In mathematical physics, many significant phenomena and dynamic processes can be represented by nonlinear partial differential equations (NLPDEs)     . Therefore, it is very important to find the solution of NLPDEs. A wealth of effective methods have been developed to explore the solutions of the NLPDEs, such as Adomian decomposition method  , Darboux and Backlund transformations  , Hirota bilinear method    , and Lie symmetry method   , etc. The Lie symmetry method for constructing explicit solutions of the NLPDEs has been regarded as one of the most widely applicable methods    . Its core idea is the invariance principle of the NLPDEs under the action of Lie point transformation group (point symmetry)  . In recent years, there has been a great deal of research and considerable development in the symmetry field of differential equations, in terms of the number of academic papers, books and new symbolic programs dedicated to this subject  -  .
At present, there is no general method for solving NLPDEs. Although the symmetry method has a wide range of applications in solving methods, it still faces many difficulties and challenges to promote its development. However, the symmetry method and other methods (e.g. generalized simple equation method  , generalized Tanh function method  , homotopy perturbation method  and power series method  , etc.) are effectively combined to reflect the complementarity of each other, which makes it possible to obtain exact solutions of some NLPDEs with physical significance, and attracts the attention and research of many scholars    .
In the present paper, based on the Lie group method, we will investigate the dispersive long wave equations
where u represents the amplitude of a surface wave, propagating along the x-axis with a horizontal velocity. It plays an important role in nonlinear physics   , considered as a good model for the study of bidirectional solitons in water waves. In   , Eckhaus and Boiti et al. presented the extensions of Equation (1) in higher-dimensional spaces. In  , Zhang J F et al. discussed its some new multi-soliton solutions and travelling wave solutions using the extended homogeneous balance method, etc.
The outline of this paper is as follows: in Section 2, the Lie symmetry analysis is performed for the dispersive long wave equations; in Section 3, the optimal systems and the similarity reductions of Equation (1) are researched employing Lie group analysis in the last section; in Section 4, the exact solutions for the reduced equation are obtained by using the power series method and the extended Tanh method; and in Section 5, a brief summary is done to the full text.
2. Lie Symmetry Analysis
We first do some preparatory work on the concept of classical Lie symmetry of general NLPDEs. Consider the kth-order scalar NLPDEs of the form
where denotes n independent variables, denotes m independent variables, and denote the partial derivatives of with respect to up to jth-order, i.e.
Suppose that the one-parameter Lie group of point transformations
where , . is an infinitesimal parameter, are some smooth function with variables .
Theorem 1.  (The Infinitesimal Invariant Formula for NLPDEs) If
is the infinitesimal generator of the one-parameter Lie group of transformations for (3), and the k-th prolongation of the infinitesimal generator is
where the prolongation of the infinitesimals satisfy the following recurrence relation
where denotes the total derivative operator defined as
That one-parameter Lie group of transformations (3) is the Lie symmetry of Equation (2), if and only if
Next, we calculate the Lie symmetry of Equation (1). With regard to the infinitesimal generator of Equation (1), it can be expressed from (4) as the following form
Applying the Theorem 1 to Equation (1), we have
By simplifying (7), we can get the following overdetermined equations about
From (8) it is easy to caculate that the only solution of this system is
where and are arbitrary constants. Accordingly, the symmetry groups of Equation (1) can be written as
The infinitesimal generators (10) correspond to a four-parameter Lie group of nontrivial point transformations acting on -space.
3. Optimal System and Symmetry Reductions
3.1. Optimal System
In this section, we study how to construct the one-dimensional optimal system of Equation (1) in order to obtain more abundant group invariant solutions. The basic method of constructing it is to simplify the expression of Lie algebra by using a variety of adjoint transformations on the most general expression of Lie algebra. The adjoint transformation is expressed as the following series form
where is a parameter, and is the usual commutator, given by
According to the method of constructing one dimensional optimal system in  , we set up the following non-zero vector field with arbitrary coefficients and , which is a Lie algebras made up of (10)
and simplify the coefficients of the vector as much as possible. Without loss of generality, suppose first that and set up , then the vector X becomes To eliminate the coefficient of , we use to act on X by means of the adjoint operation, i.e.
where the group parameter . Then continue to eliminate by using one after another to act on , the vector becomes
where the group parameters . It can be seen easily that the vector form can not be simplified much more. Secondly, suppose that and set up , the vector X becomes . To eliminate the coefficient of the vector we use to act on X by means of the adjoint operation, i.e.
Table 1. Commutator table.
Table 2. Adjoint representation table.
where the group parameter . Obviously, it can not continue to simplify by using adjoint operators. Thirdly, suppose that and set up , the vector is already the simplest form as . Last suppose that and set up , that can only be .
To summarize, we state the result that the one-dimensional optimal system of symmetry groups (10) is
where a is arbitrary constant.
3.2. Symmetry Reductions
In the present section, we present all possible similarity reduction forms of Equation (1), which is an indispensable step to solve the NLPDEs by the symmetry method.
For the symmetry , the corresponding characteristic equation is
hence we can get a similarity independent variable from (11) defined as and group invariant solutions defined as
which satisfy the following reduced equation
For other symmetries in the optimal system, the reduction method is the same as . The results are shown in Table 3.
4. Explicit Solution of the Dispersive Long Wave Equation
In the third section, we obtain the one-dimensional optimal system of Equation (1), and give the reduction equation corresponding to each symmetry in the optimal system in Table 3. The reduction equations corresponding to and can be easily solved by Mathematica, where the process is omitted. For other symmetries in the optimal system, it is very difficult to get directly through the calculation software. In this connection, we will use two methods to solve the rest of reduction equations, namely, the power series method and the extended Tanh function method.
4.1. Explicit Power Series Solutions of the Reduction Equation (A)
The power series method is a useful approach to solve higher order ordinary differential equations. A large number of solutions for ordinary differential equations can be constructed by utilizing the method.
Suppose that the power series solution is the following form
where is undetermined coefficient. Substituting (12) into (A), we get
Through comparing the coefficients of ς, we can easily get the following results
The sequence can be uniquely determined by (13) and (14) and depend on the other undetermined coefficients . It is easy to
Table 3. Reduction of the nonlinear long wave equation.
prove that the power series solution is convergent by references  , so the reduction Equation (A) has the following power series solution
And then we get the following power series solution of Equation (1)
where are arbitrary constant.
4.2. Explicit Solutions of the Reduction Equation (B) Using Extended Tanh Function Method
The extended Tanh function method is a very effective method for solving some nonlinear evolution equations proposed in recent years  . The method is based on the Tanh function expansion method and using the general Riccati equation as an auxiliary equation. It can transform the solution of complex equations into the solution of nonlinear algebraic equations by traveling wave transformation. Next we use this method to find the traveling wave solutions of the reduction Equation (B) in Table 3.
Suppose that the solution of the reduction Equation (B) can be expressed as the form
where are undetermined constants, and function satisfies
where are arbitrary constant. By solving Equation (16), we can know that the solution of function can be divided into 4 categories, and amount to 27 solutions  .
1) when and (or ),
where are two nonzero constants, and satisfy .
2) when and (or ),
where are two nonzero constants, and satisfy .
Above formula , the symbol is expressed as .
3) when and ,
where b is a arbitrary constant.
4) when and ,
where c is a arbitrary constant, and .
Considering the homogeneous equilibrium between the highest order linear term and the nonlinear term in the reduction Equation (B), we can obtain . As a result, the trial Equations (16) reduces to
In order to determine the values of undetermined coefficients , substituting (16) and (17) into the reduction Equation (B) and merging the polynomial of the same power of , and setting up each polynomial coefficient to zero, we can get the following nonlinear algebraic equations
By solving the above system with the help of Mathematic, we can get the following results
Now, substituting (18) into (19), we obtain explicit solutions of Equation (1) as follow.
where , and selecting any hyperbolic function in , for example,
The explicit solutions (19) become as
where (see Figure 1).
4.3. Explicit Power Series Solutions of the Reduction Equation (C)
In this section, we study the power series solution of the reduction Equation (C)
Figure 1. Plot of explicit solution above with .
in the form of (12). Substituting (12) into the reduction Equation (C), we get
Through comparing the coefficients of , we can easily get the following results.
Accordingly, the power series solution of the reduction Equation (C) is as follows
And then we get the following power series solution of Equation (1)
4.4. Explicit Solutions of the Reduction Equation (D) Using Extended Tanh Function Method
Using extended tanh function method, similar to the solving of the reduction Equation (B), we obtain the following results:
and selecting the following hyperbolic function
We obtain explicit solutions of the Equation (1.1) as follow
where (see Figure 2).
Figure 2. Plot of explicit solution above with .
In the field of physics and engineering mechanics, it is particularly important to solve nonlinear differential equations. In the work, the Lie group analysis method has been employed to investigate the dispersive long wave equations. Based on this method, the vector fields and symmetry reductions have been obtained for the system. Since it is difficult to solve the reduction equations directly, the power series method and the extended Tanh function method have been used to construct more explicit solutions, which can enrich the exact solutions of the dispersive long wave equations. The basic idea is efficient and powerful in solving wide classes of nonlinear differential equations.
The authors would like to express their thanks to the unknown referees for their careful reading and helpful comments. Project supported by the Natural Science Foundation of Inner Mongolia, China (Grant No. 2016MS0116).
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