Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair
American Journal of Applied Mathematics
Volume 7, Issue 5, October 2019, Pages: 137-144
Received: Sep. 10, 2019; Accepted: Sep. 23, 2019; Published: Oct. 14, 2019
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Shaimaa Salem, Department of Mathematics and Physics, Faculty of Engineering, Higher Technological Institute, 10th of Ramadan City, Egypt
Magda Kassem, Department of Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt
Samah Mohamed Mabrouk, Department of Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt
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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.
Calogero-Bogoyavlenskii-Schiff Equation, Singular Manifold Method, Lax Pair, Lie Infinitesimals, Similarity Solutions
To cite this article
Shaimaa Salem, Magda Kassem, Samah Mohamed Mabrouk, Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair, American Journal of Applied Mathematics. Special Issue: Analytical Approaches to Nonlinear Science and Applications. Vol. 7, No. 5, 2019, pp. 137-144. doi: 10.11648/j.ajam.20190705.11
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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