Applied and Computational Mathematics

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The General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method

Received: 29 June 2015    Accepted: 05 August 2015    Published: 19 August 2015
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Abstract

The general form of linearized exact solution for the Korteweg and de Vries (KdV) equation, with an arbitrary nonlinear coefficient, is derived by the simplest equation method with the Bernoulli equation as the simplest equation. It is shown that the proposed exact solution overcomes the long existing problem of discontinuity and can be successfully reduced to linearity, while the nonlinear term coefficient approaches zero. Comparison of four different soliton solutions is presented. A new phenomenon, named soliton sliding, is observed.

DOI 10.11648/j.acm.20150405.11
Published in Applied and Computational Mathematics (Volume 4, Issue 5, October 2015)
Page(s) 335-341
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), 2024. Published by Science Publishing Group

Keywords

General Form, Linearized, KdV, Simplest Equation Method, Bernoulli, Discontinuity, Soliton Sliding

References
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[3] Wazwaz, AM. Partial differential equations and solitary waves theory. The USA: Springer (2007)
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[8] Hirota, R. The direct method in soliton theory. Cambridge, Cambridge University Press (2004)
[9] Hereman, W, Nuseir, A. Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. Comput. Simulation. 43, 13-27 (1997)
[10] Ebaid, A. Exact solitary wave for some nonlinear evolution equations via Exp-function method. Phys. Lett. A 364, 213-219 (2007)
[11] Liu, S, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letter A 289, 69-74 (2001)
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[13] Wang, M, Wang, Y, Yubin, Z. An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications. Physics Letter A 303, 45-51 (2002)
[14] Kudryashov, NA. One method for finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simulat 17, 2248-2253 (2012)
[15] 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)
[16] Kudryashov, NA. Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat. 14, 3507-3529 (2009)
[17] 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)
[18] 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)
[19] Kudryashov, NA, Loguinova, NB. Extended simplest equation method for nonlinear differential equations. Applied Mathematics and Computation 205, 396-402 (2008)
[20] Mohamad, JA, Petkovic, MD, Biswas, A. Modified simple equation method for nonlinear evolution equations. Applied Mathematics and Computation 217, 869-877 (2010)
[21] Lu, D, Hong, B, Tian L. New solitary wave and periodic wave solutions for general types of KdV and KdV-Burgers equations. Commun Nonlinear Sci Numer Simulat 14, 77-84 (2009)
[22] Wazzan, L. A modified tanh-coth method for solving the KdV and KdV-Burgers’ equations. Commun Nonlinear Sci Numer Simulat 14, 443-450 (2009)
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Author Information
  • Department of Mechanical Engineering, National Cheng Kung University, Taiwan, R.O.C.

  • Department of Mechanical Engineering, Air Force Institute of Technology, Taiwan, R.O.C.

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  • APA Style

    Sen-Yung Lee, Chun-Ku Kuo. (2015). The General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method. Applied and Computational Mathematics, 4(5), 335-341. https://doi.org/10.11648/j.acm.20150405.11

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    ACS Style

    Sen-Yung Lee; Chun-Ku Kuo. The General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method. Appl. Comput. Math. 2015, 4(5), 335-341. doi: 10.11648/j.acm.20150405.11

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    AMA Style

    Sen-Yung Lee, Chun-Ku Kuo. The General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method. Appl Comput Math. 2015;4(5):335-341. doi: 10.11648/j.acm.20150405.11

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  • @article{10.11648/j.acm.20150405.11,
      author = {Sen-Yung Lee and Chun-Ku Kuo},
      title = {The General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method},
      journal = {Applied and Computational Mathematics},
      volume = {4},
      number = {5},
      pages = {335-341},
      doi = {10.11648/j.acm.20150405.11},
      url = {https://doi.org/10.11648/j.acm.20150405.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20150405.11},
      abstract = {The general form of linearized exact solution for the Korteweg and de Vries (KdV) equation, with an arbitrary nonlinear coefficient, is derived by the simplest equation method with the Bernoulli equation as the simplest equation. It is shown that the proposed exact solution overcomes the long existing problem of discontinuity and can be successfully reduced to linearity, while the nonlinear term coefficient approaches zero. Comparison of four different soliton solutions is presented. A new phenomenon, named soliton sliding, is observed.},
     year = {2015}
    }
    

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    AB  - The general form of linearized exact solution for the Korteweg and de Vries (KdV) equation, with an arbitrary nonlinear coefficient, is derived by the simplest equation method with the Bernoulli equation as the simplest equation. It is shown that the proposed exact solution overcomes the long existing problem of discontinuity and can be successfully reduced to linearity, while the nonlinear term coefficient approaches zero. Comparison of four different soliton solutions is presented. A new phenomenon, named soliton sliding, is observed.
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