JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (12): 24-31.doi: 10.6040/j.issn.1671-9352.0.2018.513

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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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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

CLC Number: 

  • 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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