# American Institute of Mathematical Sciences

October  2020, 13(10): 2803-2812. doi: 10.3934/dcdss.2020219

## 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

Received  January 2019 Published  October 2020 Early access  December 2019

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.

Citation: Ben Muatjetjeja, Dimpho Millicent Mothibi, Chaudry Masood Khalique. Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2803-2812. doi: 10.3934/dcdss.2020219
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show all references

##### 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.
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