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山东大学学报（理学版） ›› 2018, Vol. 53 ›› Issue (10): 51-60.doi: 10.6040/j.issn.1671-9352.0.2017.607

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变系数Benjamin-Bona-Mahony-Burgers方程的微分不变量和精确解

李会会,刘希强*,辛祥鹏   

  1. 聊城大学数学科学学院, 山东 聊城 252059
  • 收稿日期:2017-11-24 出版日期:2018-10-20 发布日期:2018-10-09
  • 作者简介:李会会(1991— ),女,硕士研究生,研究方向为非线性发展方程求解研究. E-mail:lihhui@163.com*通信作者简介:刘希强(1957— ),男,博士,教授,研究方向为非线性发展方程系统. E-mail:liuxiq@sina.com
  • 基金资助:
    国家自然科学基金与中国工程物理研究院基金课题资助项目(NSAF:11076015);国家自然科学基金资助项目(11505090);山东省优秀青年科学研究基金资助项目(BS2015SF009)

Differential invariants and exact solutions of variable coefficients Benjamin-Bona-Mahony-Burgers equation

LI Hui-hui, LIU Xi-qiang*, XIN Xiang-peng   

  1. School of Mathematical Science, Liaocheng University, Liaocheng 252059, Shandong, China
  • Received:2017-11-24 Online:2018-10-20 Published:2018-10-09

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摘要： 通过应用经典李群方法,得到了变系数的Benjamin-Bona-Mahony-Burgers(BBMB)方程的连续等价变换。从等价代数着手,讨论了该方程的微分不变量,发现此方程不存在零阶微分不变量,但是具有8个相互独立的一阶不变量。利用已经求得的一阶微分不变量对方程进行了群分类。在此过程中,进一步应用上述微分不变量将一般的变系数BBMB方程映射为常系数BBMB方程、Burgers方程、Benjamin-Bona-Mahony(BBM)方程,进而得到了变系数BBMB方程的一些新的精确解,并且作出了特殊变系数BBM方程、Burgers方程的精确解的图像。

关键词: 经典李群方法, 微分不变量, 群分类, 非线性发展方程, 变系数BBMB方程

Abstract: The Lie symmetry method is performed for the variable coefficients Benjamin-Bona-Mahony-Burgers(BBMB)equation and the continuous equivalence transformations are obtained. Starting with the equivalent algebra, the differential invariants of order one are constructed. It is found that there is no zero-order differential invariant for this equation, but there are eight first-order invariants that are independent of each other. Using the obtained first-order differential invariants, we make the group classification. Finally, the general variable coefficient BBMB equations are mapped to the constant coefficient BBM equation or Burgers equation or BBMB equation by the given equivalent transformation. And then a series of new exact solutions of those variable coefficient equations are obtained. The images of the exact solution of the special BBM equation with variable coefficients and Burgers equation of the exact solution are made.

Key words: classical Lie group method, nonlinear evolution equations, group classification, variable coefficients BBMB equation, differential invariants

中图分类号: 

  • O175.2
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