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Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair

Received: 10 September 2019     Accepted: 23 September 2019     Published: 14 October 2019
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Abstract

In this paper, we discussed and studied the solutions of the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. The Calogero-Bogoyavlenskii-Schiff equation describes the propagation of Riemann waves along the y-axis, with long wave propagating along the x-axis. Lax pair and Bäcklund transformation of the Calogero-Bogoyavlenskii-Schiff equation are derived by using the singular manifold method (SMM). The optimal Lie infinitesimals of the Lax pair are obtained. The detected Lie infinitesimals contain eight unknown functions. These functions are optimized through the commutator table. The eight unknown functions are evaluated through the solution of a set of linear differential equations, in which solutions lead to optimal Lie vectors. The CBS Lax pair is reduced by using the optimal Lie vectors to a system of ordinary differential equations (ODEs). The solitary wave solutions of Calogero-Bogoyavlenskii-Schiff equation Lax pair’s show soliton and kink waves. The obtained similarity solutions are plotted for different arbitrary functions and compared with previous analytical solutions. The comparison shows that we derive new solutions of Calogero-Bogoyavlenskii-Schiff equation by using the combination of two methods, which is different from the previous findings.

Published in American Journal of Applied Mathematics (Volume 7, Issue 5)

This article belongs to the Special Issue Analytical Approaches to Nonlinear Science and Applications

DOI 10.11648/j.ajam.20190705.11
Page(s) 137-144
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), 2019. Published by Science Publishing Group

Keywords

Calogero-Bogoyavlenskii-Schiff Equation, Singular Manifold Method, Lax Pair, Lie Infinitesimals, Similarity Solutions

References
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Cite This Article
  • APA Style

    Shaimaa Salem, Magda Kassem, Samah Mohamed Mabrouk. (2019). Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair. American Journal of Applied Mathematics, 7(5), 137-144. https://doi.org/10.11648/j.ajam.20190705.11

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    ACS Style

    Shaimaa Salem; Magda Kassem; Samah Mohamed Mabrouk. Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair. Am. J. Appl. Math. 2019, 7(5), 137-144. doi: 10.11648/j.ajam.20190705.11

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    AMA Style

    Shaimaa Salem, Magda Kassem, Samah Mohamed Mabrouk. Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair. Am J Appl Math. 2019;7(5):137-144. doi: 10.11648/j.ajam.20190705.11

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  • @article{10.11648/j.ajam.20190705.11,
      author = {Shaimaa Salem and Magda Kassem and Samah Mohamed Mabrouk},
      title = {Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair},
      journal = {American Journal of Applied Mathematics},
      volume = {7},
      number = {5},
      pages = {137-144},
      doi = {10.11648/j.ajam.20190705.11},
      url = {https://doi.org/10.11648/j.ajam.20190705.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190705.11},
      abstract = {In this paper, we discussed and studied the solutions of the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. The Calogero-Bogoyavlenskii-Schiff equation describes the propagation of Riemann waves along the y-axis, with long wave propagating along the x-axis. Lax pair and Bäcklund transformation of the Calogero-Bogoyavlenskii-Schiff equation are derived by using the singular manifold method (SMM). The optimal Lie infinitesimals of the Lax pair are obtained. The detected Lie infinitesimals contain eight unknown functions. These functions are optimized through the commutator table. The eight unknown functions are evaluated through the solution of a set of linear differential equations, in which solutions lead to optimal Lie vectors. The CBS Lax pair is reduced by using the optimal Lie vectors to a system of ordinary differential equations (ODEs). The solitary wave solutions of Calogero-Bogoyavlenskii-Schiff equation Lax pair’s show soliton and kink waves. The obtained similarity solutions are plotted for different arbitrary functions and compared with previous analytical solutions. The comparison shows that we derive new solutions of Calogero-Bogoyavlenskii-Schiff equation by using the combination of two methods, which is different from the previous findings.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair
    AU  - Shaimaa Salem
    AU  - Magda Kassem
    AU  - Samah Mohamed Mabrouk
    Y1  - 2019/10/14
    PY  - 2019
    N1  - https://doi.org/10.11648/j.ajam.20190705.11
    DO  - 10.11648/j.ajam.20190705.11
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 137
    EP  - 144
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20190705.11
    AB  - In this paper, we discussed and studied the solutions of the (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. The Calogero-Bogoyavlenskii-Schiff equation describes the propagation of Riemann waves along the y-axis, with long wave propagating along the x-axis. Lax pair and Bäcklund transformation of the Calogero-Bogoyavlenskii-Schiff equation are derived by using the singular manifold method (SMM). The optimal Lie infinitesimals of the Lax pair are obtained. The detected Lie infinitesimals contain eight unknown functions. These functions are optimized through the commutator table. The eight unknown functions are evaluated through the solution of a set of linear differential equations, in which solutions lead to optimal Lie vectors. The CBS Lax pair is reduced by using the optimal Lie vectors to a system of ordinary differential equations (ODEs). The solitary wave solutions of Calogero-Bogoyavlenskii-Schiff equation Lax pair’s show soliton and kink waves. The obtained similarity solutions are plotted for different arbitrary functions and compared with previous analytical solutions. The comparison shows that we derive new solutions of Calogero-Bogoyavlenskii-Schiff equation by using the combination of two methods, which is different from the previous findings.
    VL  - 7
    IS  - 5
    ER  - 

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Author Information
  • Department of Mathematics and Physics, Faculty of Engineering, Higher Technological Institute, 10th of Ramadan City, Egypt

  • Department of Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt

  • Department of Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt

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