RLW-Burgers方程的势对称及其精确解
Potential Symmetry and Exact Solutions of RLW-Burgers Equation

摘要: 通过计算RLW-Burgers方程的势对称扩大了其古典对称，并获得了RLW-Burgers方程的一系列新的精确解。首先，基于微分特征列集算法确定了RLW-Burgers方程的古典对称和势对称。其次，利用推广的Tanh函数法构造了RLW-Burgers方程的不变解，这些解分别以含任意参数的双曲函数、三角函数和有理函数表示。最后，分别选择一个势对称和古典对称的Lie变换群，将其作用于RLW-Burgers方程的不变解上获得了新的精确解，重要的是这些解都不能由方程的古典对称得到。

Abstract: We expanded the classical symmetries of RLW-Burgers equation by calculating the potential symmetries, and we obtained a series of new exact solutions of RLW-Burgers equation. Firstly, we determined the classical symmetries and potential symmetries of RLW-Burgers equation based on differential characteristic set algorithm. Secondly, we constructed the invariant solutions of Burgers equation by using the extended Tanh function method, and these solutions with arbitrary pa-rameters are expressed by the hyperbolic functions, the trigonometric functions and the rational functions, respectively. Finally, new exact solutions for RLW-Burgers equation are obtained by acting Lie transformation group of potential symmetry and the classical symmetry on the invariant solutions. It is important that these solutions can not be obtained from classical symmetries of Burgers equation.

参考文献

[1]	Bluman, G.W. and Kumei, S. (1989) Symmetries and Differential Equations. Spring-Verlag, New York, Berlin.
[2]	Sophocleous, C. and Wiltshire, R.J. (2006) Linearisation and Potential Symmetries of Certain Systems of Diffusion Equations. Physica A, 370, 329-345. http://dx.doi.org/10.1016/j.physa.2006.03.003
[3]	Qu, C.Z. (2007) Potential Symmetries to System of Nonlinear Diffusion Equation. Journal of Physics A, 40, 1757-1773. http://dx.doi.org/10.1088/1751-8113/40/8/005
[4]	Bluman, G.W. and Chaolu, T. (2005) Local and Nonlocal Symmetries for Nonlinear Telegraph Equation. Journal of Mathematical Physics, 46, 023505. http://dx.doi.org/10.1063/1.1841481
[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]	薛春荣. 扩散方程允许的势对称及其精确解. 西安: 西北大学, 2006.
[8]	康静. 非线性发展方程的势对称及其线性化[D]: [博士学位论文]. 西安: 西北大学, 2008.
[9]	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.
[10]	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.
[11]	朝鲁. 微分方程(组)对称向量的吴–微分特征列集算法及其应用. 数学物理学报, 1999, 19(3): 326-332.
[12]	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. 中国科学, 2010, 40(4): 1-18.
[13]	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. 应用数学学报, 2007, 30(5): 910-927.
[14]	Temuer, C.L. and Bai, Y.S. (2009) Differential Characteristic Set Algorithm for the Complete Symmetry Classification of Partial Differential Equations. Applied Mathematics and Mechanics, 30, 595-606. http://dx.doi.org/10.1007/s10483-009-0506-6
[15]	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. 物理学报, 2014, 63(4): 040201.
[16]	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. 应用数学, 2014, 27(4): 708-713.
[17]	Mu, M.R. and Temuer, C.L. (2014) Lie Symmetries, 1-Dimensional Optimal System and Optimal Reductions of (1+2)- Dimensional Nonlinear Schrödinger Equation. Journal of Applied Mathematics and Physics, 2, 603-620. http://dx.doi.org/10.4236/jamp.2014.27067
[18]	Benzalnln, T.B., Bona, J.L. and Mahony, J.J. (1972) Model Eq-uations for Iong Waves in Nonlinear Dlsperslve Systems. Philosophical Transactions of the Royal Society A: Mathe-matical, Physical and Engineering Sciences, 272, 47-78. http://dx.doi.org/10.1098/rsta.1972.0032
[19]	Xie, F.D., Chen, J. and Lü, Z.S. (2005) Using Symbolic Computa-tion to Exactly Solve the Integrable Broer-Kaup Equations in (2+1)-Dimensional Spaces. Communications in Theoretical Physics, 43, 585-590. http://dx.doi.org/10.1088/0253-6102/43/4/003
[20]	王明亮. Exact Solution for the RLW-Burgers Equation. 应用数学, 1988, 8(1): 51-55.
[21]	谈骏渝. RLW-Burgers方程的一类解析解. 数学的实践与认识, 2001, 31(5): 545-510.
[22]	鲍春玲, 苏道毕力格, 盖立涛. RLW-Burgers方程的对称分类及其精确行波解. 内蒙古工业大学学报(自然科学版), 2015, 34(2): 81-86.