ZHANG Ying-yuan, LIU Xi-qiang, WANG Gang-wei

?

NONCLASSICAL LIE POINT SYMMETRY AND EXACT SOLUTIONS OF THE (2+1)-DIMENSIONAL NONLINEAR EVOLUTION EQUATION

*ZHANG Ying-yuan, LIU Xi-qiang, WANG Gang-wei

(School of Mathematical Sciences, Liaocheng University, Liaocheng, Shandong 252059, China)

Employing the compatibility method and nonclassical Lie group method, we derive the nonclassical Lie point symmetry of the (2+1)-dimensional nonlinear evolution equation. Nonclassical similarity reductions of the nonlinear evolution equation are obtained by solving the corresponding characteristic equations associated with nonclassical symmetry equations. Some new exact solutions to this equation are presented.

Nonlinear evolution equation; nonclassical Lie point symmetry; similarity reductions; exact solutions

1 Introduction

In this paper, combining the compatibility method[1-3] and nonclassical Lie group approach[4,5], we consider the (2+1)-dimensional nonlinear evolution equation

The paper is organized as follows. In section 2, based on some results relating to the symmetry, the compatibility method and nonclassical Lie group approach are applied to the nonlinear evolution equation to get the nonclassical symmetry. In section 3, we use the nonclassical symmetry to get nonclassical similarity reductions of the nonlinear evolution equation. By solving the reduction equations, we get varieties of new exact solutions to the nonlinear evolution equation and generalize the corresponding results in Refs[6,8,9]. The last section is a short summary and discussion.

2 Nonclassical Lie point symmetry of the nonlinear evolution equation

The basic idea of the compatibility method is to seek the nonclassical symmetry of a given NPDE such as Eq.(2) in the form

Similarly, we can also find the nonclassical symmetry of the Eq.(2) by the nonclassical Lie group method.The constraint condition is

(8)

The vector field (8) is a nonclassical symmetry of (2) if

Solving the determining equations, we can get the nonclassical Lie point symmetry of Eq.(2)

Remark 1 To the best our knowledge, thenonclassical Lie point symmetry is completely new and has not been studied yet.

3 Similarity reductions and new exact solutions of the nonlinear evolution equation

Having determined the nonclassical symmetry (15) of the nonlinear evolution equation, nonclassical similarity variables can also be found by solving the corresponding characteristic equations

(16)

For different possibilities, we determine four independent similarity reductions of the Eq.(2) by solving Eq.(16).

Substituting Eq.(17) into Eq.(2), one can get

(18)

Therefore，Eq.(2) has the following form solution

In addition，assuming Eq. (18) has the following solution

In this section, we will consider the exact analytic solutions to the reduced equations by using the power series method. we assume that the solutions of Eq.(22) can be expressed in the form

Substituting Eq.(23) into Eq.(22), we get

Hence, the power series solution of Eq.(22) can be written as following

Combining Eq.(17) and (27), respectively, then the new exact solution of the. Eq. (2) is expressed as

Remark 2 The exact solution of the rest of Eq.(2) and the solution in the approximate form can be written in terms of the above computation. The details are omitted here.

(31)

Solving Eq.(32), we can get the following solutions of the Eq.(2)

Substituting Eq.(33) into Eq.(2)，one can get the reduction of Eq.(2) as follows

In order to obtain the exact solutions of Eq.(34)，using the Lie point transformation group further reduce to the Eq.(34).

The corresponding symmetry is

Then we can write the corresponding characteristic equations

Solving (41), we can get the following solutions of the JM equation (2)

Eq.(43) can be further simplified to

Remark 3 Allthe solutions presented in this paper for Eq.(2) have been verified by Maple software.

4 Conclusions

By applying the compatibility method and nonclassical Lie group method to the nonlinear evolution equation, we get the nonclassical Lie point symmetry of the Eq.(2). Using the obtained symmety, we find three nonclassical similarity reductions of the nonlinear evolution equation. On this basis, new cases of Eq.(2) have been derived by using the Lie point transformation group further reduction to the reduced equation. Some new exact solutions of the nonlinear evolution equation have been found by solving the reduction equations.

[1] Mostafa F E, Ahmad T A. Nonclassical Symmetries for Nonlinear Partial Differential,Equations via Compatibility. Commun[J]. Theor .Phys., 2011, 56 : 611-616 .

[2] Wan W T, Chen Y. A Note on Nonclassical Symmetries of a Class of Nonlinear Partial Differential Equations and Compatibility Commun[J]. Theor.Phys.,2009,52:398-402.

[3] Niu X H, Huang L D, Pan Z L. The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility[J]. J .Math .Anal. Appl., 2006,320: 499-509.

[4] Chen M , Liu X Q. Symmetries and Exact Solutions of the Breaking Soliton Equation[J]. Commun. Theor .Phys., 2011,56: 851-855 .

[5] Bluman G W, Cole J D. Symmetries and Differential Equations[M]. Berlin: Appl Math Sci 81. Springer, 1989.

[6] Geng X G, Cao C W, Dai H H .Quasi-periodic solutions for some (2 + 1)-dimensional integrable models generated by the Jaulent-Miodek hierarchy[J]. J. Phys. A: Math. Gen., 2001, 34: 989-993.

[7] Geng X G. Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations[J]. J .Phys. A:Math. Gen., 2003, 36: 2289-2298.

[8] Wu J P. N-soliton solution, generalized double Wronskian determinant solution and rational solution for a (2+ 1)- dimensional nonlinear evolution equation[J]. Phys. Lett. A.,2008, 373: 83-88.

[9] Wazwaz A M. Multiple kink solutions and multiple singular kink solutions for (2 +1)-dimensional nonlinear models generated by the Jaulent–Miodek hierarchy[J]. Phys. Lett .A.,2009, 373: 1844-1846.

(2+1)維非線性發(fā)展方程的非經典李點對稱和精確解

*張穎元，劉希強，王崗偉

(聊城大學數學科學學院，山東，聊城 252059)

應用相容性方法和非經典李群方法，得到了（2+1）維非線性發(fā)展方程的非經典李點對稱。通過求解非經典對稱方程的相應的特征方程組得到了非線性發(fā)展方程的非經典相似約化。進而得到了非線性發(fā)展方程的新的精確解。

非線性發(fā)展方程；非經典李點對稱；相似約化；精確解

1674-8085(2013)02-0013-07

O641

A

10.3969/J.issn.1674-8085.2013.02.003

O641

A

10.3969/j.issn.1674-8085.2013.02.003

2012-08-27

2012-11-08

Supported by National Natural Science Foundation of China and China Academy of Engineering Physics (NSAF:11076015).

*Zhang Ying-yuan(1986-), Female; Jinan Shandong; Master; research direction: the Solution of Nonlinear Evolution Equations (E-mail:zhangyingyuanok@126.com);

Liu Xi-qiang(1957-), Male; Heze Shandong;Doctor; Professor; research direction: the System of Nonlinear Evolution Equations(E-mail:liuxiq@sina.com);

Wang Gang-wei(1982-), Male; Xingtai Hebei; Master; research direction: the Solution of Nonlinear Evolution Equations (E-mail:pukai1121@163.com).