Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 805763, 7 pages

http://dx.doi.org/10.1155/2015/805763

## Symmetry Analysis and Conservation Laws of a Generalized Two-Dimensional Nonlinear KP-MEW Equation

International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

Received 2 August 2014; Accepted 20 September 2014

Academic Editor: Hossein Jafari

Copyright © 2015 Khadijo Rashid Adem and Chaudry Masood Khalique. 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

Lie symmetry analysis is performed on a generalized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation. The symmetries and adjoint representations for this equation are given and an optimal system of one-dimensional subalgebras is derived. The similarity reductions and exact solutions with the aid of -expansion method are obtained based on the optimal systems of one-dimensional subalgebras. Finally conservation laws are constructed by using the multiplier method.

#### 1. Introduction

Nonlinear evolution equations (NLEEs) have been widely used to describe natural phenomena of science and engineering. Therefore it is very important to find exact solutions of NLEEs. However, this is not an easy task. During the past few decades various integration techniques have been developed by the researchers to solve these NLEEs. Some of the well-known techniques used in the literature are the inverse scattering transform method [1], the homogeneous balance method [2], the Bäcklund transformation [3], the Weierstrass elliptic function expansion method [4], the Darboux transformation [5], the ansatz method [6, 7], Hirota’s bilinear method [8], the -expansion method [9], the Jacobi elliptic function expansion method [10, 11], the variable separation approach [12], the sine-cosine method [13], the trifunction method [14, 15], the F-expansion method [16], the exp-function method [17], the multiple exp-function method [18], and the Lie symmetry method [19–25].

The purpose of this paper is to study one such NLEE, namely, the generalized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width (KP-MEW) equation [26] that is given by Here, in (1) , , and are real valued constants. The solutions of (1) have been studied in various aspects. See, for example, the recent papers [26–28]. Wazwaz [26] used the tanh method and the sine-cosine method, for finding solitary waves and periodic solutions. Saha [27] used the theory of bifurcations of planar dynamical systems to prove the existence of smooth and nonsmooth travelling wave solutions. Wei et al. [28] used the qualitative theory of differential equations and obtained peakon, compacton, cuspons, loop soliton solutions, and smooth soliton solutions.

In this paper we obtain symmetry reductions of (1) using Lie group analysis [19–24] and based on the optimal systems of one-dimensional subalgebras. Furthermore, the -expansion method is employed to obtain some exact solutions of (1). In addition to this conservation laws will be derived for (1) using the multiplier method [29].

#### 2. Symmetry Reductions and Exact Solutions of (1)

The vector field of the form where , , and depend on , , , and , is a Lie point symmetry of (1) if whenever . Here [20] denotes the fourth prolongation of . Expanding (3) and splitting on the derivatives of , we obtain an overdetermined system of linear partial differential equations. Solving this system one obtains the following four Lie point symmetries:

##### 2.1. One-Dimensional Optimal System of Subalgebras

We now calculate the optimal system of one-dimensional subalgebras for (1) and use it to find the optimal system of group-invariant solutions for (1). We follow the method given in [20]. Recall that the adjoint transformations are given by where is the commutator defined by We present the commutator table of the Lie symmetries and the adjoint representations of the symmetry group of (1) on its Lie algebra in Tables 1 and 2, respectively. These two tables are then used to construct the optimal system of one-dimensional subalgebras for (1). As a result, after some calculations, one can obtain an optimal system of one-dimensional subalgebras given by , where , , .