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

  • * Corresponding author: Igor Freire.

    * Corresponding author: Igor Freire. 
The work of I. L. Freire is partially supported by CNPq (grant no. 308941/2013-6).
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  • 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.

    Mathematics Subject Classification: Primary: 37K05, 70H33; Secondary: 70S10.


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  • 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} $
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    [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.
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    [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.
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    [12] T. E. MogorosiI. L. FreireB. 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.
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    [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. 
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