两个非线性方程的势对称及其不变解

J4 ›› 2016, Vol. 33 ›› Issue (5): 537-544.

两个非线性方程的势对称及其不变解

王晓民¹,苏道毕力格²

内蒙古工业大学理学院, 内蒙古呼和浩特 010051

收稿日期:2015-08-31 修回日期:2015-12-16 出版日期:2016-09-28 发布日期:2016-09-28
通讯作者: 苏道毕力格 (1975-), 内蒙古人, 博士, 副教授, 从事偏微分方程求解问题的研究. E-mail:inmathematica@126.com
作者简介:王晓民(1974-), 女, 内蒙古人, 硕士, 讲师, 从事偏微分方程求解问题的研究. Email: wxmmath@sina.cn
基金资助:
国家自然科学基金项目(11571008), 内蒙古自治区自然科学基金(2014MS0114, 2014BS0105)

Potential symmetries and invariant solutions of two nonlinear equations

WANG Xiao-Min, SUDAO Bilige

College of Sciences, Inner Mongolia University of Technology, Inner Mongolia Hohhot 010051

Received:2015-08-31 Revised:2015-12-16 Published:2016-09-28 Online:2016-09-28

摘要/Abstract

摘要： 通过计算NTT方程和Burgers方程的势对称扩大了其古典对称，并获得了 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NTT方程和Burgers方程的古典对称和势对称，并确定了 Burgers 方程的两个势对称对应的单参数Lie变换群. 其次，利用推广的简单方程方法构造了 Burgers方程的不变解，这些解分别以含任意两个参数的双曲函数、三角函数和有理函数表示. 最后，将势对称对应的Lie变换群(14)作用于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.

关键词: 势对称; 微分特征列集算法; NTT方程; Burgers方程; 不变解

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

中图分类号:

O175.29

王晓民苏道毕力格. 两个非线性方程的势对称及其不变解[J]. J4, 2016, 33(5): 537-544.

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

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