American Institute of Mathematical Sciences

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

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

Igor Freire , and  Ben Muatjetjeja

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.

Keywords: Lane-Emden type systems, point symmetries, Noether symmetries, conservation laws.
Mathematics Subject Classification: Primary: 37K05, 70H33; Secondary: 70S10.
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.  Google Scholar

[2]

G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, 1989. doi: 10.1007/978-1-4757-4307-4.  Google Scholar

[3]

Y. Bozhkov and 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.  Google Scholar

[4]

Y. Bozhkov and 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.  Google Scholar

[5]

Y. Bozhkov and 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.  Google Scholar

[6]

S. Dimas and 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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[9]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition, Academic Press, New York, 2007. Google Scholar

[10]

N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley and Sons, 1999.  Google Scholar

[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.  Google Scholar

[12]

T. E. Mogorosi, I. L. Freire, B. Muatjetjeja and 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.  Google Scholar

[13]

B. Muatjetjeja and 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.  Google Scholar

[14]

B. Muatjetjeja and 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.   Google Scholar

show all references

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.  Google Scholar

[2]

G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, 1989. doi: 10.1007/978-1-4757-4307-4.  Google Scholar

[3]

Y. Bozhkov and 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.  Google Scholar

[4]

Y. Bozhkov and 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.  Google Scholar

[5]

Y. Bozhkov and 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.  Google Scholar

[6]

S. Dimas and 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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[9]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition, Academic Press, New York, 2007. Google Scholar

[10]

N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley and Sons, 1999.  Google Scholar

[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.  Google Scholar

[12]

T. E. Mogorosi, I. L. Freire, B. Muatjetjeja and 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.  Google Scholar

[13]

B. Muatjetjeja and 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.  Google Scholar

[14]

B. Muatjetjeja and 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.   Google Scholar

Table 1.  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} $
Table Options

Download as excel

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} $
[1]

Philip Korman, Junping Shi. On lane-emden type systems. Conference Publications, 2005, 2005 (Special) : 510-517. doi: 10.3934/proc.2005.2005.510
[2]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[3]

María Santos Bruzón, Tamara María Garrido. Symmetries and conservation laws of a KdV6 equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 631-641. doi: 10.3934/dcdss.2018038
[4]

Wenjing Chen, Louis Dupaigne, Marius Ghergu. A new critical curve for the Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2469-2479. doi: 10.3934/dcds.2014.34.2469
[5]

Wenxiong Chen, Congming Li. An integral system and the Lane-Emden conjecture. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1167-1184. doi: 10.3934/dcds.2009.24.1167
[6]

Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058
[7]

Jingbo Dou, Fangfang Ren, John Villavert. Classification of positive solutions to a Lane-Emden type integral system with negative exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6767-6780. doi: 10.3934/dcds.2016094
[8]

Božzidar Jovanović. Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems. Journal of Geometric Mechanics, 2018, 10 (2) : 173-187. doi: 10.3934/jgm.2018006
[9]

Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035
[10]

María Rosa, María de los Santos Bruzón, María de la Luz Gandarias. Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1331-1339. doi: 10.3934/dcdss.2015.8.1331
[11]

Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011
[12]

Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807
[13]

María-Santos Bruzón, Elena Recio, Tamara-María Garrido, Rafael de la Rosa. Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020222
[14]

Zhijie Cao, Lijun Zhang. Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020218
[15]

Martin Oberlack, Andreas Rosteck. New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 451-471. doi: 10.3934/dcdss.2010.3.451
[16]

Colin J. Cotter, Michael John Priestley Cullen. Particle relabelling symmetries and Noether's theorem for vertical slice models. Journal of Geometric Mechanics, 2019, 11 (2) : 139-151. doi: 10.3934/jgm.2019007
[17]

Marin Kobilarov, Jerrold E. Marsden, Gaurav S. Sukhatme. Geometric discretization of nonholonomic systems with symmetries. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 61-84. doi: 10.3934/dcdss.2010.3.61
[18]

Leonardo Colombo, David Martín de Diego. Optimal control of underactuated mechanical systems with symmetries. Conference Publications, 2013, 2013 (special) : 149-158. doi: 10.3934/proc.2013.2013.149
[19]

Leonardo Colombo, Fernando Jiménez, David Martín de Diego. Variational integrators for mechanical control systems with symmetries. Journal of Computational Dynamics, 2015, 2 (2) : 193-225. doi: 10.3934/jcd.2015003
[20]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (78)
  • HTML views (435)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences