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

Sen-Yung Lee; Chun-Ku Kuo

doi:doi:10.11648/j.acm.20150404.21

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

Sen-Yung Lee, Chun-Ku Kuo

Published in Applied and Computational Mathematics (Volume 4, Issue 4)

Received: 23 June 2015 Accepted: 6 August 2015 Published: 14 August 2015

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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

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[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.

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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

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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

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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

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@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}
}

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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
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VL  - 4
IS  - 4
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Author Information

Sen-Yung Lee

Department of Mechanical Engineering, National Cheng Kung University, Taiwan, R.O.C.
Chun-Ku Kuo

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

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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

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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

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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