Noether Symmetries of a Generalized Coupled Lane-Emden-Klein-Gordon-Fock System with Central Symmetry

Abstract

In this paper we carry out a complete Noether symmetry analysis of a generalized coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry. It is shown that several cases transpire for which the Noether symmetries exist. Moreover, we derive conservation laws connected with the admitted Noether symmetries. Furthermore, we fleetingly discuss the physical interpretation of the these conserved vectors.

1. Introduction

In 2017 [1], the authors studied both Lie and Noether symmetries of a Lane-Emden-Klein-Fock system with central symmetry with power functions namely,

$$\begin{array}{c}\hfill u_{tt}-u_{rr}-\frac{n}{r}{u}_r+\frac{{\gamma }^{v^q}}{r^n}=0,\\ \hfill v_{tt}-v_{rr}-\frac{n}{r}{v}_r+\frac{{\alpha }^{u^p}}{r^n}=0,\end{array}$$

(1)

where $p,n,\gamma ,\alpha ,q$ are non-zero constants. In fact, when $n=2,\gamma =\alpha =1$, system (1) becomes

$$\begin{array}{c}\hfill u_{tt}-u_{rr}-\frac{2}{r}{u}_r+\frac{v^q}{r^2}=0,\\ \hfill v_{tt}-v_{rr}-\frac{2}{r}{v}_r+\frac{u^p}{r^2}=0.\end{array}$$

(2)

System (2) has been studied in [2] for both Lie and Noether symmetries together with the associated conservation laws.

Systems of this type occur in several physical phenomena, see, for example, Refs. [1,2,3,4] and references therein. These type of system can also be viewed as a natural extension of the famous two-component generalization of the nonlinear wave equation, viz,

$$\begin{array}{c}\hfill {\displaystyle u_{tt}-u_{rr}-\frac{m}{r}{u}_r-u^p=0,}\end{array}$$

(3)

with the real-valued function $u=u(t,r)$, and p representing the interaction power while the independent variables $(t,r)$ symbolize time and radial coordinates respectively in $m\ne 0$ dimensions [4].

In 2019 [5], the authors studied the generalization of system (1) where the power functions $v^q$ and $u^p$ are replaced with arbitrary elements namely, $h(v)$ and $g(u)$ respectively. Thus system (1) becomes

$$\begin{array}{c}\hfill u_{tt}-u_{rr}-\frac{n}{r}{u}_r+\frac{h(v)}{r^n}=0,\\ \hfill v_{tt}-v_{rr}-\frac{n}{r}{v}_r+\frac{g(u)}{r^n}=0.\end{array}$$

(4)

It is worth mentioning that, if the parameter $n=0$ in system (1), then system (1) reduces to the Lane-Emden system

$$\begin{array}{c}\hfill u_{xx}+u_{yy}+v^p=0,\\ \hfill v_{xx}+v_{yy}+u^q=0,\end{array}$$

(5)

under the complex transformation $(x,y,u,v)\to (t,ir,u,v)$, where p and q are non-zero constants. This system has been extensively studied for its Noether and Lie symmetries [6]. Furthermore, if the parameter $n=0$, in system (4), then system (4) transforms to a generalized Lane-Emden system

$$\begin{array}{c}\hfill u_{xx}+u_{yy}+h(v)=0,\\ \hfill v_{xx}+v_{yy}+g(u)=0,\end{array}$$

(6)

under the aforementioned complex transformation. In [7], authors applied the classical symmetry method to investigate the symmetries of system (6).

In [5], the authors applied the method of modern group analysis to study a generalized coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry (4). Motivated by the recent results in [5], we study the aforemention system (4). To the authors’ knowledge, the method of Noether symmetry analysis has not been used in the study of a generalized Lane-Emden-Klein-Fock system with central symmetry (4). Thus in this paper, we aim to compensate for this absence by carrying out a complete Noether symmetry classification of system (4) and derive the connected conservation laws of system (4). Since system (4), has a Lagrangian structure, thus the knowledge of Noether theorem [8] gives us an elegant way to construct conservation of system (4).

The structure of this paper is as follows. Firstly, we seek to establish the admitted Noether symmetries of a generalized coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry (4) associated with the standard Lagrangian. Next, in Section 2, conservation laws connected with the admitted Noether symmetries are derived. Concluding remarks are summarised in Section 3.

2. Complete Noether Symmetries Analysis

Several authors have done much work on Noether classification for a system of PDEs. See for example [6,7,9]. Here we perform a complete Noether symmetry analysis of system (4) with respect to the standard Lagrangian. System (4) has a Lagrangian structure. This prompts the following Lemma.

Lemma 1.

The generalized coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry (4) establishes the Euler-Lagrange equations with the functional

$$\begin{array}{c}\hfill J(u,v)={\int }_0^{\infty }{\int }_0^{\infty }L(t,r,u,v,u_t,v_t,u_r,v_r)dtdr,\end{array}$$

where

$$\mathcal{L}=\frac{1}{n}\left(r^n{u}_t{v}_t-r^n{u}_r{v}_r-\int h(v)dv-\int g(u)du\right).$$

(7)

is the connected function of Lagrange.

Proof. 

The insertion of $\begin{array}{c}\hfill \mathcal{L}\end{array}$ in the Euler-Lagrange equations [6,9] gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta \mathcal{L}}{\delta u}}& ={\displaystyle {\displaystyle \frac{\partial \mathcal{L}}{\partial u}}-D_t\left(\frac{\partial \mathcal{L}}{\partial u_t}\right)-D_r\left(\frac{\partial \mathcal{L}}{\partial u_r}\right),}\hfill \\ & ={\displaystyle -\frac{g(u)}{n}-\frac{1}{n}{D}_t(r^n{v}_t)-\frac{1}{n}{D}_r(-r^n{v}_r),}\hfill \\ & ={\displaystyle -\frac{g(u)}{n}-\frac{r^n}{n}{v}_{tt}-\frac{1}{n}(-n{r}^{n-1}{v}_r-r^n{v}_{rr}),}\hfill \\ & =v_{tt}-v_{rr}-n̑r{v}_r+r^n=0,\hfill \\ \hfill {\displaystyle \frac{\delta \mathcal{L}}{\delta v}}& ={\displaystyle {\displaystyle \frac{\partial \mathcal{L}}{\partial v}}-D_t\left(\frac{\partial \mathcal{L}}{\partial v_t}\right)-D_r\left(\frac{\partial \mathcal{L}}{\partial v_r}\right),}\hfill \\ & ={\displaystyle -\frac{h(v)}{n}-\frac{1}{n}{D}_t(r^n{u}_t)-\frac{1}{n}{D}_r(-r^n{u}_r),}\hfill \\ & ={\displaystyle -\frac{h(v)}{n}-\frac{r^n}{n}{u}_{tt}-\frac{1}{n}(-n{r}^{n-1}{u}_r-r^n{u}_{rr}),}\hfill \\ & =u_{tt}-u_{rr}-\frac{n}{r}{u}_r+\frac{h(v)}{r^n}=0,\hfill \end{array}$$

Hence this complete the proof. □

Let $x=(x^1,\cdots ,x^n)$ be n independent variables and $u=(u^1,\cdots ,u^m)$m dependent variables. An operator (the sum over repeated indices is presupposed)

$$X={\xi }^i(x,u)\frac{\partial }{\partial x^i}+{\eta }^{\alpha }(x,u)\frac{\partial }{\partial u^{\alpha }}$$

(8)

is called Noether point symmetry generator of the coupled system (4) connected to the Lagrangian $\mathcal{L}$ in (7) if the Killing-type equation,

$$X^{(1)}\mathcal{L}+D_i({\xi }^i)\mathcal{L}=D_i{A}^i,$$

(9)

holds for some point-dependent potential terms $A=(A^i)$ where $A^i=A^i(t,r,u,v),i=1,2$. We now revisit the celebrated Noether Theorem [6,8], that is, corresponding to each Noether symmetry, there exist a vector $T=(T^i)$ with components

$$\begin{array}{c}\hfill {\displaystyle T^i={\xi }^i\mathcal{L}+\frac{\partial \mathcal{L}}{\delta u_i^j}({\eta }_j-u_s^j{\xi }^s)-A^i,}\end{array}$$

(10)

which is a conserved vector of system (4). The solution of (9) leads to overdetermining systems of PDEs. Solving the resulting systems of PDEs prompts the following results.

$$\begin{array}{ccc}\hfill \tau & =& a(t,r),\hfill \\ \hfill \xi & =& b(t,r),\hfill \\ \hfill {\eta }^1& =& -d_v(t,r,v)u-\frac{n}{r}bu+k(t,r),\hfill \\ \hfill {\eta }^2& =& d(t,r,v),\hfill \\ \hfill A^1& =& \frac{r^n}{n}{d}_{t}u+\frac{r^n}{n}{k}_{t}v+s(t,r),\hfill \\ \hfill A^2& =& -\frac{r^n}{n}{d}_{r}u-\frac{r^n}{n}{k}_{r}v+w(t,r),\hfill \end{array}$$

$$\begin{array}{c}(d_{v}u-k)g(u)+\frac{n}{r}bug(u)-df(v)-(b_r+a_t)\left[\int h(v)dv+\int g(u)du\right]\hfill \\ =r^n(d_{tt}-d_{rr})u+r^n(k_{tt}-k_{rr})v-n{r}^{n-1}(d_{r}u+k_{r}v)+n(s_t+w_r).\hfill \end{array}$$

(11)

A complete analysis of Equation (11) yields the following results.

Theorem 1.

Suppose $n\ne 0,h(v)andg(u)$ are arbitrary functions, then the Noether generator of a generalized coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry (4) and the associated conservation laws are given by (12)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle X_1=\frac{\partial }{\partial t},}\hfill \\ A^i=0,\hfill \\ {\displaystyle T_1^1=-\frac{r^n}{n}{u}_t{v}_t-\frac{r^n}{n}{u}_r{v}_r-\frac{1}{n}\int h(v)dv-\frac{1}{n}\int g(u)du,}\hfill \\ T_1^2=\frac{r^n}{n}{u}_t{v}_r+\frac{r^n}{n}{u}_r{v}_t.\hfill \end{array}\right.\end{array}$$

(12)

Theorem 2.

Let the elements $h(v)=\alpha v+\beta ,g(u)=\gamma u+\lambda$, with $\alpha ,\gamma ,\beta ,\lambda$ are constants, $\alpha ,\gamma \ne 0$ and n arbitrary. Then the Noether symmetries of system (4) and the connected conserved vectors are (12) and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle X_2=k(t,r)\frac{\partial }{\partial u}+f(t,r)\frac{\partial }{\partial v},A^1=\frac{r^n}{n}u{f}_t+\frac{r^n}{n}u{k}_t,A^2=-\frac{r^n}{n}u{f}_r-\frac{r^n}{n}u{k}_r,}\hfill \\ with{k}_{tt}-k_{rr}-\frac{n}{r}{k}_r+\frac{\alpha }{r^n}f=0,f_{tt}-f_{rr}-\frac{n}{r}{f}_r+\frac{\gamma }{r^n}k=0,\hfill \\ T_2^1=\frac{r^n}{n}f{u}_t+\frac{r^n}{n}k{v}_t-\frac{r^n}{n}u{f}_t-\frac{r^n}{n}v{k}_t,\hfill \\ T_2^2=\frac{r^n}{n}u{f}_r+\frac{r^n}{n}v{k}_r-\frac{r^n}{n}f{u}_r-\frac{r^n}{n}k{v}_r.\hfill \end{array}\right.\end{array}$$

(13)

Theorem 3.

Suppose that$h(v)=\gamma v^q,g(u)=\alpha u^p,\alpha ,\gamma \ne 0$. Then the Noether operators of system (4) and the associated conservation laws are as follows;

(i) if${\displaystyle n=\frac{2(q+p+2)}{(p+1)(q+1)}}$$p,q\ne 0,\pm 1$, then we have (12) and

$\left\{\begin{array}{c}{\displaystyle X_2=t\frac{\partial }{\partial t}+r\frac{\partial }{\partial r}-\frac{2}{p+1}u\frac{\partial }{\partial u}-\frac{2}{q+1}v\frac{\partial }{\partial v},}\hfill \\ A^i=0,\hfill \\ {\displaystyle T_2^1=-\frac{t{r}^n}{n}(u_t{v}_t+u_r{v}_r)-\frac{1}{n(p+1)}(\alpha t{u}^{p+1}+2r^{n}u{v}_t)-\frac{1}{n(q+1)}(\gamma t{v}^{q+1}+2r^n{u}_{t}v)-}\hfill \\ {\displaystyle \frac{r^{n+1}}{n}(v_t{u}_r+u_t{v}_r),}\hfill \\ {\displaystyle T_2^2=-\frac{r^{n+1}}{n}(u_t{v}_t+u_r{v}_r)-\frac{1}{n(p+1)}(\alpha r{u}^{p+1}-2r^{n}u{v}_r)-\frac{1}{n(q+1)}(\gamma r{v}^{q+1}-2r^n{u}_{r}v)+}\hfill \\ {\displaystyle \frac{t{r}^n}{n}(v_t{u}_r+u_t{v}_r).}\hfill \end{array}\right.$

(ii) if$p=q=-1$,$\gamma =\alpha$, n arbitrary. Here we get the generic case (12) and

$\left\{\begin{array}{c}{\displaystyle X_3=u\frac{\partial }{\partial u}-v\frac{\partial }{\partial v},}\hfill \\ A^i=0,\hfill \\ {\displaystyle T_3^1=\frac{r^n}{n}(u{v}_t-u_{t}v),}\hfill \\ {\displaystyle T_3^2=\frac{r^n}{n}(u_{r}v-u{v}_r).}\hfill \end{array}\right.$

It should be noted that in any other case one recovers (12). It should also be observed that when$p=q=1,$this falls into Theorem 2.

Theorem 4.

Let the elements$h(v)=\alpha v^p,g(u)=\gamma e^{-mu},\alpha ,\gamma ,m\ne 0$,$p\ne -1$. Then the Noether generators of system (4) and the corresponding conservation laws are;

(i) if${\displaystyle n=\frac{2}{p+1},}$$\gamma =\alpha$,$n,m$arbitrary. Here the generic case (12) extends by one Noether generator with the associated conservation laws;

$\left\{\begin{array}{c}{\displaystyle X_2=tm(p+1)\frac{\partial }{\partial t}+rm(p+1)\frac{\partial }{\partial r}+2(p+1)\frac{\partial }{\partial u}-2mv\frac{\partial }{\partial v},}\hfill \\ A^i=0,\hfill \\ T_2^1=-\frac{r^n}{n}mt{u}_r{v}_r-\frac{r^n}{n}mt{u}_t{v}_t-\frac{m\alpha }{n(p+1)}t{v}^{p+1}+\frac{\gamma t}{n}{e}^{-mu}+\frac{2r^n}{n}{v}_t-\frac{m{r}^{n+1}}{n}{u}_r{v}_t-m{r}^{n}v{u}_t-\frac{m{r}^{n+1}}{n}{v}_r{u}_t,\hfill \\ T_2^2=\frac{r^{n+1}}{n}m{u}_t{v}_t+\frac{r^{n+1}}{n}m{u}_r{v}_r-\frac{m\alpha }{n(p+1)}r{v}^{p+1}+\frac{\gamma }{n}r{e}^{-mu}-\frac{2r^n}{n}{v}_r+\frac{r^n}{n}mt{u}_t{v}_r+r^{n}mv{u}_r+\frac{r^n}{n}mt{v}_t{u}_r.\hfill \end{array}\right.$

It should be noted that in any other case one recovers (12). This analysis will also be encountered in Theorem 5.

Theorem 5.

Suppose that$h(v)=\alpha e^{\lambda v}$,$g(u)=\gamma u^q,q,\gamma ,\alpha \ne 0$. Then the Noether operators of system (4) and the associated conserved vectors are;

(i) if${\displaystyle n=\frac{2}{q+1},}$$\gamma =\alpha$,$n,\lambda$arbitrary. In this case, the generic case (12) enlarges by one operator with the following conserved vectors;

$\left\{\begin{array}{c}{\displaystyle X_2=t\lambda (q+1)\frac{\partial }{\partial t}+r\lambda (q+1)\frac{\partial }{\partial r}+2(q+1)\frac{\partial }{\partial u}-2\lambda v\frac{\partial }{\partial v},}\hfill \\ A^i=0,\hfill \\ T_2^1=-\frac{r^n}{n}\lambda t{u}_r{v}_r-\frac{r^n}{n}\lambda t{u}_t{v}_t-\frac{\gamma \lambda }{n(q+1)}t{u}^{p+1}+\frac{\alpha t}{n}{e}^{-\lambda v}+\frac{2r^n}{n}{u}_t-\frac{\lambda r^{n+1}}{n}{u}_r{v}_t-\lambda r^{n}u{v}_t-\frac{\lambda r^{n+1}}{n}{v}_r{u}_t,\hfill \\ T_2^2=\frac{r^{n+1}}{n}\lambda u_t{v}_t+\frac{r^{n+1}}{n}\lambda u_r{v}_r-\frac{\lambda \gamma }{n(q+1)}r{u}^{q+1}+\frac{\alpha }{n}r{e}^{-\lambda v}-\frac{2r^n}{n}{u}_r+\frac{r^n}{n}\lambda t{v}_t{u}_r+r^n\lambda u{v}_r+\frac{r^n}{n}\lambda t{u}_t{v}_r.\hfill \end{array}\right.$

It should be observed that in any other case one recovers (12).

The aforementioned theorems can be proved by inserting the values of $X_i$, n, $h(v)$ and $g(u)$ into Equation (11) and these will satisfy Equation (11). Moreover, substituting these values into Equation (10) one obtains the associated $T^i.$ These $T^i$ then satisfy the divergence condition.

Remark 1.

It is worth mentioning that for any case that do not fall in Theorems 2–5, the Noether algebra is one-dimentional and is generated by $X_1$. It should be noted that Theorem 2 cannot be directly obtained as a consequence of the results of [1], since the functions $h(v)$ and $g(u)$ are not linear, but affine functions, hence these give some new results. In addition, Theorems 4 and 5 exploit new forms of $h(v)$ and $g(u)$ which also lead to some new results. The cases when $h(v)$ and $g(u)$ are constants are discarded.

3. Concluding Remarks

A complete Noether symmetry classification of the generalized coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry (4) was carried out. Several functional forms of the elements $h(v)$ and $g(u)$ which resulted in Noether point symmetries were derived. Thereafter, conservation laws connected to the Noether point symmetries were obtained. Conservation laws are of undisputed significance. From the mathematical point of view, when analyzed, they can be employed to detect integrability. Although conservation laws are useful in the analysis of solutions of differential equations, we will exclude this analysis for our future work. The results of the problem under study were motivated by the recent work in [1]. However, the results derived therein were not complete since the function $h(v)$ and $g(u)$ were only considered to be power functions. However, in the present work, the function $h(v)$ and $g(u)$ were consider to be arbitrary, and this resulted in some new and more general results. The authors thank the anonymous referees whose comments helped to improve the paper.