# Lump and Lump-Type Solutions of the Generalized (3+1)-Dimensional Variable-Coefficient B-Type Kadomtsev-Petviashvili Equation

Authors: Yanni Zhang; Jing Pang
Journal: Journal of Applied Mathematics
 (2019)

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## Abstract

Based on the Hirota bilinear form of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, the lump and lump-type solutions are generated through symbolic computation, whose analyticity can be easily achieved by taking special choices of the involved parameters. The property of solutions is investigated and exhibited vividly by three-dimensional plots and contour plots.


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## Body

## 1. Introduction

The generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation [1] is(1)PBKPu=atuxxxy+ρatuxuyx+ux+uy+uzt+btuxx+uzz=0which is extended from the Kadomtsev-Petviashvili equation and can describe some interesting (3+1)-dimensional wave in fluid dynamics.In this paper, by using Hirota bilinear forms, we begin with studying the lump and lump-type solutions of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation. Exact solutions, especially the rational solutions, are important to descriptions of some physical phenomena [2–4]. Lump and lump-type solutions are special kinds of rational solutions. In recent years, there has been a growing interest in lump function solutions. Particular examples of lump and lump-type solutions are found for many nonlinear partial differential equations, such as the Kadomtsev-Petviashvili equation [5], the generalized Bogoyavlensky-Konopelchenko equation [6], the (2+1)-dimensional generalized fifth-order KdV equation [7], the generalized Kadomtsev-Petviashvili-Boussinesq equation [8], and the (3 + 1)-dimensional nonlinear evolution equation [9]. The interactions of lump solutions and interactions of other types solutions have also attracted much attention [10–13].In Section2, searching for the quadratic function solutions, we get the lump and lump-type solutions and analyze their dynamics. In the last section, some concluding remarks will be given.


## 2. Lump and Lump-Type Solutions

The Hirota bilinear form of a soliton equation needs to use appropriate variable transformation adopted in Bell polynomial theories [14, 15] to search. Based on the Bell polynomial method, under the transformation between f and u: (2)u=6ρln⁡fx.A Hirota bilinear equation can be proposed as follows: (3)BBKPf=atDx3Dy+DxDt+DyDt+DzDt+btDx2+Dz2f·f=2atffxxxy-fxxxfy+3fxxfxy-3fxxyfx+2fxtf-fxft+2fytf-fyft+2fztf-fzft+2btfxxf-fx2+fyyf-fy2+fzzf-fz2=0where f is a function of x, y, z, and t. Dx3Dy, DxDt, DyDt, Dx2, and Dz2 are the Hirota bilinear operators defined by(4)Dx1n1…DxlnlF·G=∂x1-∂x1′n1…∂xl-∂xl′nlFx1,…,xl×Gx1,…,xlx1′=x1,…xl′=xl,It is precise to find (5)PBKPu=BBKPff2xTherefore, when f solves the bilinear equation (3), u=6/ρ(ln⁡f)x will solve the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation (1).


### 2.1. Lump Solution

We apply the symbolic computation with Mathematica to show the quadratic function solutions of the (3+1)-dimensional Hirota bilinear equation (2). A direct Mathematica symbolic computation starts with (6)f=g2+h2+a5,and (7)g=x+a1y+a2z+a3t+a4,h=b0x+b1y+b2z+b3t+b4where ai(1≤i≤5) and bi(0≤i≤4) are all real parameters to be determined.With the aid of symbolic computation, substituting (3) into (2) and eliminating the coefficients of the polynomial yield the following three sets of constraining equations on the parameters: (8)a1=a1,a2=a2,a4=a4,a3=a2a2+b22+a1+1a22-1+a2b02-1-2b0b1+Abt1+a1+a22+b0+b1+b22,b0=b0,b1=b1,b2=b2,b4=b4,b3=-b21+a22+2a1a2+b22+b11-a22+b22-b0Bbt1+a1+a22+b0+b1+b22,a5=-31+b02a1+b0b11+a1+a22+b0+b1+b22ata1+2a2b0+a2-1b1-2+a1b2a1b0+b2-1+a2b1btwhere (9)A=a1b02-b22-b0-b22,B=2a1+b0b1+b0b2+1+a22-b22+b02with the condition a1+2a2b0+a2-1b1-2+a1b2[a1(b0+b2)-(1+a2)b1]≠0, and a1+b0b1<0. to guarantee the analyticity and rational localization of the solutions.Through the dependent variable transformationu=6/ρ(ln⁡f)x, substituting (6) into (3), respectively, we present three families of lump solutions for (1). To get the special lump solution of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, let us choose the following special case: a(t) and b(t) as constants at the same time which has no particular effect on the shape of the solution to this equation and choose the set of the parameters:(10)a1=-2,a2=52,a4=1,b0=1,b1=1,b2=1,b4=-1The corresponding special lump solution reads(11)u=122x-y+7/2z+1/2tx-2y+5/2z+3/2t2+x+y+z-t-12-1The obtained lump solutions of the three nonlinear evolution equations are same in the form except that the coefficients t are different. Therefore, the plots of the lump solutions of the three equations are similar and their properties are shown by giving the three-dimensional plots and the contour plots for the lump solution of the KP equation (1) (see Figure 1).Figure 1
Profiles of equation (1) with t=-0.04,0.1,3: 3d plots (top) and contour plots (bottom).


### 2.2. Lump-Type Solution

In this section, we obtain the lump-type solution by settingf into bilinear equation (2) as the quadratic function. We suppose f into bilinear equation (2) is shown in the form(12)f=g2+h2+a11,and (13)g=a1x+a2y+a3z+a4t+a5,h=a6x+a7y+a8z+a9t+a10where ai(1≤i≤11) are all real parameters to be determined.The resulting quadratic function solution presents the lump-type solutionu, under the transformation (2) to (1). The analyticity of the rational solution can be achieved and the solution u involves six parameters a1,a2, a3, a6, a7, and a8. a4, a5, and a9 are arbitrary. It yields the following set of constraining equations for the parameters:(14)a1=a1,a2=a2,a3=a3,a4=a2a32+a62-a82+a3a32+a62+a82a6+a7+a8+Cbta1+a2+a32+a6+a7+a82,a5=a5,a6=a6,a7=a7,a8=a8,a9=-a32a6+a7-a8-2a2a3a8+a6+a7+a8a62-a82+Dbta1+a2+a32+a6+a7+a82,a10=a10,a11=-3a12+a62a1a2+a6a7a1+a2+a32+a6+a7+a82ata2a6+a8-a1+a3a7+a32a6+a7+a2a6-a8-a1a7+2a8bt,where(15)C=a13-a12a2+a3+a1a32-a62-a82-2a6a7+a8,D=2a1a2a6+a3a6-a8+a12a6-a7-a8.If we choose the parameters guaranteeing a11>0, this satisfies(16)a1a7+a2a8≠0,(17)a3a7+a2a6≠0,(18)a1a2+a6a7<0.Let us choose the following special set of parameters: a1=1, a2=1, a3=1, a6=1, a7=-2, and a8=-1. The corresponding special lump-type solution reads(19)u=122x-y-4/13t+2x+y+z+8/13t+12+x-2y-z-12/13t+12+13/4We have found that all the three families of rational solutions exhibit the bright-dark lump wave structure. The property of the lump-type solution is shown by giving the three-dimensional plots and contour plots (see Figure2).Figure 2
Profiles of equation (1) with t=-1,0,5: 3d plots (top) and contour plots (bottom).


## 2.1. Lump Solution

We apply the symbolic computation with Mathematica to show the quadratic function solutions of the (3+1)-dimensional Hirota bilinear equation (2). A direct Mathematica symbolic computation starts with (6)f=g2+h2+a5,and (7)g=x+a1y+a2z+a3t+a4,h=b0x+b1y+b2z+b3t+b4where ai(1≤i≤5) and bi(0≤i≤4) are all real parameters to be determined.With the aid of symbolic computation, substituting (3) into (2) and eliminating the coefficients of the polynomial yield the following three sets of constraining equations on the parameters: (8)a1=a1,a2=a2,a4=a4,a3=a2a2+b22+a1+1a22-1+a2b02-1-2b0b1+Abt1+a1+a22+b0+b1+b22,b0=b0,b1=b1,b2=b2,b4=b4,b3=-b21+a22+2a1a2+b22+b11-a22+b22-b0Bbt1+a1+a22+b0+b1+b22,a5=-31+b02a1+b0b11+a1+a22+b0+b1+b22ata1+2a2b0+a2-1b1-2+a1b2a1b0+b2-1+a2b1btwhere (9)A=a1b02-b22-b0-b22,B=2a1+b0b1+b0b2+1+a22-b22+b02with the condition a1+2a2b0+a2-1b1-2+a1b2[a1(b0+b2)-(1+a2)b1]≠0, and a1+b0b1<0. to guarantee the analyticity and rational localization of the solutions.Through the dependent variable transformationu=6/ρ(ln⁡f)x, substituting (6) into (3), respectively, we present three families of lump solutions for (1). To get the special lump solution of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, let us choose the following special case: a(t) and b(t) as constants at the same time which has no particular effect on the shape of the solution to this equation and choose the set of the parameters:(10)a1=-2,a2=52,a4=1,b0=1,b1=1,b2=1,b4=-1The corresponding special lump solution reads(11)u=122x-y+7/2z+1/2tx-2y+5/2z+3/2t2+x+y+z-t-12-1The obtained lump solutions of the three nonlinear evolution equations are same in the form except that the coefficients t are different. Therefore, the plots of the lump solutions of the three equations are similar and their properties are shown by giving the three-dimensional plots and the contour plots for the lump solution of the KP equation (1) (see Figure 1).Figure 1
Profiles of equation (1) with t=-0.04,0.1,3: 3d plots (top) and contour plots (bottom).


## 2.2. Lump-Type Solution

In this section, we obtain the lump-type solution by settingf into bilinear equation (2) as the quadratic function. We suppose f into bilinear equation (2) is shown in the form(12)f=g2+h2+a11,and (13)g=a1x+a2y+a3z+a4t+a5,h=a6x+a7y+a8z+a9t+a10where ai(1≤i≤11) are all real parameters to be determined.The resulting quadratic function solution presents the lump-type solutionu, under the transformation (2) to (1). The analyticity of the rational solution can be achieved and the solution u involves six parameters a1,a2, a3, a6, a7, and a8. a4, a5, and a9 are arbitrary. It yields the following set of constraining equations for the parameters:(14)a1=a1,a2=a2,a3=a3,a4=a2a32+a62-a82+a3a32+a62+a82a6+a7+a8+Cbta1+a2+a32+a6+a7+a82,a5=a5,a6=a6,a7=a7,a8=a8,a9=-a32a6+a7-a8-2a2a3a8+a6+a7+a8a62-a82+Dbta1+a2+a32+a6+a7+a82,a10=a10,a11=-3a12+a62a1a2+a6a7a1+a2+a32+a6+a7+a82ata2a6+a8-a1+a3a7+a32a6+a7+a2a6-a8-a1a7+2a8bt,where(15)C=a13-a12a2+a3+a1a32-a62-a82-2a6a7+a8,D=2a1a2a6+a3a6-a8+a12a6-a7-a8.If we choose the parameters guaranteeing a11>0, this satisfies(16)a1a7+a2a8≠0,(17)a3a7+a2a6≠0,(18)a1a2+a6a7<0.Let us choose the following special set of parameters: a1=1, a2=1, a3=1, a6=1, a7=-2, and a8=-1. The corresponding special lump-type solution reads(19)u=122x-y-4/13t+2x+y+z+8/13t+12+x-2y-z-12/13t+12+13/4We have found that all the three families of rational solutions exhibit the bright-dark lump wave structure. The property of the lump-type solution is shown by giving the three-dimensional plots and contour plots (see Figure2).Figure 2
Profiles of equation (1) with t=-1,0,5: 3d plots (top) and contour plots (bottom).


## 3. Conclusion

In this paper, Based on the Hirota form of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, the lump and lump-type solutions of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation are obtained by symbolic computations and the resulting classes of solutions provide supplements to the existing lump and soliton solutions. Plots of some special lump and lump-type solutions were given which assist in describing complicated nonlinear physical phenomena in fluid mechanics and enrich the dynamic changes of high-dimensional nonlinear wave fields. Therefore, we expect that the results presented in this work will also be useful to study lump solutions in a variety of other high-dimensional nonlinear equations.



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