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《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (12): 24-31.doi: 10.6040/j.issn.1671-9352.0.2018.513

• • 上一篇    

用Riccati方程求KdV-Burgers-Kuramoto方程的显式新行波解

林府标,张千宏   

  1. 贵州财经大学数统学院, 贵州 贵阳 550025
  • 发布日期:2019-12-11
  • 作者简介:林府标(1978— ), 男, 博士, 讲师, 研究方向为李群在微分方程中的应用、微分方程精确解的构造. E-mail:linfubiao0851@163.com
  • 基金资助:
    国家自然科学基金资助项目(11761018);贵州省科技计划基金项目(黔科合基础［2019］1051);贵州省教育厅青年科技人才成长项目(黔教合KY字［2017］150);2018年度贵州财经大学校级科研基金项目资助(2018XYB04)

Solving explicit new travelling wave solutions of KdV-Burgers-Kuramoto equation by Riccati equation

LIN Fu-biao, ZHANG Qian-hong   

  1. School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, Guizhou, China
  • Published:2019-12-11

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摘要： 首先利用Riccati方程解的相关性质和试探函数法获得了Riccati方程的8种类型的显式新解析解,其次运用李群分析法得到了KdV-Burgers-Kuramoto(KBK)方程的约化方程和群不变解。最后将广义tanh函数法结合Riccati方程的8种新解析解用于KBK方程的约化方程, 找到了KBK方程的多种类型的显式新行波解。另外,把Riccati方程的这8种类型的显式新解析解结合广义tanh函数法与李群分析法可获得属于这一类方程中的其他非线性偏微分方程(组)的周期性型、幂指函数和三角函数的有理型显式新行波解。

关键词: Riccati方程, KdV-Burgers-Kuramoto方程, 李群分析法, 广义tanh函数法, 行波解

Abstract: Firstly, 8 types explicit new analytical solutions of the Riccati equation are presented by the trial function method combing with the related properties of solutions for the Riccati equation. Secondly, the reduced equations and invariant solutions of KdV-Burgers-Kuramoto(KBK)equation are given by the Lie group analysis method. Finally, the extended tanh-function method and 8 types explicit new analytical solutions of the Riccati equation are used to solve the reduced equation of KBK equation, moreover, many types explicit new travelling wave solutions of KBK equation are found. In addition, periodic types, rational types of exponential function and trigonometric function of explicit new travelling wave solutions of other similar nonlinear partial differential equations can be obtained by use of the extended tanh-function method, 8 types explicit new analytical solutions of the Riccati equation and the Lie group analysis method.

Key words: Riccati equation, KdV-Burgers-Kuramoto equation, Lie group analysis method, extended tanh-function method, travelling wave solution

中图分类号: 

  • O175.29
[1] MALFLIET W. Solitary wave solutions of nonlinear wave equations[J]. American Journal of Physics, 1992, 60(7):650-654.
[2] PARKES E J, DUFFY B R. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations[J]. Computer Physics Communications, 1996, 98(3):288-300.
[3] 范恩贵. 可积系统与计算机代数[M]. 北京: 科学出版社, 2004. FAN Engui. Integrable systems and computer algebra[M]. Beijing: Science Press, 2004.
[4] SI REN daoerji. A new application of the extended tanh-function method[J]. Journal of Inner Mongolia Normal University, 2007, 36(4):391-396, 401.
[5] LIU Shishi. Travelling wave solution for KdV-Burgers-Kuramoto equation[J]. Progress in Natural Science, 1999, 9(10):921-928.
[6] 刘式达, 刘式适, 黄朝晖, 等. KdV-Burgers-Kuramoto方程的行波解[J]. 自然科学进展, 1999, 9(10):912-918. LIU Shida, LIU Shishi, HUANG Zhaohun, et al. Travelling wave solution of KdV-Burgers-Kuramoto equation[J]. Progress in Natural Science, 1999, 9(10):912-918.
[7] SIVASHINSKY G I. Large cells in nonlinear marangoni convection[J]. Physica D Nonlinear Phenomena, 1982, 4(2):227-235.
[8] PARKES E J, DUFFY B R, ABBOTT P C. The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations[J]. Physics Letters A, 2002, 295(5/6):280-286.
[9] KUDRYASHOV N A. Exact solutions of the generalized Kuramoto-Sivashinsky equation[J]. Physics Letters A, 1990, 147(5):287-291.
[10] 李晓东, 常晶. 一类广义Kuramoto-Sivashinsky方程的Lie对称分析[J]. 黑龙江大学自然科学学报, 2015, 32(3):297-301. LI Xiaodong, CHANG Jing. Lie symmetry analysis for a generalized Kuramoto-Sivashinsky equation[J]. Journal of Natural Science of Heilongjiang University, 2015, 32(3):297-301.
[11] 毛杰健, 杨建荣. 非线性KdV-Burgers-Kuramoto方程新的行波解[J]. 兰州理工大学学报, 2006, 32(2):150-153. MAO Jiejian, YANG Jianrong. New traveling-wave solution of nonlinear KdV-Burgers-Kuramoto equation[J]. Journal of Lanzhou University of Technology, 2006, 32(2):150-153.
[12] WANG Mingliang. Solitary wave solutions for variant Boussinesq equations[J]. Physics Letters A, 1995, 199(3/4):169-172.
[13] WANG Mingliang. Exact solutions for a compound KdV-Burgers equation[J]. Physics Letter A, 1996, 213(5/6):279-287.
[14] 吴文俊. 数学机械化[M]. 北京: 科学出版社, 2003. WU Wenjun. Mathematics mechanization[M]. Beijing: Science Press, 2003.
[15] OLVER P J. Applications of Lie groups to differential equations[M]. 2nd ed. New York: Springer, 1993.
[16] OVSIANNIKOV L V. Group analysis of differential equations[M]. New York: Academic Press, 1982.
[17] MELESHKO S V. Methods for constructing exact solutions of partial differential equations: mathematical and analytical techniques with applications to engineering[M]. New York: Springer, 2005.
[18] GRIGORIEV Y N, IBRAGIMOV N H, KOVALEV V F, et al. Symmetries of integro-differential equations: with applications in mechanics and plasma physics[M]. New York: Springer, 2010.
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