Advances in Mathematical Physics

Volume 2018, Article ID 7843498, 5 pages

https://doi.org/10.1155/2018/7843498

## Lump Solutions to the (3+1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation

School of Mathematical Sciences, Yangzhou University, China

Correspondence should be addressed to Xifang Cao; nc.ude.uzy@oacfx

Received 13 August 2018; Revised 26 September 2018; Accepted 11 October 2018; Published 1 November 2018

Academic Editor: Antonio Scarfone

Copyright © 2018 Xifang Cao. 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

This paper is devoted to the study of lump solutions to the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation. First we use a direct method to construct a class of exact solutions which contain six arbitrary real constants. Then we use these solutions to generate lump solutions with four real parameters. We also determine the amplitude and velocity of these lumps.

#### 1. Introduction

Lump is a type of localized rational solutions. Lump solutions for the (2+1)-dimensional Kadomtsev-Petviashvili I equationwere first obtained by Manakov et al. [1]. Then Satsuma and Ablowitz [2] constructed more lump solutions by taking a long wave limit of the corresponding soliton solutions. More recently, by use of the bilinear form, Ma [3] generalized the results of [1, 2] and got a larger class of lump solutions. Over the past few decades, lump has been an active area in the study of nonlinear evolution equations. They can be used to describe nonlinear patterns in plasma [4], in optical media [5], in the Bose-Einstein condensate [6], and so forth. Lump solutions have been obtained for some other equations, such as the Ishimori equation [7], the Jimbo-Miwa equation [8], and the Sawada-Kotera equation [9].

Recently much literature is devoted to the study of (3+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equationwhich can be used to model fluid dynamics, plasma physics, and weakly dispersive media [10, 11]. Various methods have been applied to (2) to construct its soliton and multiple wave solutions [12–14]. In this paper, we develop another method to construct its lump solutions. We first use a direct method to obtain a class of exact solutions which contain six arbitrary real constants (see Theorem 1). Then we show that the limits of these solutions can generate lump solutions (see Theorem 4). We also determine the amplitude and velocity of these lumps.

#### 2. Exact Solutions

In this section, we seek exact solutions to the (3+1)-dimensional BKP equation (2) in the following form:where, , and are real constants to be determined. Substituting (3) into the left hand side of (2) and then factoring we get the following three determining equations:Solving for from (5) and (6) givesandSubstituting (8) and (9) into (7) yieldsTherefore, we have the following.

Theorem 1. *The (3+1)-dimensional BKP equation (2) admits the following exact solutions:where , , are arbitrary real constants, and are given by (8), (9), and (10), respectively.*

#### 3. Lump Solutions

In this section, we use (11) to construct lump solutions to the (3+1)-dimensional BKP equation (2). We first give some examples and then give a general result.

*Example 1. *Let , , , , . Then (11) becomeswhere , . As , the limit of (12) gives a lump solution to (2):where

The lump (13) attains its maximum value at and minimum value at . So its amplitude is . The profiles of the lump (13) in the -space are given in Figure 1.