Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system

Ben Muatjetjeja; Dimpho Millicent Mothibi; Chaudry Masood Khalique

doi:10.3934/dcdss.2020219

Article Contents

2020, Volume 13, Issue 10: 2803-2812. Doi: 10.3934/dcdss.2020219

This issue Previous Article A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation Next Article On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor

Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system

1.
Department of Mathematics, Faculty of Science, University of Botswana, Private Bag 22, Gaborone, Botswana
2.
Department of Mathematical Sciences, Material Science Innovation and Modelling Focus Area, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, Republic of South Africa
3.
Department of Mathematical Sciences, Sol Plaatje University, Private Bag X5008, Kimberley 8300, Republic of South Africa
4.
International Institute for Symmetry Analysis and Mathematical Modelling Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, Republic of South Africa
5.
College of Mathematics and Systems Science Shandong University of Science and Technology, Qingdao, Shandong, 266590, China

^* Corresponding author: Ben Muatjetjeja
^* Corresponding author: Ben Muatjetjeja

Received: January 2019

Early access: December 2019

Published: July 2020

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this paper we perform Lie group classification of a generalized coupled (2+1)-dimensional hyperbolic system, viz., $ u_{tt}-u_{xx}-u_{yy}+f(v) = 0,\,v_{tt}-v_{xx}-v_{yy}+g(u) = 0, $ which models many physical phenomena in nonlinear sciences. We show that the Lie group classification of the system provides us with an eleven-dimensional equivalence Lie algebra, whereas the principal Lie algebra is six-dimensional and has several possible extensions. It is further shown that several cases arise in classifying the arbitrary functions $ f $ and $ g $, the forms of which include, amongst others, the power and exponential functions. Finally, for three cases we carry out symmetry reductions for the coupled system.

Keywords:

Mathematics Subject Classification: 35J47, 35J61.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	M. Escobedo and M. A. Herrero, Boundedness and blow-up for a semilinear reaction-diffusion system, J. Differential Equations, 89 (1991), 176-202. doi: 10.1016/0022-0396(91)90118-S.
[2]	M. Escobedo and M. A. Herrero, A semilinear parabolic system in bounded domain, Ann. Mat. Pura Appl.(4), 165 (1993), 315-336. doi: 10.1007/BF01765854.
[3]	I. L. Freire and B. Muatjetjeja, Symmetry analysis of a Lane-Emden-Klein-Gordon-Fock System with central symmetry, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 667-673. doi: 10.3934/dcdss.2018041.
[4]	Y. Z. Gao and W. J. Gao, Study of solutions to initial and boundary value problem for certain systems with variable exponents, Bound. Value Probl., 76 (2013), 10 pp. doi: 10.1186/1687-2770-2013-76.
[5]	N. H. Ibragimov, CRC Handbook of lie group analysis of differential equations, CRC Press, 1-3, 1994–1996.
[6]	M. Molati and C. M. Khalique, Lie group classification of a generalized Lane-Emden type system in two dimensions, J. Appl. Math., 2012 (2012), 10 pp. doi: 10.1155/2012/405978.
[7]	B. Muatjetjeja and C. M. Khalique, Symmetry analysis and conservation laws for a coupled (2+1)-dimensional hyperbolic system, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1252-1262. doi: 10.1016/j.cnsns.2014.09.008.
[8]	L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, Inc. New York-London, 1982.
[9]	J. P. Pinasco, Blow-up for parabolic and hyperbolic problems with variable exponents, Nonlinear Anal., 71 (2009), 1094-1099. doi: 10.1016/j.na.2008.11.030.