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Mathematical Biosciences and Engineering

2006, Volume 3, Issue 1: 125-135. doi: 10.3934/mbe.2006.3.125
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Travelling wave solutions for higher-order wave equations of KDV type (III)

  • 1.
    Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004
  • 2.
    Department of Mathematics, Honghe University, Mengzi, Yunnan 661100
  • Received: 01 December 2004 Accepted: 29 June 2018 Published: 01 November 2005

  • MSC : 92D30.

  • By using the theory of planar dynamical systems to the travelling wave equation of a higher order nonlinear wave equations of KdV type, the existence of smooth solitary wave, kink wave and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are proved. In different regions of the parametric space, the sufficient conditions to guarantee the existence of the above solutions are given. In some conditions, exact explicit parametric representations of these waves are obtain.

    Citation: Jibin Li, Weigou Rui, Yao Long, Bin He. Travelling wave solutions for higher-order wave equations of KDV type (III)[J]. Mathematical Biosciences and Engineering, 2006, 3(1): 125-135. doi: 10.3934/mbe.2006.3.125

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  • By using the theory of planar dynamical systems to the travelling wave equation of a higher order nonlinear wave equations of KdV type, the existence of smooth solitary wave, kink wave and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are proved. In different regions of the parametric space, the sufficient conditions to guarantee the existence of the above solutions are given. In some conditions, exact explicit parametric representations of these waves are obtain.


  • This article has been cited by:

    1. Bin He, Bifurcations and exact bounded travelling wave solutions for a partial differential equation, 2010, 11, 14681218, 364, 10.1016/j.nonrwa.2008.11.009
    2. Xianbin Wu, Weiguo Rui, Xiaochun Hong, A Generalized KdV Equation of Neglecting the Highest-Order Infinitesimal Term and Its Exact Traveling Wave Solutions, 2013, 2013, 1085-3375, 1, 10.1155/2013/656297
    3. Weiguo Rui, Bin He, Yao Long, Can Chen, The integral bifurcation method and its application for solving a family of third-order dispersive PDEs, 2008, 69, 0362546X, 1256, 10.1016/j.na.2007.06.027
    4. Rui Weiguo, Long Yao, He Bin, Li Zhenyang, Integral bifurcation method combined with computer for solving a higher order wave equation of KdV type, 2010, 87, 0020-7160, 119, 10.1080/00207160801965321
    5. Weiguo Rui, Exact Solutions of a High-Order Nonlinear Wave Equation of Korteweg-de Vries Type under Newly Solvable Conditions, 2014, 2014, 1085-3375, 1, 10.1155/2014/714214
    6. Weiguo Rui, Can Chen, Xinsong Yang, Yao Long, Some new soliton-like solutions and periodic wave solutions with loop or without loop to a generalized KdV equation, 2010, 217, 00963003, 1666, 10.1016/j.amc.2009.09.036
    7. V. Marinakis, Higher-Order Equations of the KdV Type are Integrable, 2010, 2010, 1687-9120, 1, 10.1155/2010/329586
    8. Weiguo Rui, The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation, 2014, 76, 0924-090X, 1529, 10.1007/s11071-013-1226-8
    9. Weiguo Rui, Yao Long, Bin He, Some new travelling wave solutions with singular or nonsingular character for the higher order wave equation of KdV type (III), 2009, 70, 0362546X, 3816, 10.1016/j.na.2008.07.040
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    11. Vangelis Marinakis, New solutions of a higher order wave equation of the KdV type, 2007, 14, 1402-9251, 527, 10.1080/jnmp.2007.14.4.2
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    13. Yun-Mei Zhao, New Exact Solutions for a Higher-Order Wave Equation of KdV Type Using the Multiple Simplest Equation Method, 2014, 2014, 1110-757X, 1, 10.1155/2014/848069
    14. Huizhang Yang, Wei Li, Biyu Yang, The(G′/G)-Expansion Method and Its Application for Higher-Order Equations of KdV (III), 2014, 2014, 1110-757X, 1, 10.1155/2014/384969
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  • © 2006 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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