Lie Symmetry Reductions and Exact Solutions to the Rosenau Equation

Nonlinear partial differential equations (PDEs) arising in many physical fields such as condensed matter physics, fluid mechanics, and plasma physics and optics exhibit a rich variety of nonlinear phenomena. It is known that to find exact solutions of the PDEs is always one of the central themes in mathematics and physics. A wealth of methods have been developed to find these exact physically significant solutions of a PDE though it is rather difficult. Some of themost important methods are the inverse scattering method [1], Hirota bilinear method [2], Darboux and Bäcklund transformations [3], Lie symmetry analysis [4–6], CK method [7, 8], and so forth. As is well known, the Lie group method is a powerful and direct approach to construct exact solutions of nonlinear differential equations. Furthermore, based on the Lie group method, many types of exact solutions of PDEs can be obtained, such as the traveling wave solutions, similarity solutions, soliton wave solutions, and fundamental solutions [9, 10]. The main idea of Lie group method is to transform solutions of a system of differential equations to other solutions. Once one has determined the symmetry group of a system of differential equations, a number of applications become available. Then one can directly use the defining property of such a group and construct new solutions to the system from known ones. In the present paper, based on the Lie group method, we will consider an important equation, which is the Rosenau equation [11, 12] with the form


Introduction
Nonlinear partial differential equations (PDEs) arising in many physical fields such as condensed matter physics, fluid mechanics, and plasma physics and optics exhibit a rich variety of nonlinear phenomena.It is known that to find exact solutions of the PDEs is always one of the central themes in mathematics and physics.A wealth of methods have been developed to find these exact physically significant solutions of a PDE though it is rather difficult.Some of the most important methods are the inverse scattering method [1], Hirota bilinear method [2], Darboux and Bäcklund transformations [3], Lie symmetry analysis [4][5][6], CK method [7,8], and so forth.As is well known, the Lie group method is a powerful and direct approach to construct exact solutions of nonlinear differential equations.Furthermore, based on the Lie group method, many types of exact solutions of PDEs can be obtained, such as the traveling wave solutions, similarity solutions, soliton wave solutions, and fundamental solutions [9,10].
The main idea of Lie group method is to transform solutions of a system of differential equations to other solutions.Once one has determined the symmetry group of a system of differential equations, a number of applications become available.Then one can directly use the defining property of such a group and construct new solutions to the system from known ones.
In the present paper, based on the Lie group method, we will consider an important equation, which is the Rosenau equation [11,12] with the form where  = (, ) is the unknown real function.
Equation (1) appears in a wide variety of physical applications.It can investigate the dynamics of dense discrete systems in the case of wave-wave and wave-wall interactions that cannot be described using the well-known KdV equation.So it is important to lucubrate the exact explicit solutions and similarity reductions for this equation.A number of works have been devoted to study the Rosenau equation such as decay and scattering of small amplitude solution [13], posteriori error estimates [14], and conservative difference scheme [15].However, as the authors knew, the Lie symmetry analysis of (1) is left as open problems.
The outline of this paper is as follows: in Section 2, we perform Lie symmetry analysis for the Rosenau equation; in Section 3, we discuss the Lie symmetry group of (1); in Section 4, we deal with the similarity reductions of (1) using Lie group method and provide exact solutions of the equation based on the Lie group method; in Section 5, the exact solutions for the reduced equation are obtained by using the power series method; and in Section 6, we summarize our results and make closing remarks.

Lie Symmetry Analysis of the Rosenau Equation
In this section, we perform Lie symmetry analysis for (1) and obtain its infinitesimal generator and commutation The symmetry group of (1) will be generated by the vector field of the form (3). Applying the fourth prolongation pr (4)  to (1), we find that the coefficient functions (, , ), (, , ), and (, , ) must satisfy the symmetry condition where   ,   , and   are the coefficients of pr (4) .Furthermore, we have where   ,   are the total derivatives with respect to  and , respectively.Substituting (5) into (4), equating the coefficients of the various monomials in the first, second, and the other order partial derivatives and various powers of , we can find the following equations for the symmetry group of the Rosenau equation: Solving (6), we obtain where  1 ,  2 , and  3 are arbitrary constants.Hence, the Lie algebra of infinitesimal symmetries of (1) is spanned by the following vector fields Then, all of the infinitesimal generators of (1) can be expressed as It is easy to verify that { 1 ,  2 ,  3 } is closed under the Lie bracket.In fact, we have

Symmetry Groups of the Rosenau Equation
In this section, in order to get exact solutions from a known solution of (1), we should find the one-parameter symmetry groups   : (, , ) → ( x, t, ũ) of corresponding infinitesimal generators.To get the Lie symmetry groups, we should solve the following initial problems of ordinary differential equations: where  = (x, t, ũ),  = (x, t, ũ),  = (x, t, ũ), and  is a group parameter.
For the infinitesimal generator  =  1  1 +  2  2 +  3  3 , we will take the following different values to obtain the corresponding infinitesimal generators.
The one-parameter groups   of the above corresponding infinitesimal generators are given as follows: where  is any real number.We observe that  1 is a space translation,  3 is a time translation,  6 is a space-time translation, and the groups   ( = 2, 4, 5) are genuinely local groups of transformation.Their appearance is far from obvious from basic physical principles, but they are very important for us to study the exact solutions of PDEs.

Symmetry Reductions and Exact Solutions of the Rosenau Equation
In the previous sections, we obtained the infinitesimal generators.In this section, we will get similarity variables and their reduction equations and obtain similarity solutions by solving the reduction equations.
Case 1.For the infinitesimal generator  1 = /, we have the following similarity variables: and the group-invariant solution is  = (); that is, Substituting ( 14) into (1), we obtain the following reduction equation: where   = /.Therefore, (1) has a solution  = , where  is arbitrary constant.Obviously, the solution is not meaningful.
Case 3.For the infinitesimal generator  3 = /, we have the following similarity variables: and the group-invariant solution is  = (); that is, Substituting (20) into (1), we obtain the following reduction equation: where   = /.Therefore, (1) has a solution  = , where  is arbitrary constant.
Remark 1.Note that the reduced equations such as ( 18) and ( 24) are all higher-order nonlinear or nonautonomous ODEs; we will deal with such equations in the next section.

The Exact Power Series Solutions
In this section, we want to detect the explicit solutions expressed in terms of elementary or known functions of mathematical physics, in terms of quadratures, and so on.But this is not always the case, even for simple semilinear PDEs.However, we know that the power series can be used to solve differential equations, including many complicated differential equations with nonconstant coefficients [16].In this section, we will consider the exact analytic solutions to the reduced equations by using the power series method.
Once we get the exact analytic solutions of the reduced equations (ODEs), the exact power series solutions to the original PDEs are obtained.In this section, we will consider the exact solutions of ( 18), ( 24), ( 27), and (30).(18).Now, we seek a solution of ( 18) in a power series of the form
Hence, the power series solution of ( 18) can be written as follows: (36) Thus, the exact power series solution of (1) is where  0 ,  1 , and  2 are arbitrary constants; the other coefficients   ( ≥ 3) can be determined successively from (33) and (34).
In physical applications, it will be convenient to write the solution of (1) in the approximate form in terms of the above computation.
Advances in Mathematical Physics 5

Exact Analytic Solutions to (24).
Similarly, we seek a solution of (24) in a power series of the form (31). Substituting it into (24) and comparing coefficients, we obtain Generally, for  ≥ 0, we have In view of (39)-(42), we can get all the coefficients   ( ≥ 1) of the power series (31); for example, Thus, for arbitrary chosen constant  0 , the other terms of the sequence {  } ∞ =0 can be determined successively from (39)-(42) in a unique manner.This implies that, for (24), there exists a power series solution (31) with the coefficients given by ( 39)-(42).

Exact Analytic Solutions to
For  ≥ 1, we have (47) From ( 46) and (47), we can get all the coefficients   ( ≥ 3) of the power series (31) such as Thus, for arbitrary chosen constants  0 ,  1 , and  2 , the other terms of the sequence {  } ∞ =0 can be determined successively from ( 46) and (47) in a unique manner.This implies that, for (27), there exists a power series solution (31) with the coefficients given by ( 46) and (47).
The exact solution of (1) and the solution in the approximate form can be written in terms of the above computation.The details are omitted here.

Exact Analytic Solutions to (30).
Similarly, we seek a solution of (30) in a power series of the form (31). Substituting it into (30) and comparing coefficients, we obtain In view of (49), we can get all the coefficients   ( ≥ 4) of the power series (31) such as Thus, for arbitrary chosen constants  0 ,  1 ,  2 , and  3 , the other terms of the sequence {  } ∞ =0 can be determined successively from (49) in a unique manner.Therefore, the power series solution of (30) can be written as follows: where   ( = 0, 1, 2, 3) are arbitrary constants; the other terms  +4 ( = 0, 1, 2, . ..) are given by (49) successively.
Remark 2. By using the integration of ordinary differential equations (ODEs), we know that if we get a one-parameter symmetry group of an ODE, then we can reduce the order of the equation by one.But we note that such reduced ODEs are more complicated than the original equation in addition to some special cases.In view of this, we can see that the power series method is a useful tool of solving such higher-order nonlinear or nonautonomous ODEs.

Conclusions
In this paper, we study the symmetry reductions and exact solutions of the Rosenau equation by using the classical Lie group method.First, we perform Lie symmetry analysis for the Rosenau equation and get its infinitesimal generator and commutation table of Lie algebra.Then, we discuss the Lie symmetry groups of the Rosenau equation.Moreover, using similarity variables to obtain reduction equations and solving the reduction equation, we obtain all the group-invariant solutions to the equation.Then the exact analytic solutions are investigated by using the power series method.Especially, the similarity reductions and exact solutions are given for the first time in this paper.The physical significance of the solutions is considered from the transformation group point of view.These similarity solutions possess significant features in the nonlinear mechanics aspects of the work.