# American Institue of Mathematical Sciences

2018, 11(4): 667-673. doi: 10.3934/dcdss.2018041

## Symmetry analysis of a Lane-Emden-Klein-Gordon-Fock system with central symmetry

 Centro de Matemática, Computação e Cognição, Universidade Federal do ABC-UFABC, Avenida dos Estados, 5001, Bairro Bangu, Santo André SP, 09.210-580, Brazil International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, Republic of South Africa

* Corresponding author: Igor Freire.

B. Muatjetjeja would like to thank the Faculty Research Committee of FAST, North-West University, Mafikeng Campus for their financial support.

Received  November 2016 Revised  April 2017 Published  November 2017

Fund Project: The work of I. L. Freire is partially supported by CNPq (grant no. 308941/2013-6).

In this paper we introduce a sort of Lane-Emden system derived from the Klein-Gordon-Fock equation with central symmetry. Point symmetries are obtained and, since the system can be derived as the Euler-Lagrange equation of a certain functional, a Noether symmetry classification is also considered and conservation laws are derived from the point symmetries.

Citation: Igor Freire, Ben Muatjetjeja. Symmetry analysis of a Lane-Emden-Klein-Gordon-Fock system with central symmetry. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 667-673. doi: 10.3934/dcdss.2018041
##### References:
 [1] M. A. Abdulwahhab, Nonlinear self-adjointness and conservation laws of Klein-Gordon-Fock equation with centra symmetry, Commun Nonlinear Sci Numer Simulat, 22 (2015), 1331-1340. doi: 10.1016/j.cnsns.2014.07.025. [2] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, 1989. doi: 10.1007/978-1-4757-4307-4. [3] Y. Bozhkov, A. C. G. Martins, Lie point symmetries of the Lane-Emden systems, J. Math. Anal. Appl., 294 (2004), 334-344. doi: 10.1016/j.jmaa.2004.02.022. [4] Y. Bozhkov, I. L. Freire, Symmetry analysis of the bidimensional Lane-Emden systems, J. Math. Anal. Appl., 388 (2012), 1279-1284. doi: 10.1016/j.jmaa.2011.11.024. [5] Y. Bozhkov, I. L. Freire, On the Lane-Emden system in dimension one, Appl. Math. Comp., 218 (2012), 10762-10766. doi: 10.1016/j.amc.2012.04.033. [6] S. Dimas, D. Tsoubelis, SYM: A new symmetry-finding package for Mathematica, Proceedings of the 10th International Conference in Modern Group Analysis, Larnaca, Cyprus, 2004 (2004), 64-70. [7] S. Dimas and D. Tsoubelis, A New Heuristic Algorithm for Solving Overdetermined Systems of PDEs in Mathematica, in: 6th International Conference on Symmetry in Nonlinear Mathematical Physics, Kiev, Ukraine, 2005. [8] I. L. Freire, P. L. da Silva and M. Torrisi, Lie and Noether symmetries for a class of fourth-order Emden-Fowler equations, J. Phys. A: Math. Theor. , 46 (2013), 245206, 9 pp. doi: 10.1088/1751-8113/46/24/245206. [9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition, Academic Press, New York, 2007. [10] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley and Sons, 1999. [11] B. A. Kochetov, Lie group symmetries and Riemann function of Klein-Gordon-Fock equation with central symmetry, Commun Nonlinear Sci Numer Simulat, 19 (2014), 1723-1728. doi: 10.1016/j.cnsns.2013.10.001. [12] T. E. Mogorosi, I. L. Freire, B. Muatjetjeja, C. M. Khalique, Group analysis of a hyperbolic Lane-Emden system, Appl. Math. Comp., 292 (2017), 156-164. doi: 10.1016/j.amc.2016.07.033. [13] B. Muatjetjeja, C. M. Khalique, Lagrangian approach to a generalized coupled Lane-Emden system: Symmetries and first integrals, Commun Nonlinear Sci Numer Simulat, 15 (2010), 1166-1171. doi: 10.1016/j.cnsns.2009.06.002. [14] B. Muatjetjeja, C. M. Khalique, Lie group classification for a generalised coupled Lane-Emden system in one dimension, East. Asian. J. Appl. Math., 4 (2014), 301-311.

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##### References:
 [1] M. A. Abdulwahhab, Nonlinear self-adjointness and conservation laws of Klein-Gordon-Fock equation with centra symmetry, Commun Nonlinear Sci Numer Simulat, 22 (2015), 1331-1340. doi: 10.1016/j.cnsns.2014.07.025. [2] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, 1989. doi: 10.1007/978-1-4757-4307-4. [3] Y. Bozhkov, A. C. G. Martins, Lie point symmetries of the Lane-Emden systems, J. Math. Anal. Appl., 294 (2004), 334-344. doi: 10.1016/j.jmaa.2004.02.022. [4] Y. Bozhkov, I. L. Freire, Symmetry analysis of the bidimensional Lane-Emden systems, J. Math. Anal. Appl., 388 (2012), 1279-1284. doi: 10.1016/j.jmaa.2011.11.024. [5] Y. Bozhkov, I. L. Freire, On the Lane-Emden system in dimension one, Appl. Math. Comp., 218 (2012), 10762-10766. doi: 10.1016/j.amc.2012.04.033. [6] S. Dimas, D. Tsoubelis, SYM: A new symmetry-finding package for Mathematica, Proceedings of the 10th International Conference in Modern Group Analysis, Larnaca, Cyprus, 2004 (2004), 64-70. [7] S. Dimas and D. Tsoubelis, A New Heuristic Algorithm for Solving Overdetermined Systems of PDEs in Mathematica, in: 6th International Conference on Symmetry in Nonlinear Mathematical Physics, Kiev, Ukraine, 2005. [8] I. L. Freire, P. L. da Silva and M. Torrisi, Lie and Noether symmetries for a class of fourth-order Emden-Fowler equations, J. Phys. A: Math. Theor. , 46 (2013), 245206, 9 pp. doi: 10.1088/1751-8113/46/24/245206. [9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition, Academic Press, New York, 2007. [10] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley and Sons, 1999. [11] B. A. Kochetov, Lie group symmetries and Riemann function of Klein-Gordon-Fock equation with central symmetry, Commun Nonlinear Sci Numer Simulat, 19 (2014), 1723-1728. doi: 10.1016/j.cnsns.2013.10.001. [12] T. E. Mogorosi, I. L. Freire, B. Muatjetjeja, C. M. Khalique, Group analysis of a hyperbolic Lane-Emden system, Appl. Math. Comp., 292 (2017), 156-164. doi: 10.1016/j.amc.2016.07.033. [13] B. Muatjetjeja, C. M. Khalique, Lagrangian approach to a generalized coupled Lane-Emden system: Symmetries and first integrals, Commun Nonlinear Sci Numer Simulat, 15 (2010), 1166-1171. doi: 10.1016/j.cnsns.2009.06.002. [14] B. Muatjetjeja, C. M. Khalique, Lie group classification for a generalised coupled Lane-Emden system in one dimension, East. Asian. J. Appl. Math., 4 (2014), 301-311.
Noether symmetries of system (2) and its corresponding conservation laws. In the second column we present the constraints on $p$ and $q$.
 Generator $p$ and $q$ Components $C^i$ $X_1$ $\forall$ $\begin{array}{lcl} C^0&=&\displaystyle{\frac{r^2}{2}(u_t\,v_t+u_r\,v_r)+\frac{1}{2}\left(K(u)+M(v)\right)},\\ C^1&=&-\displaystyle{\frac{r^2}{2}(u_t\,v_r+u_r\,v_t).} \end{array}$ $X^q$ $\begin{array}{l}pq=1,\\ \\p \neq\pm1\end{array}$ $\begin{array}{lcl} C^0&=&\displaystyle{-\frac{tr^2}{2}u_tv_t-\frac{tr^2}{2}u_rv_r-\frac{r^3}{2}u_rv_t-\frac{r^3}{2}u_tv_r}\\ &&\displaystyle{-\frac{r^2uv_t}{q+1}-\frac{r^2u_tv}{q+1}-\frac{tv^{p+1}}{2(p+1)}-\frac{tu^{q+1}}{2(q+1)}},\\ C^1&=&\displaystyle{\frac{r^3}{2}u_tv_t+\frac{tr^2}{2}u_tv_r+\frac{r^3}{2}u_rv_r+\frac{tr^2}{2}u_rv_t}\\ &&\displaystyle{+\frac{r^2uv_r}{q+1}+\frac{qr^2u_rv}{q+1}-\frac{rv^{p+1}}{2(p+1)}-\frac{ru^{q+1}}{2(q+1)}.} \end{array}$ $X_3$ $-1$ $\begin{array}{lcl} C^0&=&\displaystyle{-\frac{1}{2}ruv_t-\frac{1}{2}r^2u_rv_t-\frac{1}{2}rvu_t-\frac{1}{2}r^2u_tv_r},\\ C^1&=&\displaystyle{\frac{1}{2}r^2u_tv_t+\frac{1}{2}r^2u_rv_r+\frac{1}{2}rv_ru+\frac{1}{2}rvu_r}\\ &&\displaystyle{-\frac{1}{2}\ln |v|-\frac{1}{2}\ln |u|-\ln{r}+\frac{1}{2}vu} \end{array}$ $X_4$ $p=q=-1$ $\begin{array}{lcl} C^0&=&\displaystyle{-\frac{1}{2}r^3u_tv_t-\frac{1}{2}r^3u_rv_r-\frac{1}{2}tr^2u_tv_r}\\ &&\displaystyle{-\frac{1}{2}tr^2u_rv_t-\frac{1}{2}tr^2uv_t-\frac{1}{2}tru_tv}\\ &&\displaystyle{-\frac{r}{2}\ln |v|-\frac{r}{2}\ln |u|+\frac{r}{2}uv},\\ C^1&=&\displaystyle{\frac{1}{2}tr^2u_tv_t+\frac{1}{2}tr^2u_rv_r+\frac{1}{2}r^3u_tv_r}\\&&\displaystyle{+\frac{1}{2}r^3u_rv_t+\frac{1}{2}truv_r+\frac{1}{2}tru_rv}\\ && \displaystyle{-\frac{t}{2}\ln|v|-\frac{t}{2}\ln|u|+\frac{t}{2}uv-t\ln|r|.} \end{array}$ $X^q$ $p=q=-1$ $\begin{array}{lcl} C^0&=&\displaystyle{\frac{r^2}{2}(u\,v_t-u_t\,v),}\\ C^1&=&\displaystyle{\frac{r^2}{2}(u_r\,v-u\,v_r).} \end{array}$
 Generator $p$ and $q$ Components $C^i$ $X_1$ $\forall$ $\begin{array}{lcl} C^0&=&\displaystyle{\frac{r^2}{2}(u_t\,v_t+u_r\,v_r)+\frac{1}{2}\left(K(u)+M(v)\right)},\\ C^1&=&-\displaystyle{\frac{r^2}{2}(u_t\,v_r+u_r\,v_t).} \end{array}$ $X^q$ $\begin{array}{l}pq=1,\\ \\p \neq\pm1\end{array}$ $\begin{array}{lcl} C^0&=&\displaystyle{-\frac{tr^2}{2}u_tv_t-\frac{tr^2}{2}u_rv_r-\frac{r^3}{2}u_rv_t-\frac{r^3}{2}u_tv_r}\\ &&\displaystyle{-\frac{r^2uv_t}{q+1}-\frac{r^2u_tv}{q+1}-\frac{tv^{p+1}}{2(p+1)}-\frac{tu^{q+1}}{2(q+1)}},\\ C^1&=&\displaystyle{\frac{r^3}{2}u_tv_t+\frac{tr^2}{2}u_tv_r+\frac{r^3}{2}u_rv_r+\frac{tr^2}{2}u_rv_t}\\ &&\displaystyle{+\frac{r^2uv_r}{q+1}+\frac{qr^2u_rv}{q+1}-\frac{rv^{p+1}}{2(p+1)}-\frac{ru^{q+1}}{2(q+1)}.} \end{array}$ $X_3$ $-1$ $\begin{array}{lcl} C^0&=&\displaystyle{-\frac{1}{2}ruv_t-\frac{1}{2}r^2u_rv_t-\frac{1}{2}rvu_t-\frac{1}{2}r^2u_tv_r},\\ C^1&=&\displaystyle{\frac{1}{2}r^2u_tv_t+\frac{1}{2}r^2u_rv_r+\frac{1}{2}rv_ru+\frac{1}{2}rvu_r}\\ &&\displaystyle{-\frac{1}{2}\ln |v|-\frac{1}{2}\ln |u|-\ln{r}+\frac{1}{2}vu} \end{array}$ $X_4$ $p=q=-1$ $\begin{array}{lcl} C^0&=&\displaystyle{-\frac{1}{2}r^3u_tv_t-\frac{1}{2}r^3u_rv_r-\frac{1}{2}tr^2u_tv_r}\\ &&\displaystyle{-\frac{1}{2}tr^2u_rv_t-\frac{1}{2}tr^2uv_t-\frac{1}{2}tru_tv}\\ &&\displaystyle{-\frac{r}{2}\ln |v|-\frac{r}{2}\ln |u|+\frac{r}{2}uv},\\ C^1&=&\displaystyle{\frac{1}{2}tr^2u_tv_t+\frac{1}{2}tr^2u_rv_r+\frac{1}{2}r^3u_tv_r}\\&&\displaystyle{+\frac{1}{2}r^3u_rv_t+\frac{1}{2}truv_r+\frac{1}{2}tru_rv}\\ && \displaystyle{-\frac{t}{2}\ln|v|-\frac{t}{2}\ln|u|+\frac{t}{2}uv-t\ln|r|.} \end{array}$ $X^q$ $p=q=-1$ $\begin{array}{lcl} C^0&=&\displaystyle{\frac{r^2}{2}(u\,v_t-u_t\,v),}\\ C^1&=&\displaystyle{\frac{r^2}{2}(u_r\,v-u\,v_r).} \end{array}$
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