# Lie Group Classification of a Generalized Lane-Emden Type System in Two Dimensions

Authors: Motlatsi Molati; Chaudry Masood Khalique
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

The aim of this work is to perform a complete Lie symmetry classification of a generalized Lane-Emden type system in two dimensions which models many physical phenomena in biological and physical sciences. The classical approach of group classification is employed for classification. We show that several cases arise in classifying the arbitrary parameters, the forms of which include amongst others the power law nonlinearity, and exponential and quadratic forms.


---

## Body

## 1. Introduction

For many years extensive studies using various approaches have been done on the Lane-Emden type equation(1.1)Δy+F(y)=0,
which has applications in astrophysics. Recently, the Lane-Emden type systems have attracted a lot of attention in modelling physical phenomena in biological and physical sciences [1, 2]. In many of these investigations the nonlinearity in the system is often assumed. However, the symmetry-based approach provides a systematic way to specify the nonlinearities in the models of physical importance and mathematical interest.The bidimensional Lane-Emden system [3]:
(1.2)uxx+uyy+vq=0,vxx+vyy+up=0,
is studied from the view point of both Lie and Nöether point symmetry classification where p,q∉{0,1} are arbitrary constants. In particular, the Lie point symmetry classification is obtained for the cases p≠q(p=1/q) and p=q where q≠0,±1 including the special case p=q=-1.In the current study we generalize the last system by considering the bidimensional Lane-Emden system of the form(1.3)uxx+uyy+f(v)=0,vxx+vyy+g(u)=0,
where f(v) and g(u) are nonzero arbitrary functions of their respective arguments. The underlying system (1.3) is a two-dimensional Euler-Lagrange model system in elastostatics [4].Recently, in [5] Nöether point symmetry classification of system (1.3) is performed and various forms of the arbitrary functions are obtained which include linear, power, exponential, and logarithmic types.The plan of this work is organized as follows. In Section2 we generate the classifying relations (determining equations for the arbitrary elements). The computation of the equivalence transformations is presented in Section 3. In Section 4 the Lie group classification of the underlying system is performed. Finally, we summarize our investigations in Section 5.


## 2. Generator of Symmetry Group and Classifying Relations

According to the Lie algorithm we seek the generator of Lie point symmetries for system (1.3) of the form
(2.1)X=ξ1(x,y,u,v)∂x+ξ2(x,y,u,v)∂y+η1(x,y,u,v)∂u+η2(x,y,u,v)∂v.
The application of the second prolongation of (2.1) on the underlying system yields the determining equations which are solved for ξ1,ξ2,η1, and η2, see for details [4, 6, 7]. The manual generation and manipulation of determining equations is a tiring task. Fortunately, nowadays the Lie algorithm has been implemented using the computer software packages for symbolic computation such as the  YaLie package [8] which is used in this work. Therefore, with the help of the YaLie package written in Mathematica the following determining equations are generated:
(2.2)ξu1=ξv1=0,ξu2=ξv2=0,ηuu1=ηvv1=0,ηuu2=ηvv2=0,ηuv1=ηyv1=ηxv1=0,ηuv2=ηyu2=ηxu2=0,ξy2-ξx1=0,ξy1+ξx2=0,ξxx1+ξyy1-2ηxu1=0,2ηyu1-ξyy2-ξxx2=0,ξxx1+ξyy1-2ηxv2=0,2ηyv2-ξyy2-ξxx2=0,η2f'(v)-ηv1g(u)+(2ξx1-ηu1)f(v)+ηxx1+ηyy1=0,η1g'(u)+(2ξy2-ηv2)g(u)-ηu2f(v)+ηxx2+ηyy2=0,
where subscripts denote partial differentiation with respect to the indicated variables and “prime” indicates total derivative with respect to the given argument.The manipulation of (2.2) yields the general generator of symmetry group for system (1.3) in the form
(2.3)X=a(x,y)∂x+b(x,y)∂y+[C1u+c(x,y)]∂u+[C2v+d(x,y)]∂v,
where C1,C2 are arbitrary constants and a,b,c,d are the arbitrary smooth functions which satisfy
(2.4)ax-by=0,ay+bx=0,axx+ayy=0,bxx+byy=0,(2.5)(2ax-C1)f(v)+(C2v+d)f′(v)+cxx+cyy=0,(2.6)(2by-C2)g(u)+(C1u+c)g′(u)+dxx+dyy=0.
The determining equations (2.5) and (2.6) are known as the classifying relations/equations.Since the variablesx and y do not appear explicitly in the underlying system (1.3), the principal symmetry Lie algebra admitted by this system is spanned by at least two operators, namely, X1=∂x and X2=∂y (to be established in Section 3).


## 3. Equivalence Transformations

Following the infinitesimal approach [9] we consider the generator of equivalence group of the form
(3.1)Y=ξ1∂x+ξ2∂y+η1∂u+η2∂v+μ1∂f+μ2∂g,
where ξj=ξj(x,y,u,v),ηj=ηj(x,y,u,v), and μj=μj(x,y,u,v,f,g) for j=1,2.The operator (3.1) is the generator of equivalence group for system (1.3) provided it is admitted by the extended system
(3.2)uxx+uyy+f=0,vxx+vyy+g=0,fx=fy=fu=0,gx=gy=gv=0.
We require the prolonged operator for the extended system (3.2) having the form
(3.3)Y~=Y[2]+ω11∂fx+ω21∂fy+ω31∂fu+ω41∂fv+ω12∂gx+ω22∂gy+ω32∂gu+ω42∂gv,
where Y[2] is the second prolongation of (3.1) given by
(3.4)Y[2]=ξ1∂x+ξ2∂y+η1∂u+η2∂v+μ1∂f+μ2∂g+ζ111∂uxx+ζ112∂vxx+ζ221∂uyy+ζ222∂vyy.
The variables ζji and ωki are given by the prolongation formulae
(3.5)ζxx1=Dx(Dx(η1)-uxDx(ξ1)-uyDx(ξ2))-uxxDx(ξ1)-uxyDx(ξ2),ζyy1=Dy(Dy(η1)-uxDy(ξ1)-uyDy(ξ2))-uxyDy(ξ1)-uyyDy(ξ2),ζxx2=Dx(Dx(η2)-uxDx(ξ1)-uyDx(ξ2))-uxxDx(ξ1)-uxyDx(ξ2),ζyy2=Dy(Dy(η2)-uxDy(ξ1)-uyDy(ξ2))-uxyDy(ξ1)-uyyDy(ξ2),ω11=D~x(μ1)-fxD~x(ξ1)-fyD~x(ξ2)-fuD~x(η1)-fvD~x(η2),ω21=D~y(μ1)-fxD~y(ξ1)-fyD~y(ξ2)-fuD~y(η1)-fvD~y(η2),ω31=D~u(μ1)-fxD~u(ξ1)-fyD~u(ξ2)-fuD~u(η1)-fvD~u(η2),ω12=D~x(μ2)-gxD~x(ξ1)-gyD~x(ξ2)-guD~x(η1)-gvD~x(η2),ω22=D~y(μ2)-gxD~y(ξ1)-gyD~y(ξ2)-guD~y(η1)-gvD~y(η2),ω42=D~v(μ2)-gxD~u(ξ1)-gyD~u(ξ2)-guD~u(η1)-gvD~u(η2),
respectively, where
(3.6)Dx=∂x+ux∂u+vx∂v+⋯,Dy=∂y+uy∂u+vy∂v+⋯
are the total derivative operators and the total derivative operators for the extended system are given by
(3.7)D~x=∂x+fx∂f+gx∂g+⋯,D~y=∂y+fy∂f+gy∂g+⋯,D~u=∂u+fu∂f+gu∂g+⋯,D~v=∂v+fv∂f+gv∂g+⋯.
Upon application of the prolongation (3.3), the invariance conditions of system (3.2) read
(3.8)μ1+ζ111+ζ221=0,μ2+ζ112+ζ222=0,ω11=ω21=ω31=0,ω12=ω22=ω42=0.
The solution of system (3.8) is given by
(3.9)ξ1=c5x+c6y+c7,ξ2=c5y-c6x+c8,η1=c1u+c2,η2=c3v+c4,μ1=(c1-2c5)f,μ2=(c3-2c5)g,
where c1,…,c8 are arbitrary constants.Therefore, system (1.3) has 8-dimensional equivalence Lie algebra spanned by the operators
(3.10)Y1=∂x,Y2=∂y,Y3=y∂x-x∂y,Y4=∂u,Y5=∂v,Y6=x∂x+y∂y-2f∂f-2g∂g,Y7=u∂u+f∂f,Y8=v∂v+g∂g.
The composition of the one-parameter group of transformations for each Yi yields the equivalence group for system (1.3) given by the transformations
(3.11)x-=α1x+α2y+β1,y-=-α2x+α1y+β2,u-=α3u+β3,v-=α4v+β4,f-=α5f,g-=α6g,
where α1,…,α6≠0 and β1,…,β4 are arbitrary constants.Next we use the theorem on projections of equivalence Lie algebras [9] to establish the principal Lie algebra for system (1.3). The projections of the equivalence generator (3.1) are given by
(3.12)X=pr(x,y,u,v)(Y)≡ξ1∂x+ξ2∂y+η1∂u+η2∂v,(3.13)Z=pr(u,v,f,g)(Y)≡η1∂u+η2∂v+μ1∂f+μ2∂g,
where pr(x,y,u,v) denotes projection onto the space (x,y,u,v) and pr(u,v,f,g) onto the space (u,v,f,g).An operatorX spans the principal Lie algebra provided the following condition holds:
(3.14)Z=pr(u,v,f,g)(Y)=0.
In view of (3.1) taking into account (3.9) and (3.13), equation (3.14) is recast as
(3.15)(c1u+c2)∂u+(c3v+c4)∂v+(c1-2c5)f∂f+(c3-2c5)g∂g=0.
Thus, we have
(3.16)c1=0,c2=0,c3=0,c4=0,c5=0.
Now the generator of equivalence group (3.1) reduces to
(3.17)Y=(c6y+c7)∂x+(c8-c6x)∂y,
and therefore the principal Lie algebra for system (1.3) is three-dimensional and it is spanned by the operators
(3.18)X1=∂x,X2=∂y,X3=y∂x-x∂y.Remark 3.1.
The principal Lie algebra (3.18) can be achieved alternatively by solving the resulting equations obtained from splitting the determining equations (2.5) and (2.6) with respect to the arbitrary elements and their derivatives.Our goal in Section4 is to extend the principal Lie algebra, that is, we obtain the functional forms of the arbitrary elements f(v) and g(u) which provide additional operator(s).


## 4. Group Classification

Case 1.
Following the classical approach of group classification [7], the classifying relations (2.5) and (2.6) become
(4.1)(h+kv)f′(v)+ℓf(v)+m=0,(4.2)(n+pu)g′(u)+qg(u)+r=0,
where h,k,ℓ,m,n,p,q, and r are arbitrary constants.
It is noted that the analysis of (4.1) and (4.2) is similar.
Upon the use of equivalence transformations (3.11), the classifying equations (4.1) and (4.2) take the form
(4.3)(h+α5α4β4k+α5kv)f′(v)+α5ℓf(v)+m=0,(n+α6α3β3p+α6pu)g′(u)+α6qg(u)+r=0,
provided
(4.4)h-=h+α5α4β4k,k-=α5k,ℓ-=α5ℓ,m-=m;n-=n+α6α3β3p,p-=α6p,q-=α6q,r-=r.
As an illustration in the analysis of (4.3) we consider the case k≠0(h-=0) and p≠0(n-=0). Thus, we obtain
(4.5)vf′(v)+αf(v)+β=0;α,β=0,±1,ug′(u)+γg(u)+δ=0;γ,δ=0,±1,
where ℓ=αk,m=α5kβ,q=γp, and r=α6pδ.
Consider also the casek=0(h-≠0) and p=0(n-≠0), then we have
(4.6)f'(v)+λf(v)+μ=0;λ,μ=0,±1,g′(u)+νg(u)+ρ=0;ν,ρ=0,±1,
where hλ=α5m,m=μh,nν=α6q, and r=ρn.
From (4.5)-(4.6) we obtain the functional forms of the arbitrary parameters f(v) and g(u) together with their corresponding extra operator(s) given by
(4.7)f=f0v-α,g=g0u-γ;X4=(αγ-1)x∂x+(αγ-1)y∂y+2(α-1)u∂u+2(γ-1)v∂v,(4.8)f=f~0e-λv,g=g~0e-νu,λ,ν≠0;X4=λν(y-x)∂x-λν(x+y)∂y-2λ∂u-2ν∂v,X5=λν(x2-2xy-y2)∂x+λν(x2+2xy-y2)∂y+4λ(x-y)∂u+4ν(x-y)∂v,
where f0,g0,f~0, and g~0 are nonzero arbitrary constants.
The casesk≠0(p=0) and k=0(p≠0) yield the classification results given in Table 1.Table 1
Classification results:k≠0  (p=0) and k=0  (p≠0).
f
g
Condition on const.
Extra operator(s)
f
~
0
e
-
λ
v
g
0
u
-
γ
λ
≠
0
X
4
=
λ
γ
x
∂
x
+
λ
γ
y
∂
y
+
2
λ
u
∂
u
+
2
(
γ
-
1
)
∂
v
g
0
λ
≠
0
X
4
=
λ
u
∂
u
-
∂
v,  Xc=c(x,y)∂u
f
0
v
-
1
δ
ln
u
δ
=
±
1
X
4
=
x
∂
x
+
y
∂
y
+
2
v
∂
v
g
0
u
-
γ
μ
=
±
1
X
4
=
(
1
+
γ
)
x
∂
x
+
(
1
+
γ
)
y
∂
y
+
4
u
∂
u
+
2
(
1
-
γ
)
v
∂
v
μ
v
g
~
0
e
-
ν
u
ν
≠
0
X
4
=
ν
x
∂
x
+
ν
y
∂
y
+
4
u
∂
u
-
2
ν
v
∂
v
ρ
u
ρ
=
±
1
X
4
=
u
∂
u
+
v
∂
v,  Xd=d(x,y)∂v
β
ln
v
g
0
β
=
±
1
X
4
=
x
∂
x
+
y
∂
y
+
2
u
∂
u
+
2
v
∂
v,  Xc=c(x,y)∂u
f
(
v
)
g
0
u
-
1
X
4
=
x
∂
x
+
y
∂
y
+
2
u
∂
u
g
0
X
c
=
c
(
x
,
y
)
∂
uCase 2.
Suppose thatf(v) and g(u) are nonlinear functions. Differentiation of (2.5) and (2.6) twice with respect to v and u, respectively, leads to
(4.9)(2ax-C1+2C2)f′′+(d+C2v)f′′′=0,(2by-2C1+C2)g′′+(c+C1v)g′′′=0.
Thus, the last equations prompt consideration of the following set of cases:
(4.10)f′′′(v)=0,g′′′(u)=0,(4.11)f′′′(v)=0,g′′′(u)≠0,(4.12)f′′′(v)≠0,g′′′(u)=0,(4.13)f′′′(v)≠0,g′′′(u)≠0.
Consider case (4.10) for illustration. We obtain f=Av2+Bv+C,g=A-u2+B-u+C-, respectively, where A,A-,B,B-,C and C- are arbitrary constants of integration. We make use of the equivalence transformations (3.11) in order to have simplified forms of f and g.
Firstly considerf, then
(4.14)f-=α5f=α5A(v--β4α4)2+α5B(v--β4α4)+α5C≈α5v-2+α5ABv-+α5C.
Therefore, when we drop the bars and set α5=1,β4=0 the equivalence relation for f is given by v2+σv+τ, where B=σA and τ=C. Likewise, g=u2+ϕu+ω for arbitrary constants ϕ and ω. Substitution of the forms of f and g into (2.4)–(2.6) yields
(4.15)C1=C2,a~y+b~x=0,a~yy=0,b~xx=0,(σ+2v)d-C2(2τ+σv)+cxx+cyy=0,(ϕ+2u)c-C1(2ω+ϕu)+dxx+dyy=0,
where
(4.16)a(x,y)=(C12-C2)x+a~(y),b(x,y)=(C22-C1)y+b~(x),
according to  (4.9)
The solution of (4.15) leads to the four-dimensional symmetry Lie algebra spanned by the generators
(4.17)X1=∂x,X2=∂y,X3=y∂x-x∂y,X4=x∂x+y∂y-2u∂u-2v∂v,
provided f and g are quadratic in v and u, respectively, that is,
(4.18)f=v2,g=u2.
However, the last result (4.18) is included in (4.7) for α=γ=-2.
Next, in considering case (4.11), we obtain the classification result that if f=v2 and g=eu, then the principal Lie algebra is extended by the operator
(4.19)X4=x∂x+y∂y-3∂u-v∂v.
The classification results of case (4.11) can be mapped into those of (4.12) by the use of the equivalence transformations u↦v and f↦g. The last case (4.13) does not yield the forms of f(v) and g(u) such that the principal Lie algebra is extended.Note. It should be noted that not included in the preceding classification results are the cases for which the functional forms of the arbitrary elements do not extend the principal Lie algebra, this includes amongst others the case for which both functions are of logarithmic forms. Moreover, the cases which are the same under the equivalence transformations u↦v and f↦g are also excluded. The constant coefficient case is also excluded.


## 5. Conclusion

In this work we performed the Lie symmetry classification of a generalized bidimensional Lane-Emden type system. The functional forms of the arbitrary parameters were specified via the classical method of group classification, and these include the combination of power law nonlinearity, exponential, logarithmic, quadratic, linear, and constant forms. Many cases yielded four symmetries apart from the five-dimensional symmetry Lie algebra obtained in the case for which both parameters are of exponential forms. The other cases possess infinite dimensional symmetry Lie algebra.



---

*Source: 405978-2012-11-06.xml*