Journal of Applied Mathematics
Volume 2019 (2019), Article ID 7172860, 5 pages
https://doi.org/10.1155/2019/7172860

Research Article

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

Yanni Zhang1,2 and Jing Pang1,2

¹College of Sciences, Inner Mongolia University of Technology, Hohhot, China
²Inner Mongolia Key Laboratory of Civil Engineering Structure and Mechanics, Hohhot, China

Correspondence should be addressed to Jing Pang; pang_j@imut.edu.cn

Received 9 November 2018; Accepted 23 May 2019; Published 9 June 2019

Academic Editor: Mustafa Inc

Copyright © 2019 Yanni Zhang and Jing Pang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

The generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation [1] is

which 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 Section 2, 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 and :

A Hirota bilinear equation can be proposed as follows:

where is a function of , , , and . , , , , and are the Hirota bilinear operators defined by

It is precise to find

Therefore, when solves the bilinear equation (3), 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

and

where and 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:

where

with the condition , and to guarantee the analyticity and rational localization of the solutions.

Through the dependent variable transformation , 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: and 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:

The corresponding special lump solution reads

The obtained lump solutions of the three nonlinear evolution equations are same in the form except that the coefficients 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 : 3d plots (top) and contour plots (bottom).

2.2. Lump-Type Solution

In this section, we obtain the lump-type solution by setting into bilinear equation (2) as the quadratic function. We suppose into bilinear equation (2) is shown in the form

and

where are all real parameters to be determined.

The resulting quadratic function solution presents the lump-type solution , under the transformation (2) to (1). The analyticity of the rational solution can be achieved and the solution involves six parameters , , , , and . , , and are arbitrary. It yields the following set of constraining equations for the parameters:

where

If we choose the parameters guaranteeing , this satisfies

Let us choose the following special set of parameters: , , , , , and . The corresponding special lump-type solution reads

We 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 Figure 2).

Figure 2: Profiles of equation (1) with : 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.

Data Availability

All data generated or analysed during this study are included in this published article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 11561051 and the Graduate Student Scientific Research Innovation Project of Inner Mongolia Autonomous Region under Grant No. B2018111924Z.

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