Mathematical Problems in Engineering

Volume 2015, Article ID 426363, 10 pages

http://dx.doi.org/10.1155/2015/426363

## Lie Symmetry Analysis and New Exact Solutions for a Higher-Dimensional Shallow Water Wave Equation

Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

Received 10 May 2015; Accepted 21 June 2015

Academic Editor: Chaudry Masood Khalique

Copyright © 2015 Yinghui He. 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

In our work, a higher-dimensional shallow water wave equation, which can be reduced to the potential KdV equation, is discussed. By using the Lie symmetry analysis, all of the geometric vector fields of the equation are obtained; the symmetry reductions are also presented. Some new nonlinear wave solutions, involving differentiable arbitrary functions, expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function, and trigonometric function are obtained. Our work extends pioneer results.

#### 1. Introduction

It has recently become more interesting to obtain exact solutions of nonlinear partial differential equations. These equations are mathematical models of complex physical phenomena that arise in engineering, applied mathematics, chemistry, biology, mechanics, physics, and so forth. Thus, the investigation of the traveling wave solutions to nonlinear evolution equations (NLEEs) plays an important role in mathematical physics. A lot of physical models have supported a wide variety of solitary wave solutions.

In the recent years, much efforts have been spent on finding traveling wave solution and many significant methods have been established. However, the study on nonlinear wave solution is few and there is no unified approach. In this work, we studied the nonlinear wave solution of a higher-dimensional shallow water wave equation by using Lie symmetry analysis [1–5] and extend -expansion method [6–9].

Wazwaz [10] introduced the following -dimensional equation:as a higher-dimensional shallow water wave equation. It is easy to see that (1) can be reduced to the potential KdV equation for .

In [10], Wazwaz investigated multiple soliton solutions and multiple singular soliton solutions of (1) and pointed that this equation is a completely integrable equation. In [11], Chen and Liu obtained general multiple soliton solutions and some nonlinear wave solutions of (1) by simplified Hirota’s method [12, 13] and Dynamical system approach [14, 15]. The main purpose of this paper is to investigate the vector fields, the symmetry reductions, and exact solutions to (1) by means of the combination of Lie symmetry analysis and the extended -expansion method.

The rest of this paper is organized as follows. In Section 2, the Lie symmetry analysis is performed on (1); the complete geometric vector fields of the equation are obtained. In Section 3, different types of symmetry reductions of (1) are obtained. In Section 4, some new exact explicit solutions are presented. Section 5 is a short summary and discussion.

#### 2. Lie Symmetries for (1)

First of all, let us consider a one-parameter Lie group of infinitesimal transformation: with a small parameter . The vector field associated with the above group of transformations can be written asThe symmetry group of (1) will be generated by the vector field of the form (3). Applying the fourth prolongation to (1), we find that the coefficient functions , , , , and must satisfy the symmetry conditionwhere , , , , , and are the coefficients of . Furthermore, we havewhere , , , , and are the total derivatives with respect to , , , and , respectively.

Substituting (5) into (4), combined with (1) and equating the coefficients of the various monomials in the first, second, third, and the other partial derivatives and various powers of , we can find the determining equations for the symmetry group of (1); then standard symmetry group calculations lead to the following forms of the coefficient functions:where , , , , , and are arbitrary functions on their variables; and are arbitrary constants.

Thus, in terms of the Lie symmetry analysis method, we obtain all of the geometric vector fields of (1) as follows:The symmetry of (3) can be written asIt is necessary to check that is closed under the Lie bracket. In fact, we haveThus, the Lie algebra of infinitesimal symmetries of (1) is spanned by the above eight vector fields (7), and (7) form a basis for the Lie algebra. The commutator table is given by the above commutation relations.

#### 3. Symmetry Reductions

In this section we will obtain symmetry reductions of (1) by means of the symmetry analysis. Based on the infinitesimals (6), the similarity variables are found by solving the corresponding characteristic equationsor the invariant surface conditionsWhile solving the above invariant surface conditions, one has to distinguish between cases in which some of the functions , , , , , , and , are identical to zero and cases where they are not. This leads to different relations between the similarity variables and the original variables . As a result, we obtain the following cases.

*Case 1*. Let and ; thenSolving the differential equation one can getSubstituting (13) into (1), we can reduce it to

*Case 2*. Let and ; thenSolving the differential equation one can getSubstituting (16) into (1), we can reduce it to

*Case 3*. Let and ; thenSolving the differential equation one can getSubstituting (19) into (1), we can reduce it to

*Case 4*. Let ; thenSolving the differential equation one can getSubstituting (22) into (1), we can reduce it to

*Case 5*. Let ; thenSolving the differential equation one can getSubstituting (25) into (1), we can reduce it toObviously, if , (26) becomes (23).

*Case 6*. Let ; thenSolving the differential equation one can getSubstituting (28) into (1), we can reduce it toIts solution is . So, we can obtain solution of (1) as follows:where , , and are arbitrary functions on their variables. Similarly, when , we can get the following solutions:

*Case 7*. Let , ; thenSolving the differential equation one can getSubstituting (33) into (1), we can reduce it to

*Case 8*. Let , ; thenSolving the differential equation one can getSubstituting (36) into (1), we can reduce it to

#### 4. The New Nonlinear Wave Solutions

Obviously, it is easier for us to seek the explicit solutions to the reduction equations than to solve (1). The exact solutions of the reduction equations which were proposed in our work all can be solved by extended -expansion method. By solving these reduction equations, traveling wave solutions and nontraveling wave solutions can be obtained. For simplicity, we do not discuss them in detail. In this section, by introducing a special transformation, we will give out an equivalent equation and obtain its exact solutions by extended -expansion method.

Making a transformationwhere , , , , , and are arbitrary differential functions, (1) can be reduced to the following ODE: where , , and and are nonzero constants. , , , and are arbitrary differential functions.

Since is arbitrary, (39) holds if and only ifNote that (40) comes from taking the derivative on both sides of (41). Therefore we only consider (41). Integrating (41) once, it follows thatwhere is an integral constant. If , then (42) becomes

Balancing and in (43), we obtain which gives . Suppose that (43) owns the solutions in the formwhere satisfies the following equation:where , , , , and are constant.

Substituting (44) and (45) into (43) and then setting all the coefficients of of the resulting system to zero, we can obtain the following results.

##### 4.1.

In this situation, we obtain the following set of nontrivial solutions:where , , and are arbitrary constants and and are nonzero constants.

Substituting (46) into (44), we obtain, respectively, the following solutions of (43):where and .

When , the general elliptic equation (45) is reduced to the auxiliary ordinary equation

The solutions of (50) are given in Table 1. Combining (47)–(49) with Table 1, many exact solutions of (43) can be obtained. For simplicity, we just give out the first case in Table 1; the other cases can be discussed similarly.