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

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.


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][2][3][4][5] and extend F-expansion method [6][7][8][9].
Wazwaz [10] introduced the following (3+1)-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 F-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.

Lie Symmetries for (1)
First of all, let us consider a one-parameter Lie group of infinitesimal transformation: ( The symmetry group of (1) will be generated by the vector field of the form (3). Applying the fourth prolongation pr (5)  to (1), we find that the coefficient functions , , , , and  must satisfy the symmetry condition where   ,   ,   ,   ,   , and   are the coefficients of pr (5) .Furthermore, we have where  3  =       ,   ,   ,   , 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  1 (),  2 (),  3 (),  4 (, ),  5 (, ), and  6 (, ) are arbitrary functions on their variables;  1 and  2 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 as It is necessary to check that { 1 ( 1 ),  2 ( 2 ),  3 ( 3 ),  4 ( 4 ),  5 ( 5 ),  6 ( 6 ),  7 ,  8 } is closed under the Lie bracket.In fact, we have [ Thus, 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.

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 equations While solving the above invariant surface conditions, one has to distinguish between cases in which some of the functions  1 (),  2 (),  3 (),  4 (, ),  5 (, ),  6 (, ), and  1 ,  2 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.

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 F-expansion method.By solving these reduction equations, traveling wave solutions and nontraveling wave solutions can be obtained.
When ℎ 1 = ℎ 3 = 0, 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.

ℎ
In this case, there exists three parameters , , and  such that Equation ( 80) is satisfied only if the following relations hold: Equation ( 80) is the general Riccati equation.The solutions of (80) are listed in [9].There are 24 group solutions named    , ( = 1, 2, . . ., 24), which we do not list for simplicity.

Conclusions
In this paper, employing two methods, we studied a higherdimensional shallow water wave equation (1).Firstly, the invariance property of ( 1) is presented by using the Lie symmetry analysis.Then, all of the geometric vector fields and the symmetry reductions are obtained for the first