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

Igor Freire; Ben Muatjetjeja

doi:10.3934/dcdss.2018041

Article Contents

2018, Volume 11, Issue 4: 667-673. Doi: 10.3934/dcdss.2018041

This issue Previous Article Conditional symmetries of nonlinear third-order ordinary differential equations Next Article Nonlocal and nonvariational extensions of Killing-type equations

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.
* Corresponding author: Igor Freire.

Received: October 31, 2016

Revised: March 31, 2017

Published: August 2018

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

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Abstract

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.

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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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