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  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 & Continuous Dynamical Systems - S, 2020, 13 (10) : 2803-2812. doi: 10.3934/dcdss.2020219
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.  Google Scholar

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N. H. Ibragimov, CRC Handbook of lie group analysis of differential equations, CRC Press, 1-3, 1994–1996. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

N. H. Ibragimov, CRC Handbook of lie group analysis of differential equations, CRC Press, 1-3, 1994–1996. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, Inc. New York-London, 1982.   Google Scholar
[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.  Google Scholar

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