c5n3m3c2n1,n4=−c5n32+c4n3n2+c3n22f0+3c1n14c2n1f0,m1=m2=0,where c2n1≠0, c1c5f0<0, and n2, n3 are arbitrary constants.
By substituting (38) and (39) into (4), the lump solution of (5) is as follows:(40)u=4n1n1x+n2y+n3z+n4tm3z+m4t2+n1x+n2y+n3z+n4t2+f0,where mi and nii=1,…,4 satisfy (39).Case 2.
(41)m4=∓m3n2c4+2n2m2c3+2n3m3c5+n3m2c4c2n1,n1=∓−c5m32+c4m3m2+c3m22f03c11/4,n4=±c3m22−n22+c4m2m3−n2n3+c5m32−n32c2n1,m1=0,where c2≠0, c5m32+c4m3m2+c3m22f0c1<0, and n2, n3 are arbitrary constants.
By substituting (38) and (41) into (4), the lump solution of (5) is as follows:(42)u=4n1n1x+n2y+n3z+n4tm2y+m3z+m4t2+n1x+n2y+n3z+n4t2+f0,where miandnii=1,…,4 satisfy (41). The first set parameter solution in (41) is then chosen. Figure 2 shows the spatial structures of the lump solution and the contour plots for (42) by selecting the appropriate parameter values.Figure 2
Spatial structures of lump solution (42) with parameters: c1=−1andc2=c3=c4=c5=m2=m3=n2=n3=f0=1. (a) y=0,z=0; (b) x=0,z=0; (c) z=0,t=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)Case 3.
(43)m2=Θ1±Θ22c3n1f0,m4=−c4n3Θ1±Θ2±2c3n2Θ1±Θ2+Θ32c2c3n12f0,n4=−3c1m12+n122+f0c5n32+c4n2n3+c3n22f0c2n1,where(44)Θ1=c4n3m1−c4n1m3+2c3n2m1f0,Θ2=−12c1c3c5f0n12+m123−c5f02n3m1−n1m324c3c5−c42,Θ3=−3c1c5m1n12+m122+c4c52n2n1m3−n3m1f0+c522c5m3n3n1−c5n32m1−n22m1c3f0,where c2c3n1f0≠0, Θ2≥0, and m1, m3, n2, and n3 are arbitrary constants.
By substituting (38) and (43) into (4), the lump solution of (5) is obtained as follows:(45)u=4m1m1x+m2y+m3z+m4t+4n1n1x+n2y+n3z+n4tm1x+m2y+m3z+m4t2+n1x+n2y+n3z+n4t2+f0,where miandnii=1,…,4 satisfy (43).
It is noteworthy to state that the spatial structures of the lump solutions in Case1 and Case 3 are like those in Case 2. From the above three cases, the obtained lump waves have the following asymptotic behavior: limx⟶±∞u=0, limt⟶∞u=0, limy⟶±∞u=0, and limz⟶±∞u=0.


## 4.2. Interaction Solution between Solitary and Lump Waves

To construct interaction solutions, the following form is taken:(46)f=aξ12+bξ22+kgξ3+c,with(47)ξ1=m1x+m2y+m3z+m4t,ξ2=n1x+n2y+n3z+n4t,ξ3=k1x+k2y+k3z+k4t,where a, b, k, c, mi, ni, and kii=1,…,4 are arbitrary parameters and g is an arbitrary function. By choosing different functions for g, abundant interaction solutions can be obtained [18–21].This section mainly considers the interaction between the lump and the soliton solutions and then discusses their corresponding dynamical behaviors. The following values are chosen fora=1 and b=c=0 in (46) to obtain(48)f=m1x+m2y+m3z+m4t2+kcoshk1x+k2y+k3z+k4t.By substituting (48) into (2) and combining direct but tedious calculations, 31 sets of solutions are obtained for the above undetermined parameters. To ensure the generality of parametric solutions, the following four cases are obtained.Case 4.
(49)k4=−k22c3+c4k2k3+c5k32c2k1,m4=−c4k2+2c5k3±k2c42−4c3c52m14c2c5k12,m3=±c4k2+2c5k3±k2c42−4c3c5m12c5k1,c1=m2=0,where c2c5k1≠0, c42≥4c3c5, and k, k2, k3, and m1 are arbitrary constants.
By substituting (48) and (49) into (4), the interaction solution of (5) is obtained as follows:(50)u=4m1m1x+m3z+m4t+2kk1sinhk1x+k2y+k3z+k4tm1x+m3z+m4t2+kcoshk1x+k2y+k3z+k4t,where miandkii=1,…,4 satisfy (49) and the first set parameter solution in (49) is chosen. Figure 3 shows the spatial structures of the interaction solution and the contour plots for (50) by selecting the appropriate parameter values. Interestingly, it was found that Figure 3(a) is consistent with the result of the interaction between the kink soliton and the lump waves (see Figures 1 and 2).Figure 3
Spatial structures of interaction solution (50) with parameters: c2=1,c3=1,c4=4,c5=4,andm1=k1=k2=k3=k4=1. (a) z=0,t=0; (b) y=0,z=0; (c) x=0,y=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)Case 5.
(51)k1=k2m1m2,k3=k2m3m2,k4=−k2c3m22+c4m3m2+c5m32c2m1m2,m4=−c3m22+c4m3m2+c5m32c2m1,c1=0,where c2m1m2≠0, and k, k2, m3, and cii=3,4,5 are arbitrary constants.
By substituting (48) and (51) into (4), the interaction solution of (5) is as follows:(52)u=4m1m1x+m2y+m3z+m4t+2kk1sinhk1x+k2y+k3z+k4tm1x+m2y+m3z+m4t2+kcoshk1x+k2y+k3z+k4t,where miandkii=1,…,4 satisfy (51). Figure 4 shows the spatial structures of the interaction solution and the contour plots for (52) by selecting the appropriate parameter values.Figure 4
Spatial structures of interaction solution (52) with parameters: c2=c3=c4=c5=k1=k2=1,m1=1,m2=2,andm3=3. (a) y=0,z=0; (b) x=0,z=0; (c) z=0,t=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)Case 6.
(53)k4=−c3k22+c4k3k2+c5k32c2k1,m3=m2−c4±c42−4c3c52c5,m4=m2c42−4c3c5k2c42−4c3c5∓c4k2+2c5k32c2c5k1,c1=m1=0,where c2c5k1≠0, c42≥4c3c5, and k, k2, and k3 are arbitrary constants.
By substituting (48) and (53) into (4), the interaction solution of (5) is obtained as follows:(54)u=2kk1sinhk1x+k2y+k3z+k4tm2y+m3z+m4t2+kcoshk1x+k2y+k3z+k4t,where mi,kii=1,…,4 satisfy (53). The first set parameter solution in (53) is then chosen. Figure 5 shows the spatial structures of the interaction solution and the contour plots for (54) by selecting the appropriate parameter values.Figure 5
Spatial structures of interaction solution (54) with parameters: c2=c3=c5=k1=k2=k3=k=1,c4=3,andm2=2. (a) y=0,z=0; (b) x=0,z=0; (c) z=0,t=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)Case 7.
(55)k3=−c4k2m1±c42−4c3c52k1m2−k2m12c5m1,k4=m2−c42+4c3c5k1m2−k2m1c2c5m12,m3=m2−c4±c42−4c3c52c5,c1=m4=0,where c2c5m1≠0, c42≥4c3c5, and k, k2, m1, and m2 are arbitrary constants.
By substituting (48) and (55) into (4), the interaction solution of (5) is obtained as follows:(56)u=4m1m1x+m2y+m3z+2kk1sinhk1x+k2y+k3z+k4tm1x+m2y+m3z2+kcoshk1x+k2y+k3z+k4t,where mi,kii=1,…,4 satisfies (55). The first set parameter solution in (55) is then chosen. Figure 6 shows the spatial structures of the interaction solution and the contour plots for (56) by selecting the appropriate parameter values.Figure 6
Spatial structures of interaction solution (56) with parameters: c2=c3=c5=k1=k2=k3=k=m1=1,c4=3,andm2=2. (a) y=0,z=0; (b) x=0,z=0; (c) y=0,t=0. (d–f) Contour plots corresponding to subfigures (a–c), respectively.
(a)(b)(c)(d)(e)(f)


## 5. Conclusion

In this article, a generalized (3 + 1)-dimensional NLEE whose bilinear equation is obtained by a multivariable polynomial was discussed. Based on the bilinear equation, the BBT is given and it consists of four bilinear equations involving seven arbitrary parameters except for the parameters of the equation itself. By using the BBT, two traveling wave solutions and a mixed solution were obtained. Also, theN-soliton solutions were constructed and the dynamic properties of the soliton solutions were shown graphically. Furthermore, the lump and the interaction solutions were constructed. The properties of the explicit analytic solutions were shown by using figures. The general expression of the interaction solution was also given. By determining the different forms of the arbitrary function, more abundant interaction solutions will be obtained. Hence, the interaction between the lump and other solutions, including periodic solution and kink solution, can further be explored.Finally, this article not only presents a new example for (3 + 1)-dimensional NLEE but also provides a reference for finding the lump and the interaction solutions of other generalized multidimensional NLEEs.



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*Source: 5598381-2022-01-03.xml*