derivation of conservation laws for a nonlinear wave equation modelling melt migration using lie...

Communications in Nonlinear Science and Numerical Simulation 12 (2007) 423–433

www.elsevier.com/locate/cnsns

Derivation of conservation laws for a nonlinear waveequation modelling melt migration using

Lie point symmetry generators

G.H. Maluleke, D.P. Mason *

Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics,

University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa

Received 22 March 2005; accepted 25 May 2005Available online 25 October 2006

Abstract

The derivation of conservation laws for a nonlinear wave equation modelling the migration of melt through the Earth’smantle is considered. New conserved vectors which depend explicitly on the spatial coordinate are generated using the Liepoint symmetry generators of the equation and known conserved vectors. It is demonstrated how conserved vectors thatare conformally associated with a Lie point symmetry generator can be derived more simply than by the direct method byimposing the symmetry condition on the conservation law equation.� 2006 Elsevier B.V. All rights reserved.

PACS: 02.30.Jr; 02.20.Sv

Keywords: Nonlinear wave equation; Melt migration; Lie point symmetry generator; Conservation law; Conserved vector;Conformally associated

1. Introduction

Lie point symmetry generators can be used to derive new conservation laws from known conservation laws.They can also be used to derive conserved vectors more simply than the direct method by imposing an extrainvariance condition. In this paper we will use Lie point symmetry generators to derive new conserved vectorsand to simplify the derivation of known conserved vectors for a nonlinear wave equation which models meltmigration through the Earth’s mantle.

1007-5704/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cnsns.2005.05.010

* Corresponding author. Tel.: +27 11 717 6117; fax: +27 11 717 6149.E-mail address: [email protected] (D.P. Mason).

mailto:[email protected]

424 G.H. Maluleke, D.P. Mason / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 423–433

The third order nonlinear partial differential equation

o/otþ o

oz/n 1� o

oz1

/mo/ot

� �� ¼ 0 ð1:1Þ

was proposed by Scott and Stevenson [1] and independently for m = 0 by McKenzie [2] to describe the one-dimensional migration of melt under the action of gravity through the Earth’s mantle. In Eq. (1.1), /(t,z) isthe volume fraction of melt, z is the vertical spatial coordinate and t is time. The parameters n and m are expo-nents in power laws relating the permeability of the medium and the bulk and shear viscosities of the solidmatrix to /. Scott and Stevenson [1] suggested that physical values of n and m are 2 6 n 6 5 and0 6 m 6 1. Solitary wave solutions of (1.1) have been derived [1–12].

The conservation laws for the differential equation (1.1) were first investigated by Barcilon and Ritcher [4].Their reason for starting a search for conservation laws was to investigate if the solitary wave solutions of (1.1)are solitons. Soliton equations are believed to possess an infinite number of conservation laws. Harris [13] per-formed a systematic search for conservation laws and considered all possible values of n and m. For most valuesof n and m it was shown that there exist two conservation laws. However, for two special cases, m = 1, n 5 0and n = m � 1, n 5 0, the proof is inconclusive. When n = m � 1, n 5 0, a third conservation law was derived.

The Lie point symmetry generators of the differential equation (1.1) have been derived by Nakayama et al.[14] and by Maluleke and Mason [15]. We will apply the results of Kara and Mahomed [16,17] who showedthat, even in the absence of a Lagrangian, a Lie point symmetry generator can be associated with a conservedvector for a differential equation. They also showed how a new conserved vector can be generated from a Liepoint symmetry and a known conserved vector and established a necessary condition for the generated con-served vector to be non-trivial. Kara and Mahomed showed further how to impose a symmetry condition onthe conservation law equation. Although all possible conserved vectors are not constructed by this approach,since it places a condition on the conserved vector, it is simpler than the direct method which uses only theconservation law equation.

An outline of the paper is as follows. In Section 2 the Lie point symmetry generators of Eq. (1.1) and theknown conserved vectors for Eq. (1.1) are presented in tabular form. The results of Kara and Mahomed[16,17] on the association of Lie point symmetry generators with conservation laws are stated. In Section 3the derivation of new conserved vectors from the Lie point symmetries and known conserved vectors is con-sidered. In Section 4 the conformal association of a Lie point symmetry with a conserved vector is consideredin order to broaden the search for conserved vectors. By imposing conformal invariance it is demonstratedhow many of the known conserved vectors can be derived more simply than by the direct method. The phys-ical relevance and use of conservation laws is discussed in Section 5. Finally, the conclusions are summarisedin Section 6.

2. Background results

We first state some definitions and theorems [16,17] before applying them to Eq. (1.1). The summation con-vention over repeated indices, one upper and one lower, will be used.

Consider the rth order differential equation in n independent variables and one dependent variable,

Eðx; u; uð1Þ; uð2Þ; . . . ; uðrÞÞ ¼ 0; ð2:1Þ

where x = (x1,x2, . . . ,xn) and u(k) denotes the collection of kth-order partial derivatives. The total derivativewith respect to xi is defined by

Di ¼o

oxiþ ui

o

ouþ uji

o

oujþ � � � ð2:2Þ

The equation

DiT i ¼ 0; ð2:3Þ
evaluated on the surface given by (2.1), is a conservation law for the differential equation (2.1). The vectorT = (T1, . . . ,Tn) is a conserved vector and T1, . . . ,Tn are its components.

G.H. Maluleke, D.P. Mason / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 423–433 425

A Lie point symmetry generator of (2.1) is of the form

X ¼ niðx; uÞ o

oxiþ gðx; uÞ o

ou: ð2:4Þ

The prolongation of X is

X ¼ ni o

oxiþ g

o

ouþ fi

o

ouiþ fij

o

ouijþ � � � ; ð2:5Þ

where the additional coefficients are defined by [18]

fi ¼ DiðgÞ � ujDiðnjÞ; ð2:6Þfi1...is ¼ Disðfi1...is�1

Þ � uki1...is�1DisðnkÞ; s > 1: ð2:7Þ

When X occurs it is to be understood that its prolongation to as many derivatives as required is implied. TheLie point symmetry generator X is said to be associated with the conserved vector T = (T1, . . . ,Tn) for the dif-ferential equation (2.1) if [16,17]

X ðT iÞ þ T iDkðnkÞ � T kDkðniÞ ¼ 0; i ¼ 1; 2; . . . ; n: ð2:8Þ

The action of a symmetry generator on a conserved vector yields a conserved vector [17]:

Theorem 1. If X is a Lie point symmetry generator of (2.1) and T = (T1, . . . ,Tn) is a conserved vector of (2.1),

then

T i� ¼ X ðT iÞ þ T iDkðnkÞ � T kDkðniÞ; i ¼ 1; 2; . . . ; n ð2:9Þ

are the components of a conserved vector of (2.1): that is

DiT i� ¼ 0 ð2:10Þ

on the surface defined by (2.1).

Theorem 1 gives a method of generating conservation laws for (2.1) from known symmetry generators andconserved vectors of (2.1). However, the generated conservation law may be trivial, that is, T* may be zero or alinear combination of known conserved vectors. The following theorem gives a condition for T* to be a trivialconserved vector [17].

Theorem 2. If Y is associated with a conserved vector T of (2.1) and X is a Lie point symmetry generator of (2.1),

then T* defined in terms of X by (2.9) is a trivial conserved vector of (2.1) if

adY ðX Þ ¼ ½X ; Y � ¼ bY ; ð2:11Þ

for any constant b which may be zero.

Consider now Eq. (1.1). Then u = /, x1 = t, x2 = z and

D1 ¼ Dt ¼o

otþ /t

o

o/þ /tt

o

o/tþ /zt

o

o/zþ � � � ; ð2:12Þ

D2 ¼ Dz ¼o

ozþ /z

o

o/þ /tz

o

o/tþ /zz

o

o/zþ � � � ð2:13Þ

The Lie point symmetry generators of (1.1) and their Lie algebras are given in Table 1. The Lie algebras arerequired when applying Theorem 2.

The determining equation for the conserved vectors of (1.1) is

D1T 1 þ D2T 2jð1:1Þ ¼ 0: ð2:14Þ

Table 1Lie point symmetry generators of Eq. (1.1) and their Lie algebras

Lie point symmetry generator Lie algebra

Case 1. n 5 0, �1 < m <1X 1 ¼

o

ot[X1,X2] = 0

X 2 ¼o

oz[X1,X3] = (2 � n � m)X1

X 3 ¼ ð2� n� mÞt o

otþ ðn� mÞz o

ozþ 2/

o

o/[X2,X3] = (n � m)X2

Case 2. n = 0, m 5 0 and m 6¼ 4

3X a ¼ aðtÞ o

ot[Xa,X2] = 0

X 2 ¼o

oz[Xa,X3] = 0

X 3 ¼ �mzo

ozþ 2/

o

o/[X2,X3] = �mX2

a(t) is arbitrary

Case 3. n = 0, m ¼ 43

X a ¼ aðtÞ o

ot[Xa,X2] = 0[Xa,X3] = 0[Xa,X4] = 0X 2 ¼

o

oz ½X 2;X 3� ¼ � 43 X 2

X 3 ¼ �4

3z

o

ozþ 2/

o

o/½X 2;X 4� ¼ 3

2 X 3

X 4 ¼ �z2 o

ozþ 3z/

o

o/½X 3;X 4� ¼ � 4

3 X 4

a(t) is arbitrary

Case 4. n = 0, m = 0

X a ¼ aðtÞ o

ot[Xa,X2] = 0[Xa,X3] = 0

X 2 ¼o

oz[Xa,X4] = 0[Xa,X5] = 0

X 3 ¼ /o

o/½X a;X A� ¼ X A1

[X2,X3] = 0X 4 ¼ sinhð2zÞ o

ozþ coshð2zÞ/ o

o/[X2,X4] = 2X5

[X2,X5] = 2X4

X 5 ¼ coshð2zÞ o

ozþ sinhð2zÞ/ o

o/½X 2;X A� ¼ X A2

[X3,X4] = 0X A ¼ Aðt; zÞ o

o/[X3,X5] = 0[X3,XA] = �XA

a(t) is arbitrary [X4,X5] = �2X2

A(t,z) satisfies Eq. (1.1) with n = 0, m = 0: ½X 4;X A� ¼ X A3

oAot� o3A

ot oz2¼ 0 ½X 5;X A� ¼ X A4

A1, A2, A3 and A4 satisfy (1.1) withn = m = 0

The ranges of n and m are �1 < n <1 and �1 < m <1.


Barcilon and Ritcher [4] for n = 3, m = 0 and Harris [13] for �1 < n <1, �1 < m <1, investigatedconservation laws of (1.1) using the determining equation (2.14). They looked for conserved vectors of theform

T 1 ¼ T 1ð/;/z; . . . ;/lzÞ; T 2 ¼ T 2ð/;/z; . . . ;/lz;/t;/tzÞ; ð2:15Þ

where

/lz ¼ol/ozl

: ð2:16Þ


They showed that for l P 2, T2 must be of the form

TableConser

Case A(A) �1

T1 =T2 =

Case B(B.1) m

T 1 ¼T 2 ¼

(B.2) m

T 1 ¼T2 =

(B.3) m

T 1 ¼T2 =

(B.4) m

T 1 ¼T2 =

(B.5) n

T 1 ¼T 2 ¼

(B.6) n

T 1 ¼T 2 ¼

Case C(C.1) n

T 1 ¼T 2 ¼

(C.2) n

T 1 ¼T 2 ¼

Case DDo no

n =m =

No co

T 2 ¼ Að/;/z; . . . ;/ðl�1ÞzÞ/t þ Bð/;/z; . . . ;/ðl�1ÞzÞ/zt þ Cð/;/z; . . . ;/ðl�1ÞzÞ: ð2:17Þ

The conserved vectors derived by Barcilon and Ritcher [4] and by Harris [13] are presented in Table 2. Aminor correction is made to T2 of Case B.5. For each combination of n and m, two independent conservedvectors were found. A third conserved vector was found for the case n = m � 1, m 5 1. Harris [13] provedthat except possibly for the two special cases, n = m � 1 with m 5 1 and m = 1 with n 5 0, there are nomore independent conserved vectors. So far no further conserved vectors have been found for the two spe-cial cases.

2ved vectors for the differential equation (1.1)

(l = 0) T1 = T1(/), T2 = A(/,/z)/t + B(/,/z)/zt + C(/,/z)< n <1, �1 < m <1

/m/n�m�1/z/t � /n�m/zt + /n

(l = 1) T1 = T1(/,/z), T2 = A(/,/z)/t + B(/,/z)/zt + C(/,/z)5 1, n + m 5 1, n + m 5 212 ð1� n� mÞ/�2m/2

z þ 1ð2�n�mÞ/

2�n�m

m/�2m/z/t � /1�2m/zt þ nð1�mÞ/

1�m

= 1, n 5 0, n 5 1� n

2 /�2/2z þ 1

ð1�nÞ/1�n

/�2/z/t � /�1/zt + n ln/

= 1, n = 1� 1

2 /�2/2z þ ln /

/�2/z/t � /�1/zt + ln/

= 1, n = 012 /�2/2

z þ / ln /� //�2 ln//z/t � /�1 ln//zt

+ m = 1, m 5 112 /�2m/2

z þ / ln /� /m/�2m ln //z/t � /1�2m ln //zt þ /1�m ln /� 1

ð1�mÞ/1�m

+ m = 2, m 5 1� 1

2 /�2m/2z þ ln /

m/�2m/z/t � /1�2m/zt þ ð2�mÞð1�mÞ/

1�m

(l = 2) T1 = T1(/,/z,/zz), T2 = A(/,/z)/t + B(/,/z)/zt + C(/,/z)= m � 1, m 6¼ 3

2, m 5 112 /�2m/2

zz � 16 mðmþ 2Þ/�2m�2/4

z þ 12 ð3� mÞ/1�2m/2

z þ 12ð3�2mÞ/

4�2m

16 mðmþ 5Þ/�2m�2/3

z �ðm�3Þðm�1Þð3�2mÞ /1�2m/z

h i/t � 1

2 ðmþ 1Þ/�2m�1/2z þ

ð2�mÞð3�2mÞ/

2�2mh i

/zt � 12 ðm� 1Þ/�m�1/2

z þðm�1Þð3�2mÞ/

2�m

= m � 1, m ¼ 32, n ¼ 1

2

12 /�3/2

zz � 78 /�5/4

z þ 34 /�2/2

z þ 12 / ln /� 1

2 /138 /�5/3

z � /�2/z þ 34 /�2 ln //z

� �/t � 5

4 /�4/2z þ 1

2 /�1 ln /� �

/zt � 14 /�

52/2

z þ 12 /

12 ln /

(l P 3) T1 = T1(/,/z, . . . ,/lz), T2 = A(/,/z, . . . ,/(l�1)z)/t + B(/,/z, . . . ,/(l�1)z)/zt + C(/,/z, . . . ,/(l�1)z)t exist except possibly for two special casesm � 1, m 5 11, n 5 0

nserved vectors have yet been found for the two special cases


3. Generation of new conservation laws

For each Lie point symmetry generator X in Table 1 and each conserved vector T in Table 2 we calculatedT* defined by (2.9). For some cases we found that T* = 0 and therefore X is associated with T. For other cases,T* = kT where k is a non-zero constant and therefore T* is not a new conserved vector. Sometimes T* is alinear combination of T with another conserved vector which again is not a new conserved vector. For threecases we found that T* yields a new conserved vector.

Consider first n = 0, m ¼ 43. The Lie point symmetry generators of (1.1) are listed in Case 3 of Table 1. The

known conserved vector T is given in Case A (l = 0) of Table 2. The conserved vectors generated using (2.9)are given in Table 3. We see that the action of X4 on T generates a new conserved vector T* which unlike theknown conserved vectors depends on z explicitly. It is readily verified by direct calculation that T* satisfies thedetermining equation (2.14). We observe that X2 is associated with T since T* = 0. Letting X2 = Y we see fromTable 1 that

TableGener

Case AT1 = /X a : TX 2 : TX 3 : TX 4 : T

T 1� ¼ z

X a : TX 2 : TX 3 : TX 4 : T

Case AT1 = /X a : TX 2 : TX 3 : TX 4 : TX 5 : TX A : T

B.1 (lT 1 ¼ 1

2

X a : TX 2 : TX 3 : TX 4 : TX 5 : TX A : T

adY ðX 4Þ ¼ ½X 4;X 2� ¼ �3

2X 3 6¼ bY : ð3:1Þ

Thus condition (2.11) of Theorem 2 for X4 to generate a trivial conserved vector is not satisfied. We also seefrom Table 3 that the action of the Lie point symmetry generators on the new conserved vector T* does notgenerate further conserved vectors. In fact, X4 is associated with the new conserved vector T* since T** = 0.The process of generating new conserved vectors therefore ends at the level T*.

Consider next n = 0, m = 0. The Lie point symmetry generators of (1.1) are listed in Case 4 of Table 1. Con-sider first the known conserved vector, T, given by Case A (l = 0) of Table 2 with n = m = 0. The conservedvectors generated using (2.9) are given in Table 3. We see that the action of X4, X5 and XA on T generates newconserved vectors T*. Two of the new conserved vectors depend on z explicitly and the third could depend on t

3ation of new conserved vectors for Eq. (1.1) from known conserved vectors and Lie point symmetry generators

(l = 0), n = 0, m ¼ 43

T 2 ¼ 43 /�

73/z/t � /�

43/zt

1� ¼ 0 T 2

� ¼ 01� ¼ 0 T 2

� ¼ 01� ¼ 2

3 T 1 T 2� ¼ 2

3 T 2

1� ¼ zT 1 T 2

� ¼ zT 2 þ /�43/t

/ T 2� ¼ z 4

3 /�73/z/t � /�

43/zt

� þ /�

43/t

1�� ¼ 0 T 2

�� ¼ 01�� ¼ T 1 T 2

�� ¼ T 2

1�� ¼ � 2

3 T 1� T 2

�� ¼ � 23 T 2�

1�� ¼ 0 T 2

�� ¼ 0

(l = 0) n = 0, m = 0T2 = �/zt

1� ¼ 0 T 2

� ¼ 01� ¼ 0 T 2

� ¼ 01� ¼ T 1 T 2

� ¼ T 2

1� ¼ 3/ cosh 2z T 2

� ¼ �2/t sinh 2zþ /zt cosh 2z1� ¼ 3/ sinh 2z T 2

� ¼ �2/t cosh 2zþ /zt sinh 2z1� ¼ Aðt; zÞ T 2

� ¼ � o2Aot oz

= 1), n = 0, m = 0ð/2 þ /2

z Þ T2 = �//zt1� ¼ 0 T 2

� ¼ 01� ¼ 0 T 2

� ¼ 01� ¼ 2T 1 T 2

� ¼ 2T 2

1� ¼ 2/ð/ cosh 2zþ /z sinh 2zÞ T 2

� ¼ �2//t sinh 2z1� ¼ 2/ð/ sinh 2zþ /z cosh 2zÞ T 2

� ¼ �2//t cosh 2z1� ¼ /Aðt; zÞ þ /z

oAoz T 2

� ¼ �/ztAðt; zÞ � / o2Aot oz


as well as z. The conserved vector generated by XA is referred to as the elementary conserved vector since itfollows directly from (1.1) with n = m = 0. We also see from Table 3 that X2 is associated with T since T* = 0.Let Y = X2. Then from Table 1,

adY ðX 4Þ ¼ ½X 4;X 2� ¼ �2X 5 6¼ bY ; ð3:2ÞadY ðX 5Þ ¼ ½X 5;X 2� ¼ �2X 4 6¼ bY ; ð3:3ÞadY ðX AÞ ¼ ½X A;X 2� ¼ �X A2

6¼ bY : ð3:4Þ

Hence condition (2.11) of Theorem 2 for X4, X5 and XA to generate a trivial conserved vector is not satisfied.Consider next the known conserved vector, T, given by Case B (l = 1) with n = m = 0. We see from Table 3that the action of X4, X5 and XA on T again generates new conserved vectors. The first two depend on z explic-itly and the third could depend on t as well as z. They are the only new conserved vectors generated with l = 1.By operating on the new conserved vectors for both Case A (l = 0) and Case B (l = 1) with X4, X5 and XA

using (2.9) further non-trivial conserved vectors are generated. However, we will not develop the casen = 0, m = 0 further because it is not important in the two phase flow of compacting media. The differentialequation (1.1) reduces to the linear equation

o/ot� o

3/otoz2

¼ 0: ð3:5Þ

4. Derivation of conserved vectors with an imposed invariance condition

Barcilon and Ritcher [4] and Harris [13] used the direct method to derive the conserved vectors listed inTable 2. In the direct method only the conservation law equation, (2.14), is used which plays the role of adetermining equation.

A way to simplify the derivation of conserved vectors is to impose, in addition to the conservation lawequation, an invariance condition on the unknown conserved vector [16,17]. Although all conserved vectorsmay not be constructed, since only conserved vectors which satisfy the invariance condition will be found,it is a useful procedure because of the significant simplification in the calculations that can be achieved. Wewill demonstrate this method for the derivation of conserved vectors with l = 1 and l = 0. Clearly, the weakerthe invariance condition the more inclusive will be the search procedure.

Many of the Lie point symmetry generators of Table 1 are not associated with the corresponding conservedvectors of Table 2. The invariance condition that X is associated with T is too strong and excludes from thesearch of too many conserved vectors. We consider instead the weaker symmetry condition

T i� ¼ XT i þ T iDkðnkÞ � T kDkðniÞ ¼ kT i; i ¼ 1; 2; ð4:1Þ

where k is an arbitrary constant. When (4.1) is satisfied we will say that the Lie point symmetry generator X isconformally associated with the conserved vector T. We therefore look for conserved vectors that satisfy thedual conditions, (2.14) and (4.1). Since T satisfies (2.14), it follows that T* will be a conserved vector which isconsistent with Theorem 1.

We will consider n 5 0, �1 < m <1 and search for conserved vectors of the form (2.17) with l = 1,

T 1 ¼ T 1ð/;/zÞ; T 2 ¼ Að/;/zÞ/t þ Bð/;/zÞ/zt þ Cð/;/zÞ; ð4:2Þ

which are conformally associated with the Lie point symmetry generator X3 defined in Case 1 of Table 1. Thesymmetry condition (4.1) yields the following two uncoupled equations for T1 and T2:

X ½2�3 T 1 þ ðn� m� kÞT 1 ¼ 0; ð4:3ÞX ½2�3 T 2 þ ð2� n� m� kÞT 2 ¼ 0; ð4:4Þ

where X ½2�3 is the second prolongation of X3 defined by (2.5)–(2.7)


X ½2�3 ¼ ð2� n� mÞt o

otþ ðn� mÞz o

ozþ 2/

o

o/þ ðnþ mÞ/t

o

o/tþ ð2� nþ mÞ/z

o

o/zþ 2ðnþ m� 1Þ/tt

o

o/tt

þ 2m/tzo

o/tzþ 2ð1� nþ mÞ/zz

o

o/zz: ð4:5Þ

We substitute (4.5) into (4.3) and (4.4) and split (4.4) with respect to /t and /zt. This gives the following fourfirst order quasi-linear partial differential equations:

2/oT 1

o/þ ð2� nþ mÞ/z

oT 1

o/z¼ ðkþ m� nÞT 1; ð4:6Þ

2/oAo/þ ð2� nþ mÞ/z

oAo/z¼ ðk� 2ÞA; ð4:7Þ

2/oBo/þ ð2� nþ mÞ/z

oBo/z¼ ðkþ n� m� 2ÞB; ð4:8Þ

2/oCo/þ ð2� nþ mÞ/z

oCo/z¼ ðkþ nþ m� 2ÞC: ð4:9Þ

The general solution of Eqs. (4.6)–(4.9) is

T 1 ¼ /12ðkþm�nÞkðnÞ; ð4:10Þ

A ¼ /12ðk�2Þf ðnÞ; B ¼ /

12ðkþn�m�2ÞgðnÞ; C ¼ /

12ðkþnþm�2ÞhðnÞ; ð4:11Þ

where

n ¼ /z/12ðn�m�2Þ ð4:12Þ

and k, f, g and h are arbitrary functions. From (4.2) and (4.11),

T 2 ¼ /12ðk�2Þf ðnÞ/t þ /

12ðkþn�m�2ÞgðnÞ/zt þ /

12ðkþnþm�2ÞhðnÞ: ð4:13Þ

The components (4.10) and (4.13) are substituted into the conservation law equation (2.14), /zzt is replacedusing the differential equation (1.1) and /z is expressed in terms of n and / using (4.12). The determining equa-tion is then separated with respect to the derivatives of / which yields to the following six equations:

/zt/zz :dgdn¼ 0; ð4:14Þ

/t/zz :dfdnþ mgðnÞ ¼ 0; ð4:15Þ

/zz :dhdn¼ 0; ð4:16Þ

/zt :dkdnþ 1

2ðn� m� 2Þn2 dg

dnþ f ðnÞ þ 1

2ðkþ 3m� n� 2ÞngðnÞ ¼ 0; ð4:17Þ

/t :1

2ðn� m� 2Þn dk

dnþ 1

2ðn� m� 2Þn2 df

dnþ 1

2ðk� 2Þnf ðnÞ þ gðnÞ þ mðn� m� 1Þn2gðnÞ

þ 1

2ðkþ m� nÞkðnÞ ¼ 0; ð4:18Þ

/m�1 :1

2ðn� m� 2Þn2 dh

dnþ nngðnÞ þ 1

2ðkþ nþ m� 2ÞnhðnÞ ¼ 0: ð4:19Þ

Eq. (4.14)–(4.16) are independent of k. From (4.14)–(4.17),


f ðnÞ ¼ �mc1nþ c3; gðnÞ ¼ c1; hðnÞ ¼ c2; ð4:20Þ

kðnÞ ¼ 1

4ðn� mþ 2� kÞc1n

2 � c3nþ c4; ð4:21Þ

where c1, c2, c3 and c4 are constants. Substituting (4.20) and (4.21) into (4.18) and (4.19) and splitting (4.18)with respect to powers of n gives

n2 : ½kþ nþ 3m� 4�½k� ðn� mþ 2Þ�c1 ¼ 0; ð4:22Þn : 2c1 þ ðkþ m� nÞc4 ¼ 0; ð4:23Þn0 : 2nc1 þ ðkþ nþ m� 2Þc2 ¼ 0: ð4:24Þ

There is no condition on c3.We will consider solutions with c1 5 0. It follows from (4.22) that there are two cases to consider.Case (i) k = 4 � n � 3m

Then

c2 ¼nc1

m� 1; c4 ¼

c1

nþ m� 2; ð4:25Þ

provided m 5 1 and n + m 5 2. Eqs. (4.10) and (4.13) become

T 1 ¼ �c1

1

2ð1� n� mÞ/�2m/2

z þ1

ð2� n� mÞ/2�n�m

� �� DzðxÞ; ð4:26Þ

T 2 ¼ �c1 m/�2m/z/t � /1�2m/zt þn

ð1� mÞ/1�m

� �þ DtðxÞ; ð4:27Þ

where

x ¼2c3

ð4� n� 3mÞ/12ð4�n�3mÞ if 4� n� 3m 6¼ 0;

c3 ln / if 4� n� 3m ¼ 0:

8<: ð4:28Þ

The conservation law (2.14) is trivially satisfied by T1 = �Dz(x) and T2 = Dt(x). These terms are therefore notincluded in the conserved vector. The components (4.26) and (4.27) agree with the conserved vector given inCase B.1 (l = 1) of Table 2 which requires m 5 1, n + m 5 2 and also n + m 5 1. When n + m = 1, (4.26) and(4.27) reduce to the conserved vector given in Case A of Table 2 which is of type l = 0, not l = 1. Whenn + m = 1 and l = 1 the conserved vectors are given by B.4 and B.5 of Table 2 and it can be verified that whenX = X3,

T �ð1Þ ¼ ð3� 2mÞT ð1Þ þ 2T ð0Þ; ð4:29Þ

where T(1) and T(0) are the conserved vectors for l = 1 and l = 0. Thus when n + m = 1 and l = 1 the con-served vectors do not satisfy the invariance condition (4.1) with X = X3 and therefore are not found by thissearch.

Case (ii) k = n � m + 2

Then, since n 5 0,

c2 ¼ �c1; c4 ¼ �c1 ð4:30Þ

and (4.10) and (4.13) become

T 1 ¼ �c1/� DzðXÞ; ð4:31ÞT 2 ¼ �c1½m/n�m�1/z/t � /n�m/zt þ /n� þ DtðXÞ; ð4:32Þ


where

X ¼2c3

ðn� mþ 2Þ/12ðn�mþ2Þ if n� mþ 2 6¼ 0;

c3 ln / if n� mþ 2 ¼ 0:

8<: ð4:33Þ

The components (4.31) and (4.32) agree with the conserved vector of Case A (l = 0) in Table 2.We have seen that the derivation of the conserved vectors for l = 1 and l = 0 can be performed together

starting from the same conserved vector (4.2) and the same Lie point symmetry generator (4.5).

5. Discussion

The physical relevance of the conservation laws found in this paper and those derived by Barcilon andRichter and by Harris lies in their close relationship to conserved quantities [4]. By integrating (2.14) withrespect to z from z = �1 to z = +1 we obtain

o

ot

Z 1

�1T 1 dz ¼ 0; ð5:1Þ

provided T2 vanishes at z = ±1. Thus

o

ot

Z 1

�1T 1ðt; z;/;/z; . . . ;/lzÞdz ð5:2Þ

is a conserved quantity. We must therefore be able to evaluate T1 at t = 0 and therefore T1 at t = 0 should be afunction only of the initial conditions. This is indeed the case. The components T1 of the known conservationlaws in Table 2 depend on / and the z-derivatives of / but not on the t-derivatives of /. The components T1 ofthe new conservation laws in Table 3 depend on z and / and one could depend on t and z but they do notdepend on the t-derivatives of /.

In some problems a conserved quantity plays a central role in the solution of the problem although theywere not required in deriving the group invariant solutions of Eq. (1.1) derived by Maluleke and Mason[15]. Examples in which a conserved quantity is used in the solution is the axisymmetric spreading of a thinliquid drop [19] and the spreading of a two-dimensional and axisymmetric free jet [20,21]. For the liquid dropthe conserved quantity is the total volume of the liquid drop and for the free jet it is the total momentum fluxin the direction of the jet. In these problems the Lie point symmetry which generates the solution is associatedwith the elementary conservation law for the differential equation.

Conservation laws can be used to reduce the order of differential equations although we did not investigatethat here. For second order ordinary differential equations which have a Lagrangian, the Noether point sym-metry is also a point symmetry of the corresponding conserved quantity and it can be used to reduce the orderof the differential equation more than once and in some cases the differential equation can be integrated com-pletely [22,23].

6. Conclusions

We saw how new conserved vectors can be generated from known conserved vectors and Lie point symme-try generators. Unlike the known conserved vectors the new conserved vectors depend explicitly on the spatialcoordinate z. For n = 0, m ¼ 4

3, the new conserved vector did not generate further new conserved vectors. We

found that the commutator brackets provided a useful guide in the search for new conservation laws throughTheorem 2. If the three Lie point symmetry generators in the commutator bracket relation are distinct, as in(3.1)–(3.3), then condition (2.11) of Theorem 2 is clearly not satisfied.

The calculation of conserved vectors when an invariance condition is imposed on the conservation lawequation is simpler than the direct method. However, it has the disadvantage that all possible conservationlaws may not be found. The Lie point symmetry generator, X3, is associated with a conserved vector onlyfor special values of n and m. We therefore imposed the weaker invariance condition of conformal association.This broadened the search and most, but not all, of the known conserved vectors can be derived more simply.


Eq. (4.29) gives an indication of how the invariance condition could be weakened further to become still moreinclusive. A linear combination of conserved vectors could be considered on the left hand side of (4.1).

Acknowledgements

This material is based upon work supported by the Growing Our Own Timber (GOOT) programme of theUniversity of the Witwatersrand, Johannesburg and by the National Research Foundation, Pretoria, SouthAfrica, under grant number 2053745. We thank Dr. Astri Sjoberg of the University of Johannesburg and Pro-fessor Fazal Mahomed of the University of the Witwatersrand for fruitful discussion.

References

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