Lie group classification of a generalized Lane-Emden type system in two dimensions. (English) Zbl 1282.35019

Summary: The aim of this work is to perform a complete Lie symmetry classification of a generalized Lane-Emden type system in two dimensions which models many physical phenomena in biological and physical sciences. The classical approach of group classification is employed for classification. We show that several cases arise in classifying the arbitrary parameters, the forms of which include amongst others the power law nonlinearity, and exponential and quadratic forms.


35A30 Geometric theory, characteristics, transformations in context of PDEs
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian


Full Text: DOI


[1] Q. Dai and C. C. Tisdell, “Nondegeneracy of positive solutions to homogeneous second-order differential systems and its applications,” Acta Mathematica Scientia B, vol. 29, no. 2, pp. 435-446, 2009. · Zbl 1199.34090
[2] J. Serrin and H. Zou, “Existence of positive solutions of the Lane-Emden system,” Atti del Seminario Matematico e Fisico dell’Università di Modena, vol. 46, pp. 369-380, 1998. · Zbl 0917.35031
[3] Y. Bozhkov and I. L. Freire, “Symmetry analysis of the bidimensional Lane-Emden systems,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 1279-1284, 2012. · Zbl 1298.35009
[4] P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1986. · Zbl 0588.22001
[5] B. Muatjetjeja and C. M. Khalique, “Conservation laws for a generalized coupled bidimensional Lane-Emden system,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, pp. 851-857, 2013. · Zbl 1255.35024
[6] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. · Zbl 0698.35001
[7] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, NY, USA, 1982. · Zbl 0485.58002
[8] J. M. Díaz, “Short guide to YaLie: yet another Lie mathematica package for Lie symmetries,” 2000,http://library.wolfram.com/infocenter/MathSource/4231/YaLie.ps.
[9] N. H. Ibragimov and M. Torrisi, “A simple method for group analysis and its application to a model of detonation,” Journal of Mathematical Physics, vol. 33, no. 11, pp. 3931-3937, 1992. · Zbl 0761.35104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.