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 |
Calogero-Bogoyavlenskii-Schiff Equation, Singular Manifold Method, Lax Pair, Lie Infinitesimals, Similarity Solutions
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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
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
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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Download@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}
}
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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