| Peer-Reviewed

The General Forms of the Multiple-Soliton Solutions for the Completely Integrable Equations by Using the Simplest Equation Method

Received: 23 June 2015     Accepted: 6 August 2015     Published: 14 August 2015
Views:       Downloads:
Abstract

The simplest equation method with the Burgers’ equation as the simplest equation is used to handle two completely integrable equations, the KdV equation and the potential KdV equation. The general forms of the multiple-soliton solutions are formally established. It is shown that the simplest equation method may provide us with a straightforward and effective mathematic tool for generating multiple-soliton solutions of nonlinear wave equations in fluid mechanics

Published in Applied and Computational Mathematics (Volume 4, Issue 4)
DOI 10.11648/j.acm.20150404.21
Page(s) 331-334
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

The Simplest Equation Method, Burgers’ Equation, KdV, The Potential KdV, Multiple-Soliton Solutions

References
[1] Soliman, AA. The modified extended tanh-function method for solving Burgers-type equations. Physica A 361, 394-404 (2006)
[2] Ebaid, A. Exact solitary wave for some nonlinear evolution equations via Exp-function method. Phys. Lett. A 365, 213-219 (2007)
[3] He, JH. Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700-708 (2006)
[4] Wazwaz, AM. Multipleple-front solutions for the Burgers equation and the coupled Burgers equations. Applied Mathematics and Computation 190, 1198-1206 (2007)
[5] Wazwaz, AM. The tanh method and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants. Communications in Nonlinear Science and Numerical Simulation 11, 148-160 (2006)
[6] Hirota, R. The direct method in soliton theory. Cambridge, Cambridge University Press 2004.
[7] Hereman, W. Nuseir, A. Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. Comput. Simulation. 43, 13-27 (1997)
[8] Abdou, MA. The extended F-expansion method and its application for a class of nonlinear evolution equations.Chaos Solitons Fractals 31, 95-104 (2007)
[9] Zayed, EME. Gepreel, KA. The (G_/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. J. Math. Phys. 50, 013502 (2009)
[10] Kudryashov, NA. One method for finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simulat 17, 2248-2253 (2012)
[11] Vitanov, NK. On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs:The role of the simplest equation. Commun Nonlinear Sci Numer Simulat 16, 4215-4231 (2011)
[12] Kudryashov, NA. Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat. 14, 3507-3529 (2009)
[13] Vitanov, NK. Application of simplest equations of Bernoulli and Riccati kind for obtaining exact travelling-wave solutions for a class of PDEs with polynominal nonlinearity. Commun Nonlinear Sci Numer Simulat 15, 2050-2060 (2010)
[14] Kudryashov, NA. Modified method of simplest equation:Powerful tool for obtaining exact and approximate travelling-wave solutions of nonlinear PDEs. Commun Nonlinear Sci Numer Simulat 16, 1176-1185 (2011)
[15] Kudryashov, NA. Loguinova, NB. Extended simplest equation method for nonlinear differential equations. Applied Mathematics and Computation 205, 396-402 (2008)
[16] Mohamad, JA. Petkovic, MD. Biswas, A. Modified simple equation method for nonlinear evolution equations. Applied Mathematics and Computation 217, 869-877 (2010)
[17] Peng, G. Wu, X. Wang, LB. Multipleple soliton solutions for the variant Boussinesq equations. Advances in Difference Equations 1, 1-11 (2015)
[18] Wazwaz, AM. Partial differential equations and solitary waves theory. Springer Science & Business Media, 2010.
Cite This Article
  • APA Style

    Sen-Yung Lee, Chun-Ku Kuo. (2015). The General Forms of the Multiple-Soliton Solutions for the Completely Integrable Equations by Using the Simplest Equation Method. Applied and Computational Mathematics, 4(4), 331-334. https://doi.org/10.11648/j.acm.20150404.21

    Copy | Download

    ACS Style

    Sen-Yung Lee; Chun-Ku Kuo. The General Forms of the Multiple-Soliton Solutions for the Completely Integrable Equations by Using the Simplest Equation Method. Appl. Comput. Math. 2015, 4(4), 331-334. doi: 10.11648/j.acm.20150404.21

    Copy | Download

    AMA Style

    Sen-Yung Lee, Chun-Ku Kuo. The General Forms of the Multiple-Soliton Solutions for the Completely Integrable Equations by Using the Simplest Equation Method. Appl Comput Math. 2015;4(4):331-334. doi: 10.11648/j.acm.20150404.21

    Copy | Download

  • @article{10.11648/j.acm.20150404.21,
      author = {Sen-Yung Lee and Chun-Ku Kuo},
      title = {The General Forms of the Multiple-Soliton Solutions for the Completely Integrable Equations by Using the Simplest Equation Method},
      journal = {Applied and Computational Mathematics},
      volume = {4},
      number = {4},
      pages = {331-334},
      doi = {10.11648/j.acm.20150404.21},
      url = {https://doi.org/10.11648/j.acm.20150404.21},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150404.21},
      abstract = {The simplest equation method with the Burgers’ equation as the simplest equation is used to handle two completely integrable equations, the KdV equation and the potential KdV equation. The general forms of the multiple-soliton solutions are formally established. It is shown that the simplest equation method may provide us with a straightforward and effective mathematic tool for generating multiple-soliton solutions of nonlinear wave equations in fluid mechanics},
     year = {2015}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - The General Forms of the Multiple-Soliton Solutions for the Completely Integrable Equations by Using the Simplest Equation Method
    AU  - Sen-Yung Lee
    AU  - Chun-Ku Kuo
    Y1  - 2015/08/14
    PY  - 2015
    N1  - https://doi.org/10.11648/j.acm.20150404.21
    DO  - 10.11648/j.acm.20150404.21
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
    SP  - 331
    EP  - 334
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20150404.21
    AB  - The simplest equation method with the Burgers’ equation as the simplest equation is used to handle two completely integrable equations, the KdV equation and the potential KdV equation. The general forms of the multiple-soliton solutions are formally established. It is shown that the simplest equation method may provide us with a straightforward and effective mathematic tool for generating multiple-soliton solutions of nonlinear wave equations in fluid mechanics
    VL  - 4
    IS  - 4
    ER  - 

    Copy | Download

Author Information
  • Department of Mechanical Engineering, National Cheng Kung University, Taiwan, R.O.C.

  • Department of Mechanical Engineering, National Cheng Kung University, Taiwan, R.O.C.

  • Sections