Symmetry Reduction , Exact Solutions , and Conservation Laws of the ZK-BBM Equation

Symmetry is one of the most important concepts in the area of partial differential equations, especially in integrable systems, which exist infinitely in many symmetries. These symmetry group techniques provide one method for obtaining exact and special solutions of a given partial differential equation. To find the Lie point symmetry of a nonlinear equation, some effective methods have been introduced, such as the classical Lie group approach, the nonclassical Lie group approach, the Clarkson-Kruskal CK direct symmetry method, and the compatibility method 1, 2 . The purpose of this paper is to apply the classical Lie method to the following Zakharov-Kuznetsov-Benjamin-Bona-Mahony ZK-BBM equation:


Introduction
Symmetry is one of the most important concepts in the area of partial differential equations, especially in integrable systems, which exist infinitely in many symmetries.These symmetry group techniques provide one method for obtaining exact and special solutions of a given partial differential equation.To find the Lie point symmetry of a nonlinear equation, some effective methods have been introduced, such as the classical Lie group approach, the nonclassical Lie group approach, the Clarkson-Kruskal CK direct symmetry method, and the compatibility method 1, 2 .
The purpose of this paper is to apply the classical Lie method to the following Zakharov-Kuznetsov-Benjamin-Bona-Mahony ZK-BBM equation: which was presented by Wazwaz 3,4 .Wazwaz obtained some solitons solutions, periodic solutions, and complex solutions by using the sine-cosine method and the extended tanh method.Abdou 5 used the extended F-expansion method to construct exact solutions.Song and Yang 6 employed the bifurcation method of dynamical systems to obtain traveling wave solutions.This paper is arranged as follows.In Section 2, we get the symmetries and group invariant solutions of 1.1 .In Section 3, the reductions of 1.1 are obtained, and in Section 4, we derive some new exact solutions of 1.1 .Finally, the conservation laws of 1.1 will be presented in Section 5.

Symmetry of the ZK-BBM Equation
The main task of the classical Lie method is to seek some symmetries and look for exact solutions of a given partial differential equation.The vector field of 1.1 can be expressed as From Pr 3 V F| F 0 0, it follows φ t φ x au x φ auφ x bφ xxt bφ yyx 0.

2.3
Letting the coefficients of the polynomial be zero yields a set of differential equations of the functions ξ, ζ, τ, and φ.Solving these equations, one can arrive at where c 1 , c 2 , c 3 , and c 4 are arbitrary constants.
The corresponding symmetries are The vector field can be written as In order to get some exact solutions from the known ones of 1.1 , we will find the corresponding Lie symmetry groups.To this end, solving the following initial problems: where is a parameter.From 2.7 we can obtain the Lie symmetry group g : x, y, t, u → x, y, t, u .According to different ξ, ζ, τ, and φ in σ, we have the following group by solving 2.7 : , t e , u e − − 1 a .

2.8
We can obtain the corresponding new solutions by applying the above groups g 1 , g 2 , g 3 , and g 4 as follows:

2.9
For example, taking the following periodic wave solution 3 of 1.1 where b > 0 and ξ x y − ct.
One can obtain new exact solution of 1.1 by applying u 4 as follows: where b > 0 and ξ x y − e − 1/2 − c t − e − .

Reductions of ZK-BBM Equation
Now we will reduce 1.1 by means of the symmetries 2.4 .Here we discuss the following cases.

Similarity Solutions of ZK-BBM Equation
By solving the reduced equations 3.3 , 3.6 , and 3.9 , we can get some new explicit solutions of 1.1 .Now we discuss some cases in the following.
Case 1. Assuming 3.3 has the following solution: where r η and ω η are the functions to be determined.Substituting 4.1 into 3.3 yields then the new exact solution of 1.1 is expressed as where d 1 and d 2 are constants.
Case 2. To solve 3.6 , we apply the G /G-expansion method 7 and look for the travelling wave solution of 3.6 by setting where ω kξ lη, α i are constants to be determined later.It can be seen m 2 by balancing f ξ and ff ξ in 3.4 .Suppose that the solutions of 3.4 are of the form with G ω satisfying the second-order linear ordinary differential equation where ω kξ lη, α 0 , α 1 , α 2 , λ, and μ are constants to be determined later.Substituting 4.5 into 3.6 along with 4.6 and setting the coefficients of G ω /G ω i , i 0, . . ., 4 to zero yields a system of equations with respect to α 0 , α 1 , α 2 , h, and l.Solving the corresponding algebraic equations and using 3.5 , 4.5 , and 4.6 , we get three types of traveling wave solutions of 1.1 as follows.
Case 3. Let f f ω , where ω kξ lη.Equation 3.9 becomes the following ordinary partial differential equation: Integrating twice with respect to ω in 4.10 , we get where Since solutions of 4.11 have been given 8 , we have some similarity solutions of 1.1 as follows. When

4.12
When

4.13
When

4.15
When

Conservation Laws of ZK-BBM Equation
In this section we will study the conservation laws by using the adjoint equation and symmetries of ZK-BBM equation.For 1.1 , the adjoint equation has the form v t v x auv x bv xxt bv yyx 0, 5.1 and the Lagrangian is L v u t u x auu x bu xx v t bu yy v x .

5.2
Since every Lie point, Lie-backlund and nonlocal symmetry of 1.1 , provides a conservation law for 1.1 and the adjoint equation 9 , the elements of conservation vector C 1 , C 2 , C 3 are defined by the following expression: where W α η α − ξ j u α j .The conserved vector corresponding to an operator v ξ

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The operator v yields the conservation law where the conserved vector C C 1 , C 2 , C 3 is given by 5.3 and has the components

5.6
Thus, 5.6 define components of a nonlocal conservation law for the system of 1.1 , 5.1 corresponding to any operator v admitted by 1.1 .
Let us make more detailed calculations for the operator v 1/2 y∂y t∂t − au 1 /a ∂u.For this operator, we have w 1  − au 1 /a − 1/2 y u y − tu t , w 2 1/2 v − 1/2 y v y − tv t and 5.3 written for the sixth-order Lagrangian equation 5.2 yields the conserved vector u x auu x bu xx v t bu yy v x − b v y v x vv xy 3 tv u t u x auu x bu xx v t bu yy v x t C