Potential symmetries and invariant solutions of two nonlinear equations

Abstract

Abstract: We expanded the classical symmetries of NTT equation and Burgers equation by calculating the potential symmetries, and we obtained a series of new exact solutions of Burgers equation. Firstly, we determined the classical symmetries and potential symmetries of NTT equation and Burgers equation based on differential characteristic set algorithm, and we determined one-parameter Lie transformation group of potential symmetries X_9, X_10 to Burgers equation. Secondly, we constructed the invariant solutions of Burgers equation by using the extended simplest equation method, and these solutions with double arbitrary parameters are expressed by the hyperbolic functions, the trigonometric functions and the rational functions, respectively. Finally, new exact solutions for Burgers equation are obtained by acting Lie transformation group of potential symmetry X_9 on the invariant solutions. It is important that these solutions can not be obtained form classical symmetries of Burgers equation.

Key words: potential symmetry；differential characteristic set algorithm；NTT equation；Burgers equation；invariant solution

CLC Number:

O175.29

WANG Xiao-Min, SUDAO Bilige. Potential symmetries and invariant solutions of two nonlinear equations[J]. J4, 2016, 33(5): 537-544.

References

[1]Bluman G W, Kumei S. Symmetries and Differential Equations[M]. Spring-Verlag: New York Berlin, 1989.
[2]Sophocleous C, Wiltshire R.J. Linearisation and potential symmetries of certain systems of diffusion equations[J]. Phys A, 2006,370: 329-345.
[3]Qu C Z. Potential symmetries to system of nonlinear diffusion equation[J]. J. Phys A, 2007, 40: 1757-1773.
[4]Bluman G W, Temuer Chaolu. Local and nonlocal Symmetries for nonlinear telegraph equation[J]. J. Math. phys}. 2005, 46: 023505.
[5]特木尔朝鲁, 张志勇. 一类非线性波动方程的势对称分类[J]. 系统科学与数学, 2009, 29(3): 389–411.
[6]苏道毕力格, 朝鲁. 用吴方法计算BBM-Burgers方程的势对称及其不变解[J]．内蒙古大学学报自然科学版, 2006, 37(4): 366-373.
[7]特木尔朝鲁, 白玉山. 基于吴方法的确定和分类(偏)微分方程古典和非古典对称新算法理论[J]. 中国科学A. 2010, 40(4): 1-18.
[8]特木尔朝鲁, 额尔敦布和, 郑丽霞. 扩充偏微分方程(组)守恒律和对称的辅助方程方法及微分形式吴方法的应用[J]. 应用数学学报, 2007, 30(5): 910-927.
[9]Temuer Chaolu, Bai Y S. Differential characteristic set algorithm for the complete symmetry classification of partial differential equations[J]. Appl. Math. Mech. Engl. Ed. 2009, 30(5): 595-606.
[10]苏道毕力格, 王晓民, 乌云莫日根. 对称分类在非线性偏微分方程组边值问题中的应用[J]. 物理学报, 2014, 63(4): 040201.
[11]苏道毕力格,王晓民,鲍春玲. 利用对称方法求解非线性偏微分方程组边值问题的数值解[J]. 应用数学, 2014, 27(4): 708-713.
[12]朱永平, 吉飞宇, 陈晓艳. 广义KdV-Burgers方程的势对称和不变解[J]. 纯粹数学与应用数学, 2013, 29(2): 164-171.
[13]张红霞, 郑丽霞, 杜永胜. Benney方程的势对称和不变解[J]. 动力学与控制学报, 2008, 6(3): 219-222.
[14]Sudao Bilige, Temuer Chaolu. An extended simplest equation method and its application to several forms of the fifth-order KdV equation[J]. Appl. Math. Comput. 2010, 216: 3146-3153.
[15]Sudao Bilige, Temuer Chaolu, Wang X.M. Application of the extended simplest equation method to the coupled Schrodinger-Boussinesq equation[J]. Appl. Math. Comput. 2013, 224: 517-523.
[16]He Y H, Zhao Y M, Long Y. Exact Solutions for a New Nonlinear KdV-Like Wave Equation Using Simplest Equation Method and Its Variants[J]. J. Appl. Math. 2014, 578362.
[17]Zhao Y M. New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using the Multiple Simplest Equation Method[J]. J. Appl. Math. 2014, 848069.
[18]Sudao Bilige, Wang X M. Application of the extended simplest equation method to the Whitham-Broer-Kaup-Like equations[J]. Chinese Journal of Quantum Electronics(量子电子学报), 2014, 31: 141-148.