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
[1] | R. Hirota, The direct method in soliton theory, Cambridge University Press, 2004. |
[2] | J. Hietarinta, A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, Journal of Mathematical Physics, 28 (1987) 1732-1742. |
[3] | J. P. Wang, A list of 1+1 dimensional integrable equations and their properties, Journal of Nonlinear Mathematical Physics, 9 (2002) 213-233. |
[4] | R. Yamilov, Symmetries as integrability criteria for differential difference equations, Journal of Physics A: Mathematical and General, 39 (2006) R541. |
[5] | S. M. Mabrouk, A. S. Rashed, Analysis of (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation via Lax pair investigation and group transformation method, Computers and Mathematics with Applications, 74 (2017) 2546-2556. |
[6] | R. Saleh, S. M. Mabrouk, M. Kassem, Truncation method with point transformation for exact solution of Liouville Bratu Gelfand equation, Computers and Mathematics with Applications, 76 (2018) 1219-1227. |
[7] | P. Estévez, J. Prada, Singular manifold method for an equation in 2+1 dimensions, Journal of Nonlinear Mathematical Physics, 12 (2005) 266-279. |
[8] | S. M. Mabrouk, Chase-Repulsion analysis for (2+1)-Dimensional Lotka-Volterra System, International Journal of Engineering Research & Technology, 8 (2019) 875-879. |
[9] | F. Engui, Z. Hong qing, L. Gang, Bäcklund transformation, lax pairs, symmetries and exact solutions for variable coefficient KdV equation, Acta Physica Sinica 7 (1998) 649. |
[10] | B. Cheng Lin, Extended homogeneous balance method and Lax pairs, Backlund transformation, Communications in Theoretical Physics, 37 (2002) 645. |
[11] | J. Ji, J. Wu, J. Zhang, Homogeneous balance method for an inhomogeneous KdV equation: Backlund transformation and Lax pair, Int. J. Nonlinear Sci, 9 (2010) 69-71. |
[12] | A. C. Newell, M. Tabor, Y. Zeng, A unified approach to Painlevé expansions, Physica D: Nonlinear Phenomena, 29 (1987) 1-68. |
[13] | H.-Q. Zhang, B. Tian, J. Li, T. Xu, Y.-X. Zhang, Symbolic-computation study of integrable properties for the (2+1)-dimensional Gardner equation with the two-singular manifold method, IMA journal of applied mathematics, 74 (2008) 46-61. |
[14] | M. Singh, Multi soliton solutions, bilinear Backlund transformation and Lax pair of nonlinear evolution equation in (2+1)-dimension, Computational Methods for Differential Equations, 3 (2015) 134-146. |
[15] | S.-J. Yu, K. Toda, T. Fukuyama, N-soliton solutions to a-dimensional integrable equation, Journal of Physics A: Mathematical and General, 31 (1998) 10181. |
[16] | K. Toda, Y. Song-Ju, T. Fukuyama, The Bogoyavlenskii-Schiff hierarchy and integrable equations in (2+1) dimensions, Reports on Mathematical Physics, 44 (1999) 247-254. |
[17] | O. Bogoyavlenskiĭ, Overturning solitons in new two-dimensional integrable equations, Mathematics of the USSR-Izvestiya, 34 (1990) 245. |
[18] | A. M. Wazwaz, Multiple soliton solutions for the Bogoyavlenskii’s generalized breaking soliton equations and its extension form, Applied Mathematics and Computation, 217 (2010) 4282-4288. |
[19] | A. M. Wazwaz, Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equations, Physica Scripta, 81 (2010) 035005. |
[20] | M. Bruzon, M. Gandarias, C. Muriel, J. Ramirez, S. Saez, F. Romero, The Calogero–Bogoyavlenskii–Schiff equation in 2+1 dimensions, Theoretical and mathematical physics, 137 (2003) 1367-1377. |
[21] | M. Shakeel, S. T. Mohyud-Din, Improved (G′/G)-expansion and extended tanh methods for (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff equation, Alexandria Engineering Journal, 54 (2015) 27-33. |
[22] | G. H. Xu, S. H. Ma, J. P. Fang, Investigation of (G'/G)-Expansion Method and Exact Solutions for the (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff System, in: Applied Mechanics and Materials, Trans Tech Publ, 2013, pp. 235-239. |
[23] | A. Malik, F. Chand, H. Kumar, S. Mishra, Exact solutions of the Bogoyavlenskii equation using the multiple (G′ G)-expansion method, Computers & Mathematics with Applications, 64 (2012) 2850-2859. |
[24] | T. Xia, S. Xiong, Exact solutions of (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation with symbolic computation, Computers & Mathematics with Applications, 60 (2010) 919-923. |
[25] | B. Li, Y. Chen, Exact analytical solutions of the generalized Calogero-Bogoyavlenskii-Schiff equation using symbolic computation, Czechoslovak journal of physics, 54 (2004) 517-528. |
[26] | M. N. Alam, C. Tunc, An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator–prey system, Alexandria Engineering Journal, 55 (2016) 1855-1865. |
[27] | A. Cesar, S. Gmez, Exact solution of the Bogoyavlenskii equation using the improved tanh–coth method, Nonlinear Dynamics, 70 (2015) 13-24. |
[28] | G. Moatimid, R. M. El-Shiekh, A.-G. A. Al-Nowehy, Exact solutions for Calogero–Bogoyavlenskii–Schiff equation using symmetry method, Applied Mathematics and Computation, 220 (2013) 455-462. |
[29] | A. M. Wazwaz, Multiple-soliton solutions for the Calogero–Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equations, Applied Mathematics and Computation, 203 (2008) 592-597. |
[30] | P. Estévez, P. Gordoa, The singular manifold method: Darboux transformations and nonclassical symmetries, Journal of Nonlinear Mathematical Physics, 2 (1995) 334-355. |
[31] | J. Weiss, The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative, Journal of Mathematical Physics, 24 (1983) 1405-1413. |
[32] | J. Weiss, M. Tabor, G. Carnevale, The Painlevé property for partial differential equations, Journal of Mathematical Physics, 24 (1983) 522-526. |
[33] | R. Saleh, M. Kassem, S. Mabrouk, Exact solutions of Calgero-Bogoyavlenskii-Schiff equation using the singular manifold method after Lie reductions, Mathematical Methods in the Applied Sciences, 40 (2017) 5851-5862. |
[34] | R. Kumar, Application of Lie-group theory for solving Calogero–Bogoyavlenskii–Schiff equation, IOSR J Math, 124 (2016) 144-147. |
[35] | M. Gandarias1, M. Bruzon1, Symmetry group analysis and similarity solutions of the CBS equation in (2+1) dimensions, in: PAMM: Proceedings in Applied Mathematics and Mechanics, Wiley Online Library, 2008, pp. 10591-10592. |
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
@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 -