Wave Equations in Bianchi Space-Times · tt − aβ N te− a/N xu x βte − a/N xu xx βte a/N xu yy βt 2L 1− 2L/m e− a/N xu zz 0. 3.2 In 17 , some aspects of the wave equation

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 765361, 12 pagesdoi:10.1155/2012/765361

Research ArticleWave Equations in Bianchi Space-Times

S. Jamal,1 A. H. Kara,1 and R Narain2

1 Centre for Differential Equations, Continuum Mechanics, and Applications, School of Mathematics,University of the Witwatersrand, Johannesburg 2001, South Africa

2 School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa

Correspondence should be addressed to S. Jamal, [email protected]

Received 31 July 2012; Accepted 26 September 2012

Academic Editor: Fazal M. Mahomed

Copyright q 2012 S. Jamal et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We investigate the wave equation in Bianchi type III space-time. We construct a Lagrangian ofthe model, calculate and classify the Noether symmetry generators, and construct correspondingconserved forms. A reduction of the underlying equations is performed to obtain invariantsolutions.

1. Introduction

The study of partial differential equations (PDEs) in terms of Lie point symmetries iswell known and well established [1–5], where these symmetries can be used to obtain,inter alia, exact analytic solutions of the PDEs. In addition, Noether symmetries are alsowidely investigated and are associated with PDEs that possess a Lagrangian. Noether [6]discovered the interesting link between symmetries and conservation laws showing that forevery infinitesimal transformation admitted by the action integral of a system there existsa conservation law. Investigations have been devoted to understand Noether symmetries ofLagrangians that arise from certain pseudo-Riemannian metrics of interest [7, 8]. Recently, astudy was aimed at understanding the effect of gravity on the solutions of the wave equationby solving the wave equation in various space-time geometries [9].

In [10], the Bianchi universes were investigated using Noether symmetries. Theauthors of [11] studied the Noether symmetries of Bianchi type I and III space-times in scalarcoupled theories. Therein, they obtained the exact solutions for potential functions, scalarfield, and the scale factors, see also [12].

We pursue an investigation of the symmetries of the wave equation in Bianchi IIIspace-time. We construct solutions of these equations and find conservation laws associatedwith Noether symmetries. The plan of the paper is as follows.

2 Journal of Applied Mathematics

In Section 2, we discuss the procedure to obtain an expression representing Noethersymmetries and conservation laws. In Section 3, we derive and classify strict Noethersymmetries of the Bianchi III space-time. Also in Section 3, we briefly describe the relationof Noether symmetries to conservation laws. We then illustrate the reduction of the waveequation and obtain invariant solutions.

2. Definitions and Notation

We briefly outline the notation and pertinent results used in this work. In this regard, thereader is referred to [13].

The convention that repeated indices imply summation is used. Let x =(x1, x2, . . . , xn) ∈ Rn be independent variable with xi, and let u = (u1, u2, . . . , um) ∈ Rm

be the dependent variable with coordinates uα. Furthermore, let π : Rn+m → Rn be theprojection map π(x, u) = x. Also, suppose that s : χ ⊂ Rn → U ⊂ Rn+m is a smoothmap such that π ◦ s = 1χ, where 1χ is the identity map on χ. The r-jet bundle Jr(U) isgiven by the equivalence classes of sections of U. The coordinates on Jr(U) are denoted by(xi, uα, . . . , uα

i1···ir ), where 1 ≤ i1 ≤ · · · ≤ ir ≤ n and uαi1···ir corresponds to the partial derivatives

of uα with respect to xi1 , . . . , xir . The partial derivatives of u with respect to x are connectedby the operator of total differentiation

Di =∂

∂xi+ uα

i

∂

∂uα+ uα

ij

∂

∂uαj

+ · · · , i = 1, . . . , n, (2.1)

as

uαi = Di(uα), uα

ij = DjDi(uα), . . . (2.2)

The collection of all first-order derivatives uai will be denoted by u(1). Similarly, the collections

of all higher order derivatives will be denoted by u(2), u(3), . . ..The r-jet bundle on U will be written as Jr(U) = {(x, u, u(1), . . . , u(r))/(x, u) ∈ U}. We

now review the space of differential forms on Jr(U). To this end, let Ωrk(U) be the vector

space of differential k-forms on Jr(U) with differential d. A smooth differential k-form onJr(U) is given by

ω = fi1,i2,...,ik dxi1 ∧ dxi2 ∧ · · · ∧ dxik , (2.3)

where each component fi1,i2,...,ik ∈ Ωr0(U). Note that for differential functions f ∈ Ωr

0(U),

Df = Djfdxj , (2.4)

where D is the total differential or the total exterior derivative. Moreover, the total exteriorderivative of ω is

Dω = Dfi1,i2,...,ik dxi1 ∧ dxi2 ∧ · · · ∧ dxik , (2.5)

Journal of Applied Mathematics 3

and by invoking (2.4) one has

Dω = Dfi1,i2,...,ik dxj ∧ dxi1 ∧ dxi2 ∧ · · · ∧ dxik . (2.6)

The total differential D has properties analogous to the algebraic properties of the usualexterior derivative d:

D(ω ∧ υ) = Dω ∧ υ + (−1)kω ∧Dυ, (2.7)

for ω a k-form and υ an l-form and D(Dω) = 0. Also, it is known that if D(Dω) = 0, then ωis a locally exact k-form, that is, ω = Dυ for some (k − 1)-form υ, [14].

2.1. Action of Symmetries

Consider an rth-order system of partial differential equations of n independent variables andm dependent variables:

Eβ(x, u, u(1), . . . , u(r))= 0, β = 1, . . . , m. (2.8)

Definition 2.1. A conserved form of (2.8) is a differential (n − 1)-form,

ω = Ti(x, u, u(1), . . . , u(r−1))(

∂

∂xi�(dx1 ∧ · · · ∧ dxn

)), (2.9)

defined on Jr−1(U) if

Dω = 0 (2.10)

is satisfied on the surface given by (2.8).

Remark 2.2. When Definition 2.1 is satisfied, (2.10) is called a conservation law for (2.8).It is clear that (2.10) evaluated on the surface (2.8) implies that

DiTi = 0 (2.11)

on the surface given by (2.8), which is also referred to as a conservation law of (2.8). Thetuple T = (T1, . . . , Tn), Tj ∈ Ωr−1

0 (U), j = 1, . . . , n, is called a conserved vector of (2.8).

We now review some definitions and results relating to Euler-Lagrange, Lie, andNoether operators ([15, 16] and references therein).

Let A =⋃p

r=0 Ωr0(U) for some p < ∞. Then A is the universal space of differential

functions of finite orders.


Consider a symmetry operator given by the infinite formal sum:

X = ξi∂

∂xi+ ηα ∂

∂uα+∑

s≥1ζαi1···is

∂

∂uαi1···is

, (2.12)

where ξi, ηα ∈ A, and the additional coefficients are determined uniquely by the prolongationformulae

ζαi1···is = Di1 · · ·Dis(Wα) + ξjuα

ji1...is, s > 1. (2.13)

In (2.13),Wα is the Lie characteristic function given by

Wα = ηα − ξjuαj . (2.14)

In particular, a symmetry operator of the form X = Qα∂/∂uα + · · · , where Qα ∈ A, is called acanonical or evolutionary representation of X, and Qα is called its characteristic.

An operatorX is said to be a Noether symmetry corresponding to a Lagrangian L ∈ A,if there exists a vector Bi = (B1, . . . , Bn), Bi ∈ A, such that

X(L) + LDi

(ξi)= Di

(Bi). (2.15)

If Bi = 0, (i = 1, . . . , n), then X is referred to as a strict Noether symmetry corresponding toa Lagrangian L ∈ A. This case is also obtained by setting the Lie derivative on the n-formLdx1 ∧ · · · ∧ dxn in the direction of X to zero, that is,

LXLdx1 ∧ · · · ∧ dxn = X

(Ldx1 ∧ · · · ∧ dxn

)= 0, (2.16)

where L is the Lie derivative operator.In view of the above discussions and definitions, the Noether theorem [6] is

formulated as follows.

Noether’s Theorem

For any Noether symmetry X corresponding to a given Lagrangian L ∈ A, there correspondsa vector Ti = (T1, . . . , Tn), T i ∈ A, defined by

Ti = Bi −Ni(L), i = 1, . . . , n, (2.17)

which is a conserved current of the Euler-Lagrange equations δL/δuα = 0, α = 1, . . . , m,where δ/δuα is the Euler-Lagrange operator given by

δ

δuα=

∂

∂uα+∑

s≥1(−1)sDi1 · · ·Dis

∂

∂uαi1···is

, α = 1, . . . , m, (2.18)


and the Noether operator associated with the operator X is given by

Ni = ξi +Wα δ

δuαi

+∑

s≥1Di1 · · ·Dis(W

α)δ

δuαii1···is

, i = 1, . . . , n, (2.19)

in which the Euler-Lagrange operators with respect to derivatives of uα are obtained from(2.18) by replacing uα by the corresponding derivatives, for example,

δ

δuαi

=∂

∂uαi

+∑

s≥1(−1)sDj1 · · ·Djs

∂

∂uαij1···js

, i = 1, . . . , n, α = 1, . . . , m. (2.20)

3. Bianchi III Space-Time

Consider the Bianchi III metric:

ds2 = −β2dt2 + t2L(dx2 + e−2ax/Ndy2

)+ t2L/mdz2. (3.1)

The wave equation in (3.1) takes the form [17]

− 1β(2L + 1)t2Le−(a/N)xut − 1

βt2L+1e−(a/N)xutt

− aβ

Nte−(a/N)xux + βte−(a/N)xuxx

+ βte(a/N)xuyy + βt2L+1−(2L/m)e−(a/N)xuzz = 0.

(3.2)

In [17], some aspects of the wave equation on the Bianchi metric were studied. Themultiplier method [1]was adopted to determine some of the conserved densities. This lengthyprocedure ultimately leads to the construction of only three symmetries and its associatedconserved vectors.

In this paper, we investigate the wave equation on the Bianchi III metric usingNoether’s theorem and the method of differential forms. We obtain a wide range of resultsand also perform symmetry reductions of the wave equation for some cases to obtaininvariant solutions. For the purposes of Sections 3.1 and 3.2, we denote the Lagrangian by L.

3.1. The Strict Noether Symmetries of (3.2)

We classify the cases that yield strict Noether symmetries (gauge is zero) of (3.2), via theLagrangian

L = −β2te−(a/N)xux

2 − β

2te(a/N)xuy

2 − β

2t2L+1−(2L/m)e−(a/N)xuz

2 +12β

t2L+1e−(a/N)xut2. (3.3)

Many of the calculations have been left out as they are tedious—the details are available tothe reader in a number of texts that have been cited here.


The principle Noether algebra is

X1 = ∂y,

X2 = ∂z,

X3 = ∂u,

X4 =N

a∂x + y∂y,

X5 =−2ayN

∂x +1N2

(−a2y2 +N2e2ax/N

)∂y.

(3.4)

Furthermore, specific cases of L and m give rise to the symmetries X1 to X5 from above, andsome additional symmetries.

Case 1 (L = 1, m = 1/3). The additional symmetries are,

X6 = −2z∂z + t∂t,

X7 = −(

4z2t4 + β2

4t4

)

∂z + zt∂t.(3.5)

Case 2 (L = 1, m = −1). The additional symmetries are,

X8 = −z∂z − t

2∂t + u∂u. (3.6)

Case 3 (L = 1, m = 1). The additional symmetries are,

X9 = −t∂t + u∂u. (3.7)

Table 1 contains the conserved forms corresponding to each Noether symmetryXi (i =1, . . . , 9), that is, it lists the three form ω. The four form is Dω, and Dω vanishes on thesolutions of (3.2). Thus

ω = −Ttdx ∧ dy ∧ dz + Tzdx ∧ dy ∧ dt − Tydx ∧ dz ∧ dt + Txdy ∧ dz ∧ dt, (3.8)

so that

DtTt +DxT

x +DyTy +DzT

z = 0 (3.9)

in (3.2).


3.2. Symmetry Reduction and Invariant Solutions

We briefly show how the order of the (1+3) wave equation (3.2) can be reduced. Ultimatelythe equation with four independent variables is reduced to an ordinary differential equation.

3.2.1. Reduction—Using the Principle Noether Algebra

We begin reducing (3.2) using X2 followed by X4. The characteristic equations are

adx

N=

dt

0=

dy

y=

du

0. (3.10)

Integrating yields s = ye−(a/N)x and (3.2) is reduced to

−1βt2Lutt − 1

β(2L + 1)t2Lut + βt

(

1 +a2

N2s2)

uss + 2sa2

N2βtus = 0, (3.11)

with u = u(s, t).A Lagrangian of (3.11) is

L = −1βt2L+1

u2t

2+ βt

(

1 +a2

N2s2)

u2s

2. (3.12)

It turns out that we if we let L = 1, N = 1 in (3.11), we can obtain its Noether symmetries,namely,

X∗1 = t∂t − u∂u, X∗

2 = ∂u. (3.13)

We reduce (3.11)with X∗1, and the characteristic equations are

dt

t=

ds

0=

du

−u. (3.14)

Integrating yields Y = tu and (3.11) is reduced to the ordinary differential equation:

1βY + β

(1 + s2a2

)Y ′′ + 2a2βsY ′ = 0, (3.15)


Table

1:Con

served

Form

ω.

Tt=

e−ax/Nt1+2L(−

uyut+uuty)

2β,

Tx=

1 2e−

ax/Ntβ(u

yux−u

uxy),

X1

Ty=

e−ax/Nt−

2L/m(e

2ax/NNt1+(2L/m) β

2 uy2+u(N

t1+2Lβ2 u

zz−t

2L/m(atβ

2 ux+N(−

tβ2 u

xx+t2L((1+2L

)ut+tu

tt)))))

2Nβ

,

Tz=

1 2e−

ax/Nt1+(2L(−

1+m)/

m) β(u

zuy−u

uyz)

Tt=

e−ax/Nt1+2L(−

uzut+uutz)

2β,

Tx=

1 2e−

ax/Ntβ(u

zux−u

uxz),

X2

Ty=

1 2ea

x/Ntβ(u

zuy−u

uyz),

Tz=

e−ax/Nt−

2L/m(N

t1+2Lβ2 u

z2+t2L/mu(e

2ax/NNtβ

2 uyy−a

tβ2 u

x+N(tβ2 u

xx−t

2L((1+2L

)ut+tu

tt))))

2Nβ

X3

Tt=−e

−ax/Nt1+2Lut

β,

Tx=e−

ax/Ntβux,

Ty=ea

x/Ntβuy,

Tz=e−

ax/Nt1+2L

−(2L

/m) βuz

Tt=−e−

ax/Nt1+2L(ayuyut+Nuxut−u

(ayuty+Nutx))

2aβ

,

Tx=

1 2aβ

( e−a

x/Nt−

2L/m( t

1+(2L/m) β

2 ux

( ayuy+Nux

)

X4

+u( N

t1+2Lβ2 u

zz+t2L/m( e

2ax/NNtβ

2 uyy−a

tβ2 u

x−a

tyβ2 u

xy−N

t2Lut−2

LNt2Lut−N

t1+2Lutt

)))),

Ty=

12a

Nβe−

ax/Nt−

2L/m( e

2ax/NNt1+(2L/m) β

2 uy

( ayuy+Nux

)

+u( a

Nt1+2Lyβ2 u

zz−t

2L/m( a

e2ax/NNtβ

2 uy+a2 tyβ2 u

x+N( e

2ax/NNtβ

2 uxy+ay( −

tβ2 u

xx+t2L((1+2L

)ut+tu

tt)))))),

Tz=

e−ax/Nt1+(2L(−

1+m)/

m) β(u

z(ayuy+Nux)−u

(ayuyz+Nuxz))

2a

Tt=

12N

2 β

( e−a

x/Nt1+2L( −( e

2ax/NN

2−a

2 y2)uyut+2a

Nyuxut+u((e2

ax/NN

2−a

2 y2)uty−2

aNyutx

))) ,

Tx=

12N

2 βe−

ax/Nt−

2L/m( t

1+(2L/m) β

2 ux

((e2

ax/NN

2−a

2 y2)uy−2

aNyux

)

+u( −

2aNt1+2Lyβ2 u

zz+t2L/m( −

2ae2

ax/NNtβ

2 uy−2

ae2

ax/NNtyβ2 u

yy+2a

2 tyβ2 u

x−e

2ax/NN

2 tβ2 u

xy+a2 ty2 β

2 uxy

X5

+2a

Nt2Lyut+4a

LNt2Lyut+2a

Nt1+2Lyutt

))) ,

Journal of Applied Mathematics 9Ta

ble

1:Con

tinu

ed.

Ty=

12N

3 βe−

ax/Nt−

2L/m( e

2ax/NNt1+(2L/m) β

2 uy

((e2

ax/NN

2−a

2 y2)uy−2

aNyux

)

+u( N

t1+2L( e

2ax/NN

2−a

2 y2)β2 u

zz+t2L/m( 2a2 e

2ax/NNtyβ2 u

y+at(e2

ax/NN

2+a2 y

2)β2 u

x

+N( 2ae2

ax/NNtyβ2 u

xy+( e

2ax/NN

2−a

2 y2)( tβ2 u

xx−t

2L((1+2L

)ut+tu

tt)))))),

Tz=

e−ax/Nt1+(2L(−

1+m)/

m) β(u

z((e2

ax/NN

2−a

2 y2 )uy−2

aNyux)−u

((e2

ax/NN

2−a

2 y2 )uyz−2

aNyuxz))

2N2

Tt=

12N

t2β

( e−a

x/N( N

t5ut(2z

uz−t

ut)+u( N

β2 u

zz+t4( e

2ax/NNβ2 u

yy−a

β2 u

x+Nβ2 u

xx−2

Ntu

t−2

Ntzutz

)))),

X6

Tx=

1 2e−

ax/Ntβ(−

2zuzux+tu

xut+u(2zuxz−t

utx)),

Ty=

1 2ea

x/Ntβ(−

2zuzuy+tu

yut+u(2zuyz−t

uty)),

Tz=

12N

t3β

( e−a

x/N( N

β2 u

z(−

2zuz+tu

t)+u( 2Nβ2 u

z−2

e2ax/NNt4zβ2 u

yy+2a

t4zβ2 u

x−2

Nt4zβ2 u

xx+6N

t5zut−N

tβ2 u

tz+2N

t6zutt

)))

Tt=

18N

t2β

( e−a

x/N( N

tut(( 4t4z2+β2)uz−4

t5zut)

+u( 4Nβ2 u

z+4N

zβ2 u

zz+4e

2ax/NNt4zβ2 u

yy−4

at4zβ2 u

x

+4N

t4zβ2 u

xx−8

Nt5zut−4

Nt5z2 u

tz−N

tβ2 u

tz

))) ,

X7

Tx=

e−ax/Nβ(−

(4t4z2+β2 )uzux+4t

5 zuxut+((4t

4 z2+β2 )uxz−4

t5zutx))

8t3

,

Ty=

eax/Nβ(−

(4t4z2+β2 )uzuy+4t

5 zuyut+u((4t

4 z2+β2 )uyz−4

t5zuty))

8t3

,

Tz=

18N

t7βe−

ax/N( −

Nβ2 u

z

((4t

4 z2+β2)uz−4

t5zut)

+t4u( 8Nzβ2 u

z−e

2ax/NNβ2(4t

4 z2+β2)uyy+4a

t4z2 β

2 ux+aβ4 u

x−4

Nt4z2 β

2 uxx−N

β4 u

xx

+12Nt5z2 u

t−N

tβ2 u

t−4

Ntzβ2 u

tz+4N

t6z2 u

tt+Nt2β2 u

tt))

Tt=

14N

β

( e−a

x/Nt2( −

Ntu

t(2z

uz+tu

t)+u( N

t4β2 u

zz+e2

ax/NNβ2 u

yy−a

β2 u

x+Nβ2 u

xx−2

Ntu

t+2N

tzutz

))) ,

X8

Tx=

1 4e−

ax/Ntβ(2zuzux+tu

xut−u

(2zuxz+tu

tx)),

Ty=

1 4ea

x/Ntβ(2zuzuy+tu

yut−u

(2zuyz+tu

ty)),

Tz=

14N

β

( e−a

x/Nt(Nt4β2 u

z(2zuz+tu

t)−u( 2Nt4β2 u

z−2

e2ax/Nnzβ2 u

yy+2a

zβ2 u

x−2

Nzβ2 u

xx+6N

tzut+Nt5β2 u

tz+2N

t2zutt

)))

Tt=

12N

β

( e−a

x/Nt2( −

Nt2ut2+u( N

β2 u

zz+e2

ax/NNβ2 u

yy−a

β2 u

x+Nβ2 u

xx−2

Ntu

t))),

X9

Tx=

1 2e−

ax/Nt2β(u

xut−u

utx),

Ty=

1 2ea

x/Nt2β(u

yut−u

uty),

Tz=

1 2e−

ax/Nt2β(u

zut−u

utz)


with Y = Y (s), and which has a solution in terms of special functions, that is,

Y (s) = C1 LegendreP

⎡

⎢⎣−aβ +

√−4 + a2β2

2aβ, ias

⎤

⎥⎦

+ C2 LegendreQ

⎡

⎢⎣−aβ +

√−4 + a2β2

2aβ, ias

⎤

⎥⎦,

(3.16)

where C1, C2 are arbitrary constants, LegendreP[n, x] refers to the Legendre polynomialPn(x), and Q[n, z] refers to the Legendre function of the second kind Qn(z).

3.2.2. Reduction—Case 1, L = 1, m = 1/3.

We reduce (3.2) using X1 followed by X6 from Case 1. The characteristic equations are

dz

−2z =dt

t=

dx

0=

du

0. (3.17)

Integrating yields r = t2z and (3.2) is reduced to

−8βre

−ax

N ur −aβ

Ne−ax

N ux + βe−ax

N uxx + e−ax

N

(β2 − 4r2

β

)

urr = 0, (3.18)

with u = u(r, x).A Lagrangian of (3.18) is

L = βe−ax/Nu2x

2+ e−ax/N

(β2 − 4r2

β

)u2r

2. (3.19)

Hence, we obtain the Noether symmetries of (3.18), namely,

X∗3 = ∂x +

a

2Nu∂u, X∗

4 = ∂u. (3.20)

We reduce (3.18)with X∗3, and the characteristic equations are

dx

1=

dr

0=

2Ndu

au. (3.21)

Integrating yields Z = e−ax/2Nu and (3.18) is reduced to the ordinary differential equation:

−8βrZ′ +

(β2 − 4r2

β

)

Z′′ − a2β

4N2Z = 0, (3.22)


with Z = Z(r), and which has a solution in terms of special functions, that is,

Z(r) = C1 LegendreP

⎡

⎢⎣−2N +

√4N2 − a2β2

4N,2rβ

⎤

⎥⎦

+ C2 LegendreQ

⎡

⎢⎣−2N +

√4N2 − a2β2

4N,2rβ

⎤

⎥⎦,

(3.23)

where, as before, C1, C2 are arbitrary constants, LegendreP[n, x] refers to the Legendrepolynomial Pn(x), and Q[n, z] refers to the Legendre function of the second kind Qn(z).

4. Conclusion

We classified the Noether symmetry generators, determined some conserved forms, andreduced some cases of the underlying equations associated with the wave equation on theBianchi III manifold. The first reduction done above involved the principle Noether algebra,whilst the second dealt with a particular case. To obtain other reductions, one needs toconclude a three-dimensional subalgebra of symmetries to reduce to an ordinary differentialequation whose solution would be an invariant solution invariant under the subalgebra.Alternatively, a lower dimensional subalgebra can be used to reduce to a partial differentialequation which may be tackled using other methods. The final solution in this case will beinvariant only under the lower dimensional algebra. In general, the procedure performedabove is the most convenient.

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