# Exact solutions and conservation laws of a $(3+1)$-dimensional B-type Kadomtsev-Petviashvili equation

- Mufid Abudiab
^{1}and - Chaudry Masood Khalique
^{2}Email author

**2013**:221

https://doi.org/10.1186/1687-1847-2013-221

© Abudiab and Khalique; licensee Springer 2013

**Received: **3 May 2013

**Accepted: **26 June 2013

**Published: **23 July 2013

## Abstract

In this paper we study a $(3+1)$-dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation. This equation is an extension of the well-known Kadomtsev-Petviashvili equation, which describes weakly dispersive and small amplitude waves propagating in quasi-two-dimensional media. We first obtain exact solutions of the BKP equation using the multiple-exp function and simplest equation methods. Furthermore, the conservation laws for the BKP equation are constructed by using the multiplier method.

## Keywords

## 1 Introduction

It is well known that many phenomena in science and engineering, especially in fluid mechanics, solid state physics, plasma physics, plasma waves and biology, are described by the nonlinear partial differential equations (NLPDEs). Therefore the investigation of exact solutions of NLPDEs plays an important role in the study of NLPDEs. For this reason, during the last few decades, researchers have established several methods to find exact solutions to NLPDEs. Some of these methods include the inverse scattering transform method [1], the Bäcklund transformation [2], the Darboux transformation [3], the Hirota bilinear method [4], the $({G}^{\prime}/G)$-expansion method [5], the homogeneous balance method [6], the variable separation approach [7], the tri-function method [8, 9], the sine-cosine method [10], the Jacobi elliptic function expansion method [11, 12], the exp-function expansion method [13] and the Lie symmetry method [14–16].

where *α* is a real-valued constant. This is a nonlinear wave equation in three spatial $(x,y,z)$ and one temporal coordinate $(t)$.

was studied in [18–20] by different approaches. In [17] a new form of the $(3+1)$-dimensional BKP equation given by (1.1) was investigated and it was shown, using the simplified form of the Hirota method, that one- and two-soliton solutions exist for (1.1). Also, specific constraints were developed that guarantee the existence of multiple soliton solutions for (1.1).

In this paper we employ the multiple exp-function method [21] and the simplest equation method [22, 23] to obtain some exact solutions of (1.1). In addition to this, conservation laws are constructed for (1.1) using the multiplier method [24].

## 2 Exact solutions of (1.1)

In this section we employ two methods of solution.

### 2.1 Exact solutions using the multiple exp-function method

where *θ* is any root of ${\theta}^{2}+{k}_{1}^{2}\theta +{k}_{1}^{2}+1=0$.

and *θ* is any root of $2{\theta}^{2}-({m}_{2}-6)\theta +2{m}_{2}^{2}+6{m}_{2}+12=0$.

### 2.2 The simplest equation method

In this subsection we use the simplest equation method and obtain exact solutions of (1.1). This method was introduced by Kudryashov [22] and modified by Vitanov [23]. The simplest equations we use in this paper are the Bernoulli and Riccati equations. Their solutions can be written in elementary functions. For details, see, for example, [25].

where $G(\nu )$ satisfies the Bernoulli and Riccati equations, *M* is a positive integer that can be determined by balancing procedure as in [23] and ${A}_{0},\dots ,{A}_{M}$ are parameters to be determined.

#### 2.2.1 Solutions of (1.1) using the Bernoulli equation as the simplest equation

where $\nu ={k}_{1}x+{k}_{2}y+{k}_{3}z+{k}_{4}t+{k}_{5}$ and *C* is a constant of integration.

#### 2.2.2 Solutions of (1.1) using the Riccati equation as the simplest equation

where $\nu ={k}_{1}x+{k}_{2}y+{k}_{3}z+{k}_{4}t+{k}_{5}$ and *C* is a constant of integration.

## 3 Conservation laws

In this section we construct conservation laws for $(3+1)$-dimensional generalized B-type Kadomtsev-Petviashvili equation (1.1). The multiplier method will be used [15, 24, 26]. First we recall some results that will be used in the computation of conserved vectors.

### 3.1 Preliminaries

*k*th-order system of PDEs given by

*n*independent variables $x=({x}^{1},{x}^{2},\dots ,{x}^{n})$ and

*m*dependent variables $u=({u}^{1},{u}^{2},\dots ,{u}^{m})$. Here ${u}_{(1)},{u}_{(2)},\dots ,{u}_{(k)}$ denote the collections of all first, second, … ,

*k*th-order partial derivatives. That is, ${u}_{i}^{\alpha}={D}_{i}({u}^{\alpha}),{u}_{ij}^{\alpha}={D}_{j}{D}_{i}({u}^{\alpha}),\dots $ , respectively, where the total derivative operator with respect to ${x}^{i}$ is given by

*n*-tuple $T=({T}^{1},{T}^{2},\dots ,{T}^{n})$, ${T}^{j}\in \mathcal{A}$, $j=1,\dots ,n$, where is the space of differential functions, is a conserved vector of (3.1) if ${T}^{i}$ satisfies

and equation (3.3) defines a local conservation law of system (3.1).

*α*, is defined as

After obtaining the multipliers, we can calculate the conserved vectors by using a homotopy formula [24].

### 3.2 Construction of conservation laws for (1.1)

*C*is an arbitrary constant and

*f*is any solution of ${f}_{zz}-{f}_{tz}-{f}_{ty}=0$. Corresponding to the above multiplier, we obtain the following conserved vectors:

**Remark 1** Due to the presence of the arbitrary function *f* in the multiplier, one can obtain infinitely many conservation laws.

## 4 Concluding remarks

In this paper we studied $(3+1)$-dimensional generalized B-type Kadomtsev-Petviashvili equation (1.1). Exact solutions of the BKP equation were found using two distinct methods, namely the multiple-exp function method and the simplest equation method. Also, the conservation laws for the BKP equation were derived by using the multiplier method.

## Declarations

### Acknowledgements

MA and CMK would like to thank the organizing committee of the International Conference on the Theory, Methods and Application of Nonlinear Equations, held at Texas A&M University-Kingsville, USA, for their kind hospitality during the conference.

## Authors’ Affiliations

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.