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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 837241, 9 pages
doi:10.5402/2012/837241
Research Article

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

Wenbin Zhang,1 Jiangbo Zhou,2 and Sunil Kumar3

1Taizhou Institute of Science and Technology, (NUST), Taizhou, Jiangsu 225300, China
2Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China
3Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand 831014, India

Received 22 April 2012; Accepted 4 July 2012

Academic Editors: A. Aghamohammadi and S. del Campo

Copyright © 2012 Wenbin Zhang et al. 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

Employing the classical Lie method, we obtain the symmetries of the ZK-BBM equation. Applying the given Lie symmetry, we obtain the similarity reduction, group invariant solution, and new exact solutions. We also obtain the conservation laws of ZK-BBM equation of the corresponding Lie symmetry.

1. 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: 𝑢 𝑡 + 𝑢 𝑥 + 𝑎 𝑢 𝑢 𝑥 𝑢 + 𝑏 𝑥 𝑥 𝑡 + 𝑢 𝑦 𝑦 𝑥 = 0 , ( 1 . 1 ) 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.

2. 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 𝜕 𝑉 = 𝜉 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) 𝜕 𝜕 𝑥 + 𝜁 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) 𝜕 𝜕 𝑦 + 𝜏 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) 𝜕 𝜕 𝑡 + 𝜂 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) , 𝜕 𝑢 ( 2 . 1 ) and its third prolongation can be given as P r ( 3 ) 𝑉 = 𝑉 + 𝜙 𝑡 𝜕 𝜕 𝑢 𝑡 + 𝜙 𝑥 𝜕 𝜕 𝑢 𝑥 + 𝜙 𝑥 𝑥 𝑡 𝜕 𝜕 𝑢 𝑥 𝑥 𝑡 + 𝜙 𝑦 𝑦 𝑥 𝜕 𝜕 𝑢 𝑦 𝑦 𝑥 , ( 2 . 2 ) where 𝜙 𝑡 , 𝜙 𝑥 , 𝜙 𝑥 𝑥 𝑡 , 𝜙 𝑦 𝑦 𝑥 are given explicitly in terms of 𝜉 , 𝜁 , 𝜏 , 𝜙 , and the derivatives of 𝜙 .

From P r ( 3 ) 𝑉 𝐹 | 𝐹 = 0 = 0 , it follows 𝜙 𝑡 + 𝜙 𝑥 + 𝑎 𝑢 𝑥 𝜙 + 𝑎 𝑢 𝜙 𝑥 + 𝑏 𝜙 𝑥 𝑥 𝑡 + 𝑏 𝜙 𝑦 𝑦 𝑥 = 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 𝜉 = 𝑐 3 1 , 𝜁 = 2 𝑐 1 𝑦 + 𝑐 4 , 𝜏 = 𝑐 1 𝑡 + 𝑐 2 , 𝜙 = − 𝑐 1 𝑎 𝑢 + 1 𝑎 , ( 2 . 4 ) where 𝑐 1 , 𝑐 2 , 𝑐 3 , and 𝑐 4 are arbitrary constants.

The corresponding symmetries are 𝜎 = 𝑐 3 𝑢 𝑥 + 1 2 𝑐 1 𝑦 + 𝑐 4 𝑢 𝑦 + 𝑐 1 𝑡 + 𝑐 2 𝑢 𝑡 − 𝑐 1 𝑎 𝑢 + 1 𝑎 𝑢 . ( 2 . 5 ) The vector field can be written as 𝜈 = 𝑐 3 𝜕 + 1 𝜕 𝑥 2 𝑐 1 𝑦 + 𝑐 4 𝜕 + 𝑐 𝜕 𝑦 1 𝑡 + 𝑐 2 𝜕 𝜕 𝑡 − 𝑐 1 𝑎 𝑢 + 1 𝑎 𝜕 . 𝜕 𝑢 ( 2 . 6 ) 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: 𝑑 𝑑 𝜖 𝑥 , 𝑦 , 𝑡 , 𝑢 = ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) , 𝑥 , 𝑦 , 𝑡 , 𝑢 | 𝜖 = 0 = ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) , ( 2 . 7 ) where 𝜖 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): 𝑔 1 ∶ 𝑥 , 𝑦 , 𝑡 , 𝑢 ⟶ 𝑔 ( 𝑥 + 𝜖 , 𝑦 , 𝑡 , 𝑢 ) , 2 ∶ 𝑥 , 𝑦 , 𝑡 , 𝑢 𝑔 ⟶ ( 𝑥 , 𝑦 + 𝜖 , 𝑡 , 𝑢 ) , 3 ∶ 𝑥 , 𝑦 , 𝑡 , 𝑢 ⟶ 𝑔 ( 𝑥 , 𝑦 , 𝑡 + 𝜖 , 𝑢 ) , 4 ∶ 𝑥 , 𝑦 , 𝑡 , 𝑢 ⟶ 𝑥 , 𝑦 + 𝑒 ( 1 / 2 ) 𝜖 , 𝑡 + 𝑒 𝜖 , 𝑢 + 𝑒 − 𝜖 − 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: 𝑢 1 𝑢 = 𝑓 ( 𝑥 − 𝜖 , 𝑦 , 𝑡 ) , 2 𝑢 = 𝑓 ( 𝑥 , 𝑦 − 𝜖 , 𝑡 ) , 3 𝑢 = 𝑓 ( 𝑥 , 𝑦 , 𝑡 − 𝜖 ) , 4 = 𝑓 𝑥 , 𝑦 − 𝑒 − ( 1 / 2 ) 𝜖 , 𝑡 − 𝑒 − 𝜖 − 1 𝑎 + 𝑒 − 𝜖 . ( 2 . 9 ) For example, taking the following periodic wave solution [3] of (1.1) 𝑢 ( 𝑥 , 𝑦 , 𝑡 ) = 3 ( 𝑐 − 1 ) 2 𝑎 s e c 2 1 2 √ 𝑏 𝜉 , ( 2 . 1 0 ) where 𝑏 > 0 and 𝜉 = 𝑥 + 𝑦 − 𝑐 𝑡 .

One can obtain new exact solution of (1.1) by applying 𝑢 4 as follows: 𝑢 ( 𝑥 , 𝑦 , 𝑡 ) = 3 ( 𝑐 − 1 ) 2 𝑎 s e c 2 1 2 √ 𝑏 𝜉 − 1 𝑎 + 𝑒 − 𝜖 , ( 2 . 1 1 ) where 𝑏 > 0 and 𝜉 = 𝑥 + 𝑦 − 𝑒 − ( 1 / 2 ) 𝜖 − 𝑐 ( 𝑡 − 𝑒 − 𝜖 ) .

3. Reductions of ZK-BBM Equation

Now we will reduce (1.1) by means of the symmetries (2.4). Here we discuss the following cases.

Case 1. Letting 𝑐 1 ≠ 0 , 𝑐 2 = 0 , 𝑐 3 = 0 , 𝑐 4 = 0 . Then 1 𝜎 = 2 𝑐 1 𝑦 𝑢 𝑦 + 𝑐 1 𝑡 𝑢 𝑡 − 𝑐 1 𝑎 𝑢 + 1 𝑎 𝑢 . ( 3 . 1 ) Solving the differential equation 𝜎 = 0 , one can get 𝑦 𝜉 = 𝑥 , 𝜂 = 2 𝑡 1 , 𝑢 = 𝑡 1 𝑓 ( 𝜉 , 𝜂 ) − 𝑎 . ( 3 . 2 ) Substituting (3.2) into (1.1), we have − 𝑓 − 𝜂 𝑓 𝜂 + 𝑓 𝑓 𝜉 + 𝑏 − 𝑓 𝜉 𝜉 − 𝜂 𝑓 𝜉 𝜉 𝜂 + 2 𝑓 𝜂 𝜉 + 4 𝜂 𝑓 𝜂 𝜂 𝜉 = 0 . ( 3 . 3 )

Case 2. Letting 𝑐 1 = 0 , 𝑐 2 ≠ 0 , 𝑐 3 ≠ 0 , 𝑐 4 ≠ 0 . Then 𝜎 = 𝑐 3 𝑢 𝑥 + 𝑐 4 𝑢 𝑦 + 𝑐 2 𝑢 𝑡 . ( 3 . 4 ) Solving the differential equation 𝜎 = 0 , one can get 𝑐 𝜉 = 𝑥 − 3 𝑐 2 𝑐 𝑡 , 𝜂 = 𝑦 − 4 𝑐 2 𝑡 , 𝑢 = 𝑓 ( 𝜉 , 𝜂 ) . ( 3 . 5 ) Substituting (3.5) into (1.1), one can reduce (1.1) to the following equation: 𝑐 1 − 3 𝑐 2 𝑓 𝜉 − 𝑐 4 𝑐 2 𝑓 𝜂 + 𝑎 𝑓 𝑓 𝜉 − 𝑐 + 𝑏 3 𝑐 2 𝑓 𝜉 𝜉 𝜉 − 𝑐 4 𝑐 2 𝑓 𝜉 𝜉 𝜂 + 𝑓 𝜂 𝜂 𝜉 = 0 . ( 3 . 6 )

Case 3. Letting 𝑐 1 = 0 , 𝑐 2 ≠ 0 , 𝑐 3 = 0 , 𝑐 4 ≠ 0 . Then 𝜎 = 𝑐 4 𝑢 𝑦 + 𝑐 2 𝑢 𝑡 . ( 3 . 7 ) Solving the differential equation 𝜎 = 0 , one can get 𝑐 𝜉 = 𝑥 , 𝜂 = 𝑦 − 4 𝑐 2 𝑡 , 𝑢 = 𝑓 ( 𝜉 , 𝜂 ) . ( 3 . 8 ) Substituting (3.8) into (1.1), we have 𝑓 𝜉 − 𝑐 4 𝑐 2 𝑓 𝜂 + 𝑎 𝑓 𝑓 𝜉 − 𝑐 + 𝑏 4 𝑐 2 𝑓 𝜉 𝜉 𝜂 + 𝑓 𝜂 𝜂 𝜉 = 0 . ( 3 . 9 )

4. 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: 𝑓 = 𝑟 ( 𝜂 ) 𝜉 + 𝜔 ( 𝜂 ) , ( 4 . 1 ) where 𝑟 ( 𝜂 ) and 𝜔 ( 𝜂 ) are the functions to be determined. Substituting (4.1) into (3.3) yields 1 𝑓 = 1 + 𝑐 1 𝜂 𝜉 + 𝜔 ( 𝜂 ) , ( 4 . 2 ) then the new exact solution of (1.1) is expressed as 𝑢 1 = 1 𝑎 𝑡 + 𝑑 1 𝑦 2 𝑥 + 2 𝑏 𝑑 1 l n 𝑡 + 𝑑 1 𝑦 2 − 4 𝑏 𝑑 1 l n 𝑦 + 𝑑 2 − 8 𝑏 𝑡 𝑑 1 𝑎 𝑡 + 𝑑 1 𝑦 2 2 − 1 𝑎 , ( 4 . 3 ) 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 𝑓 = 𝑁 𝑖 = 1 𝛼 𝑖 𝐺 ( 𝜔 ) 𝐺 ( 𝜔 ) 𝑖 + 𝛼 0 , 𝛼 𝑚 ≠ 0 , ( 4 . 4 ) where 𝜔 = 𝑘 𝜉 + 𝑙 𝜂 , 𝛼 𝑖 are constants to be determined later. It can be seen 𝑚 = 2 by balancing 𝑓 𝜉 and 𝑓 𝑓 𝜉 in (3.4). Suppose that the solutions of (3.4) are of the form 𝑓 = 𝛼 2 𝐺 ( 𝜔 ) 𝐺 ( 𝜔 ) 2 + 𝛼 1 𝐺 ( 𝜔 ) 𝐺 ( 𝜔 ) + 𝛼 0 , 𝛼 2 ≠ 0 ( 4 . 5 ) with 𝐺 ( 𝜔 ) satisfying the second-order linear ordinary differential equation 𝐺 ( 𝜔 ) + 𝜆 𝐺 ( 𝜔 ) + 𝜇 𝐺 ( 𝜔 ) = 0 , ( 4 . 6 ) where 𝜔 = 𝑘 𝜉 + 𝑙 𝜂 , 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.

When 𝜆 2 − 4 𝜇 > 0 , 𝑢 2 = 3 𝑏 𝑎 𝑐 2 𝜆 2 𝑐 − 4 𝜇 3 𝑘 + 𝑐 4 𝑙 − 𝑐 2 𝑙 2 𝑘 𝑑 1 s i n ℎ 𝜔 + 𝑑 2 c o s ℎ 𝜔 𝑑 1 c o s ℎ 𝜔 + 𝑑 2 + s i n ℎ 𝜔 2 𝑏 𝑎 𝑐 2 𝜆 2 − 4 𝜇 − 𝑐 3 𝑘 − 𝑐 4 𝑙 + 𝑐 2 𝑙 2 𝑘 + 1 𝑎 𝑐 2 𝑐 4 𝑙 𝑘 − 𝑐 2 + 𝑐 3 . ( 4 . 7 )

When 𝜆 2 − 4 𝜇 < 0 , 𝑢 3 = − 3 𝑏 𝑎 𝑐 2 4 𝜇 − 𝜆 2 𝑐 3 𝑘 + 𝑐 4 𝑙 − 𝑐 2 𝑙 2 𝑘 𝑑 1 s i n ℎ 𝜔 + 𝑑 2 c o s ℎ 𝜔 𝑑 1 c o s ℎ 𝜔 + 𝑑 2 + s i n ℎ 𝜔 2 𝑏 𝑎 𝑐 2 4 𝜇 − 𝜆 2 − 𝑐 3 𝑘 − 𝑐 4 𝑙 + 𝑐 2 𝑙 2 𝑘 + 1 𝑎 𝑐 2 𝑐 4 𝑙 𝑘 − 𝑐 2 + 𝑐 3 . ( 4 . 8 )

When 𝜆 2 − 4 𝜇 = 0 , 𝑢 4 = 1 2 𝑏 𝑎 𝑐 2 𝑐 3 𝑘 + 𝑐 4 𝑙 − 𝑐 2 𝑙 2 𝑘 𝑑 1 s i n ℎ 𝜔 + 𝑑 2 c o s ℎ 𝜔 𝑑 1 c o s ℎ 𝜔 + 𝑑 2 + 1 s i n ℎ 𝜔 𝑎 𝑐 2 𝑐 4 𝑙 𝑘 − 𝑐 2 + 𝑐 3 , ( 4 . 9 ) where 𝜔 = 𝑘 𝑥 + 𝑙 𝑦 − ( ( 𝑘 𝑐 3 + 𝑙 𝑐 − 4 ) / 𝑐 2 ) 𝑡 .

Case 3. Let 𝑓 = 𝑓 ( 𝜔 ) , where 𝜔 = 𝑘 𝜉 + 𝑙 𝜂 . Equation (3.9) becomes the following ordinary partial differential equation: 𝑐 𝑘 − 4 𝑐 2 𝑙 𝑓 + 𝑎 𝑘 𝑓 𝑓 − 𝑐 + 𝑏 4 𝑐 2 𝑘 2 𝑙 + 𝑘 𝑙 2 𝑓 = 0 . ( 4 . 1 0 )
Integrating twice with respect to 𝜔 in (4.10), we get 𝑓 2 = 𝐴 1 𝑓 3 + 𝐴 2 𝑓 2 + 𝐴 3 𝑓 , ( 4 . 1 1 ) where 𝐴 1 = 𝑐 2 𝑎 / 3 𝑏 𝑙 ( 𝑐 4 𝑘 − 𝑐 2 𝑙 ) ,   𝐴 2 = ( 𝑐 2 𝑘 − 𝑐 4 𝑙 ) / 𝑏 𝑘 𝑙 ( 𝑐 4 𝑘 − 𝑐 2 𝑙 ) , and 𝐴 3 is a constant.

Since solutions of (4.11) have been given [8], we have some similarity solutions of (1.1) as follows.

When 𝐴 1 = 4 𝑚 2 , 𝐴 2 = − 4 ( 𝑚 2 + 1 ) , 𝐴 3 = 4 , 𝑢 5 = 𝑠 𝑛 2 𝑐 𝑘 𝑥 + 𝑙 𝑦 − 4 𝑐 2 𝑙 𝑡 . ( 4 . 1 2 )

When 𝐴 1 = 4 𝑚 2 , 𝐴 2 = − 4 ( 2 𝑚 2 − 1 ) , 𝐴 3 = 4 ( 1 − 𝑚 2 ) , 𝑢 6 = 𝑐 𝑛 2 𝑐 𝑘 𝑥 + 𝑙 𝑦 − 4 𝑐 2 𝑙 𝑡 . ( 4 . 1 3 )

When 𝐴 1 = − 4 , 𝐴 2 = 4 ( 2 − 𝑚 2 ) , 𝐴 3 = − 4 ( 1 − 𝑚 2 ) , 𝑢 7 = 𝑑 𝑛 2 𝑐 𝑘 𝑥 + 𝑙 𝑦 − 4 𝑐 2 𝑙 𝑡 . ( 4 . 1 4 )

When 𝐴 1 𝑓 < 0 , 𝐴 2 > 0 , 𝐴 3 = 0 , 𝑢 8 𝐵 = − 2 𝐵 1 s e c h 2 √ 𝐵 2 2 𝑐 𝑘 𝑥 + 𝑙 𝑦 − 4 𝑐 2 𝑙 𝑡 . ( 4 . 1 5 )

When 𝐴 1 𝑓 > 0 , 𝐴 2 > 0 , 𝐴 3 = 0 , 𝑢 9 = 𝐵 2 𝐵 1 c s c h 2 √ 𝐵 2 2 𝑐 𝑘 𝑥 + 𝑙 𝑦 − 4 𝑐 2 𝑙 𝑡 . ( 4 . 1 6 )

When 𝐴 2 < 0 , 𝐴 3 = 0 , 𝑢 1 0 = 𝐵 2 𝐵 1 s e c 2 √ 𝐵 2 2 𝑐 𝑘 𝑥 + 𝑙 𝑦 − 4 𝑐 2 𝑙 𝑡 . ( 4 . 1 7 )

5. 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 + 𝑎 𝑢 𝑣 𝑥 + 𝑏 𝑣 𝑥 𝑥 𝑡 + 𝑏 𝑣 𝑦 𝑦 𝑥 = 0 , ( 5 . 1 ) and the Lagrangian is 𝑢 𝐿 = 𝑣 𝑡 + 𝑢 𝑥 + 𝑎 𝑢 𝑢 𝑥 + 𝑏 𝑢 𝑥 𝑥 𝑣 𝑡 + 𝑏 𝑢 𝑦 𝑦 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: 𝐶 𝑖 = 𝜉 𝑖 𝐿 + 𝑊 𝛼 𝜕 𝐿 𝜕 𝑢 𝛼 𝑖 − 𝐷 𝑗 𝜕 𝐿 𝜕 𝑢 𝛼 𝑖 𝑗 + 𝐷 𝑗 𝐷 𝑘 𝜕 𝐿 𝜕 𝑢 𝛼 𝑖 𝑗 𝑘 + 𝐷 𝑗 ( 𝑊 𝛼 ) 𝜕 𝐿 𝜕 𝑢 𝛼 𝑖 𝑗 − 𝐷 𝑘 𝜕 𝐿 𝜕 𝑢 𝛼 𝑖 𝑗 𝑘 + 𝐷 𝑗 𝐷 𝑘 ( 𝑊 𝛼 ) 𝜕 𝐿 𝜕 𝑢 𝛼 𝑖 𝑗 𝑘 ( 𝑖 = 1 , 2 , 3 ) , ( 5 . 3 ) where 𝑊 𝛼 = 𝜂 𝛼 − 𝜉 𝑗 𝑢 𝛼 𝑗 .

The conserved vector corresponding to an operator 𝑣 = 𝜉 1 𝜕 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) 𝜕 𝑥 + 𝜉 2 𝜕 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) 𝜕 𝑦 + 𝜉 3 𝜕 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) 𝜕 𝜕 𝑡 + 𝜂 ( 𝑥 , 𝑦 , 𝑡 , 𝑢 ) . 𝜕 𝑢 ( 5 . 4 ) The operator 𝑣 yields the conservation law 𝐷 𝑡 𝐶 1 + 𝐷 𝑥 𝐶 2 + 𝐷 𝑦 𝐶 3 = 0 , ( 5 . 5 ) where the conserved vector 𝐶 = ( 𝐶 1 , 𝐶 2 , 𝐶 3 ) is given by (5.3) and has the components 𝐶 1 = 𝑤 1 𝑣 ( 1 + 𝑎 𝑢 ) − 𝐷 𝑥 𝑏 𝑣 𝑣 𝑡 + 𝑘 𝑢 𝑦 𝑦 𝑣 𝑤 2 + 𝑏 𝑣 𝑣 𝑡 𝐷 𝑥 𝑤 1 , 𝐶 2 = 𝜉 2 𝐿 − 𝐷 𝑦 𝑘 𝑣 𝑣 𝑥 𝑤 1 + 𝑏 𝑣 𝑣 𝑥 𝐷 𝑦 𝑤 1 , 𝐶 3 = 𝜉 3 𝐿 + 𝑤 1 + 𝑏 𝑢 𝑥 𝑥 𝑣 𝑤 2 . ( 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 − ( ( 𝑎 𝑢 + 1 ) / 𝑎 ) 𝜕 𝑢 . For this operator, we have 𝑤 1 = − ( 𝑎 𝑢 + 1 ) / 𝑎 − ( ( 1 / 2 ) 𝑦 ) 𝑢 𝑦 − 𝑡 𝑢 𝑡 , 𝑤 2 = ( 1 / 2 ) 𝑣 − ( ( 1 / 2 ) 𝑦 ) 𝑣 𝑦 − 𝑡 𝑣 𝑡 and (5.3) written for the sixth-order Lagrangian equation (5.2) yields the conserved vector 𝐶 1 1 = − 𝑢 + 𝑎 + 1 2 𝑦 𝑢 𝑦 + 𝑡 𝑢 𝑡 𝑣 + 𝑎 𝑢 𝑣 − 𝑏 𝑣 𝑥 𝑣 𝑡 − 𝑏 𝑣 𝑣 𝑡 𝑥 + 𝑏 𝑢 𝑦 𝑦 𝑣 1 2 1 𝑣 − 2 𝑦 𝑣 𝑦 − 𝑡 𝑣 𝑡 − 𝑏 𝑣 𝑣 𝑡 𝑢 𝑥 + 1 2 𝑦 𝑢 𝑦 𝑥 + 𝑡 𝑢 𝑡 𝑥 , 𝐶 2 = 1 2 𝑢 𝑦 𝑣 𝑡 + 𝑢 𝑥 + 𝑎 𝑢 𝑢 𝑥 + 𝑏 𝑢 𝑥 𝑥 𝑣 𝑡 + 𝑏 𝑢 𝑦 𝑦 𝑣 𝑥 𝑣 − 𝑏 𝑦 𝑣 𝑥 + 𝑣 𝑣 𝑥 𝑦 ⋅ 1 𝑢 + 𝑎 + 1 2 𝑦 𝑢 𝑦 + 𝑡 𝑢 𝑡 − 𝑏 𝑣 𝑣 𝑥 𝑢 𝑦 + 1 2 𝑢 𝑦 + 1 2 𝑦 𝑢 𝑦 𝑦 + 𝑡 𝑢 𝑡 𝑦 , 𝐶 3 𝑢 = 𝑡 𝑣 𝑡 + 𝑢 𝑥 + 𝑎 𝑢 𝑢 𝑥 + 𝑏 𝑢 𝑥 𝑥 𝑣 𝑡 + 𝑏 𝑢 𝑦 𝑦 𝑣 𝑥 − 1 𝑢 + 𝑎 + 1 2 𝑦 𝑢 𝑦 + 𝑡 𝑢 𝑡 + 𝑏 𝑢 𝑥 𝑥 𝑣 1 2 1 𝑣 − 2 𝑦 𝑣 𝑦 − 𝑡 𝑣 𝑡 . ( 5 . 7 )

Acknowledgments

This work was supported by the Startup Fund for Advanced Talents of Jiangsu University (no. 09JDG013), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), the National Nature Science Foundation of China (no. 71073072), the Nature Science Foundation of Jiangsu (no. BK 2010329), the Project of Excellent Discipline Construction of Jiangsu Province of China, the Taizhou Social Development project (no. 2011213), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

References

  1. X. P. Xin, X. Q. Liu, and L. L. Zhang, “Explicit solutions of the Bogoyavlensky-Konoplechenko equation,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3669–3673, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. X. P. Xin, X. Q. Liu, and L. L. Zhang, “Symmetry reduction, exact solutions and conservation laws of the modifed kadomtzev-patvishvili-II equation,” Chinese Physics Letters, vol. 28, no. 2, Article ID 020201, 2011.
  3. A. M. Wazwaz, “Compact and noncompact physical structures for the ZK-BBM equation,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 713–725, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. M. Wazwaz, “The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equations,” Chaos, Solitons and Fractals, vol. 38, no. 5, pp. 1505–1516, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. M. A. Abdou, “The extended F-expansion method and its application for a class of nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 95–104, 2007. View at Publisher · View at Google Scholar
  6. M. Song and C. Yang, “Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation,” Applied Mathematics and Computation, vol. 216, no. 11, pp. 3234–3243, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. M. Wang, X. Li, and J. Zhang, “The G′/G-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008. View at Publisher · View at Google Scholar
  8. A. Elhanbaly and M. A. Abdou, “Exact travelling wave solutions for two nonlinear evolution equations using the improved F-expansion method,” Mathematical and Computer Modelling, vol. 46, no. 9-10, pp. 1265–1276, 2007. View at Publisher · View at Google Scholar
  9. Z. L. Yan and X. Q. Liu, “Symmetry reductions and explicit solutions for a generalized Zakharov-Kuznetsov equation,” Communications in Theoretical Physics, vol. 45, no. 1, pp. 29–32, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH