Characterisation of the Algebraic Properties of First Integrals of Scalar Ordinary Differential Equations of Maximal Symmetry

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 212, 349᎐374Ž. 1997 ARTICLE NO. AY975506

G. P. Flessas,* K. S. Govinder,† and P. G. L. Leach‡ View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by Elsevier - Publisher Connector Department of Mathematics, Uni¨ersity of the Aegean, Karlo¨assi, 83 200, Samos, Greece

Submitted by Thanasis Fokas

Received March 10, 1995

We undertake a study of the first integrals of linear nth order scalar ordinary differential equations with maximal symmetry. We establish patterns for the first integrals associated with these equations. It is shown that second and third order equations are the pathological cases in the study of higher order differential equations. The equivalence of contact symmetries for third order equations to non-Cartan symmetries of second order equations is highlighted. ᮊ 1997 Academic Press

1. INTRODUCTION

A scalar ordinary differential equation

Žn. ExŽ.,y,yЈ,..., y s0,Ž. 1

*E-mail address: [email protected]. † Permanent address: Department of Mathematics and Applied Mathematics, University of Natal, Durban, 4041, Republic of South Africa. E-mail address: [email protected]. ‡ Permanent address: Department of Mathematics and Applied Mathematics, University of Natal, Durban, 4041, Republic of South Africa. Member of the Centre for Theoretical and Computational Chemistry, University of Natal and associate member of the Centre for Nonlinear Studies, University of the Witwatersrand, Johannesburg, Republic of South Africa. E-mail address: [email protected].

349

0022-247Xr97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 350 FLESSAS, GOVINDER, AND LEACH where Ј denotes differentiation of the dependent variable, y, with respect to the independent variable, x, and yŽn. the nth derivative, possesses a Lie symmetry ѨѨ Gs␰q␩ Ž.2 Ѩx Ѩy if Gw nx E 0,Ž. 3

niy1 Ѩ n Ži.Žij1.Žij. GwxsGqÝÝ␩y yq␰y .4Ž. ½5ž/jѨyŽi. is1js0

ŽThe extension is needed to give the infinitesimal transformations in the derivatives up to yŽn. induced by the infinitesimal transformations which G produces in x and y..Ž. The symmetries of 1 constitute a Lie algebra under the operation of taking the Lie Bracket

wxG12,GsGG 12yGG 21.5Ž. A first integral ofŽ. 1 associated with the symmetryŽ. 2 is a function, fxŽ ,y,yЈ,..., yŽny1.., in which the dependence on yŽ ny1. is nontrivial, satisfying the two conditions

n 1 Gw y x f s 06Ž. and df s 0.Ž. 7 dx Es0 The association of f with G, as stated inŽ. 6 and Ž. 7 , means that f is a first integral ofŽ. 1 . Equally a first integral, fx Ž ,y,yЈ,..., yŽny1..Ž., of 1 has a symmetry of the form ofŽ. 2 if

n 1 Gw y x f s 0.Ž. 8 In recent years a number of papers have been devoted to the algebraic properties of first integrals of scalar ordinary differential equations associated with their symmetries. The number of symmetries associated with a differential equation depends upon its internal structure up to an upper limit which is fixed by the order of the equation and the type of symmetry under consideration. In this paper we are concerned with point and LIE ALGEBRAS OF FIRST INTEGRALS 351 contact symmetries only. The symmetryŽ. 2 is a Lie point symmetry if the coordinate functions ␰ and ␩ are functions of x and y only. It is a contact symmetry if ␰ and ␩ depend upon x, y, and yЈ subject to the constraint thatwx 2, p. 94

Ѩ␩ Ѩ␰ s yЈ Ž.9 ѨyЈ ѨyЈ which means that the first extension of G also depends upon x, y, and yЈ only. A point symmetry is always a contact symmetry. Note that we do not restrict ␰ to be a function of x only which is the case for the so-called Cartan symmetrieswx 3 . Lie showed that the maximum number of point symmetries of a scalar ordinary differential equation was infinite for equations of the first order wx4, p. 114 , eight for equations of the second order wx 4, p. 405 , and n q 4 for equations of the nth orderwx 5, p. 298 . He also wx 6 classified all the invariance algebras of dimension one, two, and three for second order equations. Mahomed and Leachwx 1 showed that higher order linear equations could have n q 1ornq2 point symmetries instead of the n q 4 for the maximal symmetry case.Ž The contributions of Krause and Michelwx 7 should also be noted.. Whenever reference is made to linear equations, we include nonlinear equations which are linearisable by a pointŽ. resp. contact transformation when point Ž. resp. contact symmetries are being considered. Lie also showed that second order equations possessed an infinite number of contact symmetrieswx 2, p. 84 and third order at most tenwx 2, p. 241 . For higher orderŽ. in standard form it has been shownwx 8 that equations of maximal symmetry only admit n q 4 contact symmetries. Abraham-Shrauner et al.wx 9 showed that the maximal number of contact symmetries of third order equations could be found in equations which did not have sevenŽ. the maximum point symmetries in addition to the Kummer-Schwarz equation given by Liewx 2, p. 148 . Leach and Mahomedwx 10 discussed the algebraic properties of the first integrals of equations of maximal point symmetry which are represented by the single equivalence class

yЉ s 0. Ž.10

It was found that the two functionally independent first integrals

I1s xyЈ y y Ž.11

I2syЈ Ž.12 352 FLESSAS, GOVINDER, AND LEACH

each possessed three symmetries with the three-dimensional algebra A3, 3 in the Mubarakzyanov classificationwx 11 . Furthermore the related integral

I1 I3s Ž.13 I2 also possessed three symmetries with the same algebra. In the case ofŽ. 13 the three symmetries are

Ѩ G1s y Ž.14 Ѩy ѨѨ 2 G2sxy q y Ž.15 Ѩx Ѩy Ѩ G3sy .16Ž. Ѩx

G1 is the homogeneity symmetry and for any integral to possess it the integral must be homogeneous of degree zero in y. G23and G are the non-Cartan symmetries ofŽ. 10 . Note that, although the integrals have nine symmetries in all, there are only eight linearly independent symmetries. The interesting algebraic structure of the symmetries associated with the linear integrals ofŽ. 10 led to the study of the corresponding algebras of third order equations with four, five, and seven symmetrieswx 12 and nth order equations with n q 1, n q 2, and n q 4 symmetrieswx 13 . The pattern of the second order equations was not maintained. The demonstra- tion by Abraham-Shrauner et al.wx 9 that contact symmetries are the appropriate ones to be used in treatments of third order equations led to a studywx 14 of the algebraic properties of the contact symmetries associated with the integrals of third order equations with the symmetry algebra spŽ.4 which was more extensive than the comparable study of second order equations by Leach and Mahomedwx 10 . Second and third order ordinary differential equations have properties which are peculiar to each and which mark them off from the general nth order equation. This is the case even for the representative equations of maximal symmetry. For

yЉ s 017Ž. there are eight point symmetries, six of which are of Cartan form, and two of which are not. In the case of

.yٞ s 018Ž LIE ALGEBRAS OF FIRST INTEGRALS 353 the seven point symmetries are all of Cartan form, but there are three additional intrinsicallyŽ i.e., contain derivatives when the equation is in standard form. contact symmetries and the consideration of third order equations without contact symmetries is as incomplete as the consideration of second order equations without the non-Cartan point symmetries. However,

Žn. y s 019Ž. has only n q 4 point symmetries all of which are of Cartan typewx 5, p. 298 . It is this variation in the type, rather than in the numberŽ apart from the fact that it is maximal. , of symmetries which suggests that the generic behaviour for ordinary differential equations of maximal symmetry is not to be found at the second or third order, but at the fourth order. It is the intention of this paper to demonstrate this feature and to describe the generic properties of the algebras of the symmetries of first integrals of scalar ordinary differential equations thereby to increase awareness of the algebraic features of first integrals of these equations. In this sense the paper is a theoretical discussion. However, there are possible practical applications. The symmetries of higher order nonlinear ordinary differential equations are very difficult to calculate even with symbolic manipulation codes. The calculation of symmetries of first integrals of those equations is somewhat easier. In the case that the nonlinear equation belongs to one of the equivalence classes of the equations discussed here and one has the good fortune to light upon an appropriate first integral the number of symmetries and their algebra will indicate the path to linearisation and consequent solution. That the algebra is a proper subalgebra of the algebra of the differential equation makes the identifica- tion of the correct transformation all that much easier. A simple example is given in Section 6.

2. METHODOLOGY

There is a certain ambiguity in the treatment of the symmetries of the first integrals. In the case of the second order equation,Ž. 10 , we recalled that the three first integrals,Ž.Ž. 11 , 12 , and Ž. 13 , each had three point symmetries and that the algebras were isomorphic. However, this is not the case for any combination, other thanŽ. 13 , of Ž. 11 and Ž. 12wx 10 . We do seek those first integrals which have, in some sense, a maximal number of symmetries. To make comparison from one order to another feasible it is necessary to use as much of the common structure for the algebras of the differential equations as is possible. To this end we use the form given by 354 FLESSAS, GOVINDER, AND LEACH

ŽŽ . . Mahomed and Leachwx 1 , nA1 [s sl 2, R [ A1 , to which the two non- Cartan symmetries are to be added when n s 2 and the three purely Ž contact symmetries are to be added when n s 3. We use [s to denote the semidirect sum.. To each of these symmetries there corresponds a first integral and it is the symmetries of these integrals which we consider. For an nth order equation there are n y 1 integrals associated with each symmetry of the nth order equation and the first integral mentioned above is an arbitrary function of these. We select n y 1 of these arbitrary independent functions in such a way as to have the maximum number of symmetries possible. In the case ofŽ. 18 there are three functionally independent first integrals which we take to bewx 14

1 2 I1s 2xyЉyxyЈ q y Ž.20

I2sxyЉ y yЈ Ž.21

I3syЉ Ž.22 which have been shownwx 12 to possess maximal symmetry. In terms of I1, I23, and I the integrals associated with the symmetries ofŽ. 18 are given wx14 in Table I. Note that each symmetry has two integralsŽ denoted by p and q. since the equation is of the third order. In Table II we list the symmetrieswx 14 associated with each integral given in Table I in terms of the symmetries ofŽ.Ž 18 and combinations thereof . and the corresponding algebra according to the Mubarakzyanov classificationwx 11 . We observe that the algebras are either three-dimensional or four-dimensional. The 1 latter is always A4, 9, but the former are either A3, 4 ŽŽalso known as E 1, 1. ,

TABLE I The First Integrals Associated with the Ten Contact Symmetries of yٞ s 0

Symmetry pq

GII123 GII213 GII312 12 GII431I3y2I2 12 GII521I3y2I2 12 GII611I3y2I2 GI72rII33rI1 Ž.12 GI82rIII313y2I2rI3 Ž.12 GI93rIII113y2I2rI1 Ž.12 GI10 1rIII 2 1 3y2 I 2rI 2 LIE ALGEBRAS OF FIRST INTEGRALS 355

TABLE II The Contact Symmetries and Algebras of the Integrals

Integral Symmetries Nonzero Lie brackets Algebra

1 IX1 11sGX 2 wx11,X 13sXA 12 4, 9 X12sGX 3 wx11,X 14 syX11 X13sGX 7 wx12,X 14 sy2X12 X14sG 6yGX 4 wx13,X 14 syX13 ŽŽ .. IX2 21sGX 1 wx21,X 23sXAE 21 3, 4 1, 1 X22sGX 3 wx22,X 23 syX22 X23s G 6 1 IX3 31sGX 1 wx31,X 34s2XA 31 4, 9 X32sGX 2 wx32,X 33 syX31 X33sGX 5 wx32,X 34sX 32 X34sG 6qGX 4 wx33,X 34sX 33 ŽŽ .. I4sI 3rIX 2 41sGX 1 wx41,X 42sXAE 41 3, 4 1, 1 X42sGX 4 wx42,X 43sX 43 X43s G 8 ŽŽ .. I5sI 3rIX 1 51sGX 4 wx51,X 52 syXAsl52 3, 8 2, R X52sGX 2 wx51,X 53sX 53 X53sGX 9 wx52,X 53 sy2X51 ŽŽ .. I6sI 2rIX 1 61sGX 4 wx61,X 62 syXAE62 3, 4 1, 1 X62sGX 3 wx61,X 63sX 63 X63s G 10 1 2 ŽŽ .. I7132sIIy2IX 715sGXwx717271,XsXAsl 3,82, R X72sGX 6 wx71,X 73s2X 72 X73sGX 7 wx72,X 73sX 73 1 I8sI 7rIX 3 81sGX 5 wx81,X 82sXA 81 4, 9 X82sG 6yGX 4 wx81,X 84s2X 83 X83sGX 8 wx82,X 83 syX83 X84sGX 9 wx82,X 84 syX84 ŽŽ .. I9sI 7rIX 2 91sGX 8 wx91,X 92sXAE 91 3, 4 1, 1 X92sGX 6 wx92,X 93sX 93 X93s G 10 1 I10sI 7rIX 1 101sG 6qGX 4wx 101,X 102sXA 102 4, 9 X102sGX 7 wx101,X 103sX 103 X103sGX 9 wx101,X 104s2X 104 X104sGX 10 wx102,X 103 sy2X104 356 FLESSAS, GOVINDER, AND LEACH

the algebra of the pseudo-Euclidean group in the plane.Ž or A3, 8 much better known as slŽ..2, R . This already indicates two departures from the results forŽ. 17 in that the dimensions of the algebras are not the same and also the three-dimensional algebras differ not only from that of the second order case, A3, 3, but within themselves. Perhaps even more surprising is 1 that A3, 3 is not a subalgebra of A4, 9 wx15 . With the example of the third order equation before us we shall present the results for the fourth order equation of maximal symmetry, viz.

i y ¨ s 0,Ž. 23 in Sections 3 and 4. In Section 5 our discussion addresses, among a number of observations, the matter of the general equation

Žn. y s 0.Ž. 24

3. SYMMETRIES AND THEIR INTEGRALS

EquationŽ. 23 is the representative equation of the fourth order with the maximum number of symmetries which is eight with the algebraic struc- ŽŽ.. ture A1 [ sl 2, R [s 4 A1. The symmetries are Ѩ G1 s Ѩ y Ѩ G2 s x Ѩ y Ѩ 1 2 G3 s 2 x Ѩ y 1 Ѩ 3 G4 s x 6 Ѩ y Ѩ G5 s Ѩ x Ѩ 3 Ѩ G6 s x q y Ѩ x 2 Ѩ y ѨѨ 2 G7 sxq3xy Ѩ x Ѩ y Ѩ G8s y .25Ž. Ѩy LIE ALGEBRAS OF FIRST INTEGRALS 357

The first four symmetries are called the solution symmetries since the coefficient of ѨrѨ y is a solution of the original differential equation.Ž In this respect the linear superposition principle is one of the banes of linear differential equations. It is necessary to be able to solve the equation before most of the symmetries can be determined. In this respect nonlinear equations are more amenable to treatment.. G57through G are the elements of slŽ.2, R appropriate toŽ. 23 . For an nth order equation of maximal symmetry they have the formwx 1 Ѩ n y 1 Ѩ G s axŽ. q aЈŽ.xy ,26Ž. Ѩx 2 Ѩy where axŽ.is one of the three solutions of the self-adjoint equation Ž.n 1! q 1X aٞqBan2Јq2Ban2 s0,Ž. 27 Ž.ny2!4! y y

Žny2. where Bny2 is the coefficient of y when the equation is cast into normal form. The final symmetry, G8, follows from the homogeneity of the differential equation in y and its derivatives which happens to coincide with linearity in this case.

In calculating the first integrals associated with each of G18through G according to Eqs.Ž. 6 and Ž. 7 we find that four functionally independent linear first integrals occur. To make the reportage of our results more compact we express all other integrals in terms of them. They are

1132 I1 s62xyٞy xyЉqxyЈ y y 12 I2 s2 xyٞyxyЉ q yЈ

I3 s xyٞ y yЉ

.I4s yٞ.28Ž With each symmetry there will be associated three functionally independent first integrals. This follows from the solutions to the two first order partial differential equationsŽ. 6 and Ž. 7 for the first integral associated with a particular symmetry. InŽ. 6 there are five variables, x, y, yЈ, yЉ, and yٞ, and so four characteristics. This means thatŽ. 7 has four variables and hence three characteristics each of which is a first integral. The integrals belonging to the symmetries are listed in Table III. In comparison with the integrals listed in Table IŽ less those associated . with the purely contact symmetries, G810y G for yٞ s 0 we note a number of similarities and differences. The solution symmetries, G14y G , simply form a permutation with three of the linear integrals.Ž The labelling was chosen to highlight this feature.. The homogeneity symmetry, G8, has 358 FLESSAS, GOVINDER, AND LEACH

TABLE III i The First Integrals Associated with the Eight Point Symmetries of y ¨ s 0

Symmetry First integrals

GII123 I 4 GII234 I 1 GII341 I 2 GII412 I 3 11232 GIII542432y23IIII3431yIyII4 11 33y GIIIIII61424 34 24232 GIII711321y33IIII2321yIyII4 GI81rII22rII3 3rI4 the three independent ratios of the linear integrals. Any other independent set of three could equally be chosen. The ratios are the only integrals possible for G8 since any integral associated with it must be of zero degree in y. These are the anticipated generalisations of the corresponding results for yٞ s 0 and, indeed, for yЉ s 0. ForŽ. 24 we may infer that each of the n solution symmetries will have n y 1 of the n functionally independent linear integrals associated with it and that, by a suitable choice of labels as was made forŽ. 18 and Ž. 23 , the one symmetry label and the n y 1 integral labels will be a cyclic permutation of the integers 1 through n. Equally confidently we infer that the homogeneity symmetry, Gnq4 , will have n y 1 independent ratios of the linear integrals associated with it. In the case of the representation of slŽ.2, R , which is common to all linear equations of maximal symmetry, the situation is not so clear. To give more scope for observation we list the corresponding relationships for yЉ s 0. They are Ѩ G32s I Ѩx ѨѨ 1 G41sxq2yII2Ž.29 ѨxѨy ѨѨ 2 1 G51sxq2xy I , ѨxѨy where, in the spirit adopted forŽ. 18 and Ž. 23 , the sl Ž2, R .symmetries have been listed after the solution symmetries and

I1 s xyЈ y y

I2s yЈ.30Ž. LIE ALGEBRAS OF FIRST INTEGRALS 359

One point should be made before we continue. Since the first and third of the two symmetries in slŽ.2, R are related by the transformation 1 y x y y Ž.31 ª x ª xny1 and the second is invariant underŽ. 31 , we need only consider the first two symmetries, invariance under translation in the independent variable and a self-similar transformation.Ž The integrals of the so-called conformal symmetry follow from those of invariance under translations in x by the application ofŽ.. 31 . There are two problems. The first is the form of the expressions for the more complicated integrals associated with ѨrѨ x. Under the labelling scheme adopted, In can always be taken as the first representative. The first representative for the self-similar symmetry differs from the even order equations to the odd order equation. This provides the necessary hint and we have

PROPOSITION 1. One of the integrals associated with the self-similar symmetry Ѩ n y 1 Ѩ G s x q y Ž.32 Ѩx 2 Ѩy of Žn. y s 033Ž. is

J11s IIn Ž.34 when n is e¨en and

J1s IŽnq1.r2 Ž.35 when n is odd, where the numbering of the functionally independent linear first integrals of Ž.33 is according to the scheme

k nyi Ž.1 y nyiykŽny1yk. IisÝ xy ,is1, n.36Ž. Ž.n i k! ks0yy

The result follows trivially from Proposition 3. Associated with the self-similar symmetry there is another problem. We choose this symmetry so that the subalgebra slŽ.2, R occurs naturally within the list of symmetries of the equation. However, the homogeneity symmetry, Gnq4 , can be added without changing the nature of the self-similar symmetry. 360 FLESSAS, GOVINDER, AND LEACH

4. INTEGRALS AND THEIR SYMMETRIES

The Lie point symmetries associated with each of the first integrals listed in Table III are calculated following the prescription ofŽ. 6 using the symbolic code Program LIEwx 16 . The integrals, symmetries, nonzero Lie Brackets, and algebras are given in Table IV. The symmetries are written as Xij in which the label i refers to the integral number and the label j to the number of the symmetry within the integral’s algebra. The relationship of these symmetries to those of the differential equationŽ. 23 is also given.

5. DISCUSSION

The linear integrals, as expectedwx 13 , have either five or four symmetries. Three of these are solution symmetries and, in the way the labelling has been arranged, the solution symmetry not included is the one of the number of the integral.Ž The action of the omitted symmetry on the integral is a constant, q1 for I24and I and y1 for I13and I due to the way the integrals have been defined.. The integrals with five symmetries have either G57or G which are equivalent under the transformation

1 y xyy.3Ž.7 ªx ªx3

The remaining symmetry is a combination of the self-similar G6 and the homogeneity G8 of the form

2i y 5 Xis G68q G , i s 1,4.Ž. 38 2

Similar combinations occur for some of the other integrals. This does suggest that the choice of the form of the self-similar symmetry is at our disposal and that we should not be constrained by the form of G6 as it occurs in the list of symmetries forŽ. 23 . This has some impact on the expressions for the integrals associated with the self-similar symmetry. If we take it to be of the form

ѨѨ Gsxyay ,39Ž. Ѩx Ѩy LIE ALGEBRAS OF FIRST INTEGRALS 361

TABLE IV The Point Symmetries and Algebras of the Integrals

Integral Symmetries Nonzero Lie brackets Algebra

1 IX1 11sGX 2 wx11,X 14syXA 11 4, 9 X12sGX 3 wx11,X 15s4X 12 X13sGX 4 wx12,X 14 sy2X12 33 X14sG 6y22GX 8 wx12,X 15sX 13 X15sGX 7 wx14,X 15sX 15 IX2 21sGX 1 wx21,X 24sXAE 21 3, 4 ŽŽ1, 1 .. X22sGX 3 wx22,X 24syX 22 X23sGX 4 wx23,X 24 sy2X23 1 X24s G 6y 2 G 8 1 IX3 31sGX 1 wx31,X 34s2XA 31 4, 9 X32sGX 2 wx32,X 34sX 32 X33sGX 4 wx33,X 34syX 33 1 X34s G 6q 2 G 8 1 IX4 41sGX 1 wx41,X 45s3XA 41 4, 9 X42sGX 2 wx42,X 44syX 41 X43sGX 3 wx42,X 45s2X 42 X44sGX 5 wx43,X 44 sy2X42 3 X45sG 6q2 GX 8 wx43,X 45sX 43 wxX44,X 45sX 44 I5sI 1rIX 2 51sGX 3 wx51,X 52sXAE 51 3, 4 ŽŽ1, 1 .. X52sGX 4 wx52,X 53sX 52 X53s G 8 I6sI 2rIX 3 61sGX 1 wx61,X 63sXAD 61 3, 3 Ž.ms T2 X62sGX 4 wx62,X 63sX 62 X63s G 8 I7sI 3rIX 4 71sGX 1 wx71,X 73sXAE 71 3, 4 ŽŽ1, 1 .. X72sGX 2 wx72,X 73sX 73 X 73s G 8 1 2 1r2 I8243sIIy2IX 811sGXwx818381,XsXA 3,5 1 X82sG 5 wxX82,X 83s2 X 82 11Ž. X83s22G 6qG 8 13 I9sIII 2 3 4y3 IX 3 91sGX 5 wx91,X 92sX 91 2A1 2 1 yII14 X92sG 6q2G 8 I10sII 1 4 X101sGX 2 wx101,X 103sXAE 101 3, 4 ŽŽ1, 1 .. X102sGX 3 wx102,X 103syX 102 X103s 2G 6 1r3 y1r3 I11sII 1 3 X111sGX 1 wx111,X 113sXA 111 3, 5 1 X112sGX 3 wx112,X 113s3 X 112 2 X113s 3 G 6 y1r3 1r3 I12sII 1 2 X121sGX 1 wx121,X 123sXA 121 3, 5 1 X122sGX 2 wx122,X 123s3 X 122 2 X123s 3 G 6 2 2 1r2 I13sII 1 3y3 IX 2 131sGX 4 wx131,X 133sXA 131 3, 5 1 X132sGX 7 wx132,X 133s2 X 133 11Ž. X133 sy22G6y G 8 413 I14sIII 1 2 3y32IX 2 141sG 6yGX 8 wx141,X 142sX 142 2A1 2 yII14 X142sG 7 362 FLESSAS, GOVINDER, AND LEACH where a is a parameter, the three integrals are

ya rŽaq3. J14s I I 1 Ž.40

yŽaq1.rŽaq3. J24sI I2 Ž.41

yŽaq2.rŽaq3. J34sI I3.42Ž.

Equivalently other combinations may be taken. For example, if I1 were taken as the integral to have the fractional power associated with it, we would have

yŽ aq3.r a K11s I I 4 Ž.43

yŽaq2.ra K21sI I 3 Ž.44

yŽaq1.ra K31sI I 2.45Ž. In the cases of the four forms of the self-similarity symmetry found in the algebras we find the following sets of integrals

X14: J 1s I 1

y1r3 J24s I I 2

y2r3 J34sII 3

1r2 X24:J 1sI 4I 1

J22s I

y1r2 J34sII 3 Ž.46 2 X34:J 1sI 4I 1

J242s II

J343sII

X44:J 1sI 4

y2r3 J21s I I 2

y1r3 J31sII 3. The symmetries of the ratio integrals all have the same algebras. There are three symmetries with the algebra A3, 3 which represents dilatations and translations in the plane. Two of the symmetries are the solution symmetries of subscript not one of the two integrals in the ratio and the third is the expected homogeneity symmetry, G8. We note that this algebra is the same as for the ratio integral ofŽ. 17 although that algebra used LIE ALGEBRAS OF FIRST INTEGRALS 363 non-Cartan symmetries. The three ratio integrals ofŽ. 18 are different.

Although they are three-dimensional, the algebras are A3, 4 Žthe algebra of Ž.. the pseudo-Euclidean group E 1, 1 for the ratios I12rI and I 32rI . For ŽŽ.. I13rIit is A 3,8or sl 2, R . These algebras involved one truly contact symmetry. Given the conflicting information from the three equations as far as the algebras of their ratio integrals are concerned it is not immediately evident what the general situation is. However, if we consider just the Cartan symmetries, we find the following pattern for the algebras of the k ratio integrals. If we denote a ratio integral by Rij, where i and j are the labels of the two linear integrals comprising the ratio integral and k is the order of the equation, we have Ѩ 2 R12 :Ä4Gsy ½5Ѩy ѨѨ 3 Rij:Ä4Gsqsm ;m/i,j ½5½Ѩy Ѩy 5 ѨѨѨ 4 Rij:Ä4Gsqsm,s n ;m,n/i,j, ½5½Ѩy Ѩy Ѩy 5 where smnand s are solutions ofŽ. 18 or Ž. 23 . Thus we are led to

PROPOSITION 2. The number of symmetries associated with the ratio of any two linear first integrals of an nth order scalar ordinary differential equation is n y 1, where n G 4. The symmetries consist of the homogeneity symmetry yŽ.ѨrѨ y and n y 2 solution symmetries. The Lie algebra of the Ž. symmetries is n y 2 A1 [s A1. Proof. Since the linear integrals are each homogeneous of degree one in y, the ratio integral is of degree zero in y and so possesses yŽ.ѨrѨ y as a symmetry.

Let the linear integrals be labelled I1 y In and the solution symmetries G1 y Gn in such a way that

w ny1x GijI s 0, i / j Ž wny1x . as noted above GIiis"1 depending on the value of i . Then

wny1x wny1x IGjijkjikŽ.II IGŽ.I wny1x y Gi s 2 ž/Ik Ik s0 provided

w ny1x wny1x Ž.GijkjikIIyIGŽ.I s0. Ž.47 364 FLESSAS, GOVINDER, AND LEACH

If i / j, k, this is true. Hence there are at least n y 2 solution symmetries for IjkrI . There are also at most n y 2 solution symmetries since j / k Ž.Ž. and, if i s k say , 47 would require Ij to be zero. The symmetries of the Ž.Ž . ratio integral comprise solution symmetries of the form sxm ѨrѨy, Ž. Žn. where sxm is one of the fundamental solutions of y s 0, and the homogeneity symmetry, yŽ.ѨrѨ y . The algebra of the solution symmetries Ž. admitted by the ratio integral is n y 2 A1 and, since ѨѨ Ѩ sxmmŽ. ,y ssxŽ. , ѨyѨyѨy Ž. the algebra is n y 2 A1 [s A1. Q.E.D. We have now been able to explain the algebras of the linear and ratio integrals in general. It now remains to deal with those associated with the three symmetries of slŽ.2, R . It is well to recall them. For yЉ s 0 we have Ѩ G31s , J s yЈ Ѩx ѨѨ 1 G421sxq2y,JsyЈŽ.xyЈ y y s JJ3 ѨxѨy ѨѨ 2 G53sxqxy , J s xyЈ y y. ѨxѨy

J13and J are the two linear integrals and have three Cartan symmetries whereas J2 has only one Cartan symmetry. They are Ѩ J111: X s Ѩy Ѩ X12 s Ѩ x ѨѨ X13 sxqy Ѩ x Ѩ y ѨѨ 1 J221:Xsxq2y Ѩx Ѩy Ѩ J331:Xsx Ѩy ѨѨ 2 X32 sxqxy Ѩ x Ѩ y Ѩ X33 s x . Ѩ x LIE ALGEBRAS OF FIRST INTEGRALS 365

The algebras are A3, 3 for the symmetries of J1and J 3 and A1 for that of J2 . The situation for the third order equation presents an anomaly due to the behaviour of the symmetries of slŽ.2, R and the solution integrals under the transformationŽ. 35 for which

G57ª GG 66ªGG 75ªG

I13ªII 22ªII 31ªI.

This does not have much effect on the single integrals, I12, I , and I 3, but it 12 does on the quadratic integral, II13y2I 2, which is invariant under the transformationŽ.Ž 37 . The reader will appreciate the resemblance to the generalised Kummer-Schwarz equationwx 17 .. This is why this integral possesses the slŽ.2, R algebra of its symmetries in contrast to the A3, 3, 1r2 A3, 4 , and A3, 5 found for the integrals of the second and fourth order equations. In fact this labelling is a little misleading. The nonzero Lie 1r2 Brackets of the three algebras, A3, 3, A 3, 4 , and A3, 5 , are respectively

wxG13,GsG 1, wxG 23,GsG 2

wxG13,GsG 1, wxG 23,GsyG 2

wxG13,GsG 1, wxG 23,GsaG 2,0-<

wxG13,GsG 1, wxG 23,Gs␣G 2,0-<<␣ F1 to avoid confusion with the more standard notation. In this way we obviate a plethora of labels for what is essentially the one algebra. The numerical coefficient other than 1 depends upon the order of the equation and the number of the integral being considered. 2 We have noted that for n G 4 the treatment of the ѨrѨ x and x Ž.ѨrѨ x qŽ.Ž.ny1xy ѨrѨ y symmetries and their associated integrals is equivalent under the transformationŽ. 37 . Consequently the only two symmetries with which we must deal are ѨrѨ x and xŽ.ŽŽ..Ž.ѨrѨ x q n y 1 r2 y ѨrѨ y .Itis not surprising that these should be the most complex of all the symmetries. In fact there is insufficient information in the cases considered and we add Ž . Ž i. the results for y ¨ s 0 and y ¨ s 0. We first note some of the properties of the linear integrals as they are defined inŽ. 36 . For the linear integrals Žn. of y s 0 we have

w ny1x nqj GIijsyŽ.1 ␦ij, i,js1, n Ž.48 366 FLESSAS, GOVINDER, AND LEACH

for Gi a solution symmetry,

Ij1,1jn1 wny1x q FFy GIn1 js Ž.49 q ½0, j s n,

Ѩ Ѩ where Gnq1 s r x, and n 1 2 j w ny1x q y GIn2 jjs I,50Ž. q 2 Ѩ Ѩ 1Ž.Ѩ Ѩ where Gnq2 s x r x q 2 n y 1 y r y. This accounts for the two mem- bers of slŽ.2, R with which we must treat. In addition we have

w ny1x GIs1 jjswxnyjI,51Ž. Ž. where Gs1 s xѨrѨ x q n y 1 yѨrѨ y, and

w ny1x GIs2 jjswxnyjy1I,52Ž.

Ѩ Ѩ Ž.Ѩ Ѩ where Gnq2 s x r x q n y 2 y r y. The necessity for considering Gs1 and Gs2 becomes evident shortly. As we do not have labels for all of the algebras to be considered, we simply label them by the two subalgebras they contain. The first subalgebra consists of solution symmetries and the second of elements of slŽ.2, R .

In Table V we list the integrals associated with Gnq1 for n s 4, 5, and 6 and the algebras associated with the integrals. The A2 algebra consists of

TABLE V i¨¨ ¨i Integrals and Associated Algebras of Gnq1 for y s 0, y s 0, and y s 0

Equation Integrals Algebras

i¨ y s 0 I413 A [sA2 12 II24y2IA 3 1[sA2 1 32 III234y3I 3yII 14 A2 ¨ ys0 I514A[sA2 12 II35y2I 4 2A1[s A2 1 32 III345y3I 4yII 25 A1[s A2 1124 23 28III345y I 4yIII 245qII 15 A2 ¨i ys0I615A[sA2 12 II46y2I 5 3A1[sA2 1 32 III456y3I 5yII 36 2A1[sA2 1124 23 28III456y I 5yIII 356qII 26 A1[sA2 11352224 63III456y 0I 5yIII 356qIII 256yII 16 A 2 LIE ALGEBRAS OF FIRST INTEGRALS 367

Gnq1 and Gs1 for the Gnq1 first integral and Gnq1 and Gs2 for the subsequent integrals. We note that all of the integrals have both Gnq1 and Gnq2 as symmetries. The number of solution symmetries decreases as the number of linear integrals included in each integral increases. To take the Ž¨ . case of y s 0, for example, I5 is invariant under four solution symmetries, the second integral contains three of the linear integrals and these can only have two solution symmetries in common. Hence the number of these drops from four to two. The third integral introduces the linear integral I2 and the number of solution symmetries is reduced to one. This single symmetry is lost in the fourth integral when I1 is introduced. The pattern for the algebraic structures of the integrals associated with

Gnq1 is clear from Table V. It is also evident that, as the order of the equation increases by one, an additional integral is added to the list for the previous equation and that the subscripts for the existing ones are in- creased by one. Hence the problem of determining the integrals is reduced to finding a homogeneous polynomial of degree n y 1in I1 yIn which is w ny1x invariant under the action of Gnq1 . We turn now to the self-similar symmetry, Gnq2 . The integrals and the algebras of their symmetries are listed in Table VI, also for the three i i equations y ¨ s 0, y¨ s 0, and y¨ s 0. Recalling that in the case of yЉ s 0 the integral was II12with the single Cartan symmetry G4s xѨrѨ x 1 q2yѨrѨyand for yٞ s 0 the integral was I23with the anomalous A ,4 algebra, the pattern for the self-similar symmetry is clear to see. We have

PROPOSITION 3. For

Žn. y s 0, n G 453Ž.

TABLE VI i¨¨ ¨i Integrals and Associated Algebras of Gnq2 for y s 0, y s 0, and y s 0

Equation Integrals Algebras

i¨ Ž. ys0 II14 2A1[sGA63,4 1r31Ž.r3 II24 2A1[sGA63,5 y1r31Ž.r3 II34 2A1[sGA63,5 ¨ ys0 II15 3A1[sG7 1r2 II25 3A1[sG7 I314A[sG7 y1r2 II45 3A1[sG7 ¨i ys0 II16 4A1[sG8 3r5 II26 4A1[sG8 1r5 II36 4A1[sG8 y1r5 II46 4A1[sG8 y3r5 II56 4A1[sG8 368 FLESSAS, GOVINDER, AND LEACH there are n y 1 integrals associated with the symmetry

Ѩ n y 1 Ѩ G s x q y Ž.54 Ѩx 2 Ѩy of the form

Žnq1y2i.rŽny1. Jiins II , is1,...,ny1.Ž. 55

Proof. We first confirm thatŽ. 54 is a symmetry ofŽ. 53 . The nth extension ofŽ. 54 is Ž using Ž.. 4

Ѩ n n 1 Ѩ n y Ži. Gw x s x qyÝ iy .56Ž. Ѩx ž/2 ѨyŽi. is0

Operating onŽ. 53 with Ž. 56 gives

n 1 y Žn. y ny s0,Ž. 57 ž/2 which, givenŽ. 53 , is identically satisfied. For the determination of the first integrals associated withŽ. 54 we require the Ž.n y 1 st extension of G, viz.

Ѩ ny1 n 1 Ѩ n 1 y Ži. Gw y x s x qyÝ iy .5Ž.8 Ѩx ž/2 ѨyŽi. is0

If we assume the form

Žn 1. I s fxŽ.,y,yЈ,..., y y Ž.59 for the first integral, the associated Lagrange’s system of

n 1 Gw y x I s 060Ž. is

dx dy dyŽi. ssиии s Ži. xnŽ.y1yr2 Ž.Ž.ny1r2yiy dyŽny1. s иии s Žn1. .61Ž. yŽ.ny1yyr2 LIE ALGEBRAS OF FIRST INTEGRALS 369

The first set of Ž.n characteristics comprises Ž taking combinations of the first and ith terms inŽ.. 61

Ž1yn.r2 u1 s x y

Ž3yn.r2 u2 s x yЈ . .

Ž2 iyny1.r2 Žiy1. ui s x y Ž.62 Ž2iynq1.r2Ži. uiq1sx y . .

Žny3.r2Žny2. uny1sx y Žny1.r2Žny1. unsx y .

The first integralŽ. 59 now has the form

I s guŽ.i, is1,...,n,63Ž.

where the ui are given inŽ. 62 . The final requirement that

dI 064Ž. Žn. s dx y s0 gives the linear first order partial differential equation

ny1 2i n 1 Ѩ f y q Ž2 i n 1. 2 1 Ži.Ž2in1.2Ži1. Ý xyy q r y qxyyqrq s0, ž/2 Ѩu is0 iq1 Ž.65 where we have substituted for yŽn.fromŽ. 53 . MultiplyingŽ. 65 by x and usingŽ. 62 we obtain

n 2i y n y 1 Ѩ f Ý uiiq u 1 s 066Ž. ž/2 qѨu is1 i 370 FLESSAS, GOVINDER, AND LEACH with the associated Lagrange’s system

du1

Ž.1 y nu12r2qu

du2 dui ssиии s Ž.3 y nu23r2qu Ž.2iyny1uiir2quq1

duiq1 duny1 ssиии s Ž.2i y n q 1 uiq1r2 q uniq2 Ž.y3uny1r2qun

dun s .67Ž. Ž.ny1r2un

The second set of Ž.n y 1 characteristics is obtained by taking combinations of the ith and final terms inŽ. 67 . Starting with i s n y 1 we have to solve

duny1 n y 3 uny1 2 sq.68Ž. dunn n y 1 uny1

This linear equation is easily integrated to

Ž3yn.rŽny1. 2rŽny1. uuny1nnnsu q¨y1,6Ž.9 where ¨ny1 is the constant of integration which we take to be the first characteristic, i.e.,

Ž3yn.rŽny1. 2rŽny1. ¨ny1s uuny1nnyu .7Ž.0

Substituting for ui fromŽ. 62 we have

Ž.Ž.3nn12 Ž.n1 Žny3.r2Žny2.Žny1.r2Žny1.Žyry ny1.r2Žny1.ry ¨ny1sxyxyŽ.Ž.yxy

Žn1.ŽŽ.Ž.3ynrny1 n2.Žn1. sŽ.yyy Žyyxy y .

Ž3yn.rŽny1. syIIny1n ,7Ž.1 where we have usedŽ. 36 to determine the Ii, which is justŽ. 55 with i s n y 1. In general the equation to be solved for the ith characteristic is

duiii2i y n y 1 u 2 u q1 sq.72Ž. dunnn n y 1 uny1u LIE ALGEBRAS OF FIRST INTEGRALS 371

Ž.Ž. We can write uiq1 in terms of ¨ini s i q 1,...,ny1 and u . Thus 72 is always a linear equation in ui. However, it is not at all obvious as to how the general formula for uiq1 can be determined as a repeated substitution needs to be effected. With the use of a symbolic manipulation packageŽ for example, Mathematicawx 18.Ž it can be verified, from 67. , thatŽ 55. holds. For our purposes we simply demonstrate thatŽ. 54 is a symmetry ofŽ. 55 . To this end we operate onŽ. 55 with the Žn y 1 . st extension ofŽ. 54 , viz.

w ny1x G Ji

w ny1x Žnq1y2i.rŽny1. s G Ž.IIin n 1 2i wny1x Žnq1y2i.rŽny1.Žqy 2y2i.rŽny1.wny1x sŽ.GIIin q IIin Ž. G In ny1 n12i Ž2y2i.rŽny1. wny1xwqy ny1x sIGIIniŽ.niqIG Ž In . ny1

Žn1.Ž.Ž.2y2irny1 sŽ.yy

k nyiŽ.1 y Žnik.Žn 1 k. =Ý Ž.nyiykxyy yyy ½Ž.nyiyk! ks0 2kn1 yqŽnik.Žn 1 k. q xyyyyy ž/2

k n12iny1Ž.11n qy y Žnik.Žyn1k. qÝxyyy yy ž/ny1 Ž.nyiyk!2 ž/5 ks0

s0.Ž. 73

Q.E.D.

Each of the integrals, Ji, has n y 1 symmetries with the ‘‘algebra’’ Ž. ny2A1 [snGq2unless n is an odd integer in which case JŽnq1.r2has Ž. the ‘‘algebra’’ n y 1 A1 [snG q2. The algebraic properties follow directly from the symmetries of the constituent linear integrals. The solution symmetries are those apart from the twoŽ. one if n is odd associated with the twoŽ. one integrals in the expression for Ji. We note that the structure given for the integrals is not unique. For example, we could equally use 1r3 y1r3 Ži¨ . II41,II 31 and II21 for y s 0. 372 FLESSAS, GOVINDER, AND LEACH

6. CONCLUSION

In this paper we have treated at length the integrals and the algebras of the symmetries associated with them in the case of scalar ordinary differential equations of maximal symmetry. The pattern for the general equa- Žn. tion y s 0 has been established with the exception of a formula for the integral associated with the symmetry ѨrѨ x of homogeneous degree n y 1. To elaborate this formula does not appear to be a feasible proposition. We have seen that the cases n s 2 and n s 3 are anomalous and that the pattern is established at n s 4 when the only symmetries are of Cartan form. However, there is still a distinction between equations of even and odd degree of a number theoretic origin. In the context of linear equations this problem has, in a sense, been the easier one since equations of Žn. maximal symmetry are equivalent to y s 0 and so the solution symmetries are trivial to determine. This is not the case with linear equations of lower symmetry, particularly in the case of the linear equation of least symmetry. That one had to contend with the additional complications of the slŽ.2, R subalgebra did compensate for the ease of solution of the equations. In the Introduction we promised a simple example to illustrate the potential for application of the ideas and results contained in this paper. We do this with the nonlinear equation

3 yЉ q 3 yyЈ q y s 074Ž. which has been the object of many studiesŽ for example,wx 19᎐21. when the three is replaced by a parameter, k. A first integral for this equation is

1 xy 1 2 y I s 2x q 2.75Ž. yЈqy

Program LIEwx 16 shows that the first integralŽ. 75 has the three symmetries

ѨѨ 2232 G1sŽ.Žyx y x y y y 2 yxqyx . Ѩx Ѩy ѨѨ 2 G2sŽ.1yxy y y Ž.1 y xy Ѩx Ѩy Ѩ 3 Ѩ 1123 22 G3syŽ.22xq yx q 1 y 2 xy q yx Ž.76 Ѩxž/2Ѩy LIE ALGEBRAS OF FIRST INTEGRALS 373

ŽŽ.incidently the calculation of the symmetries of 75 by LIE proceeds considerably easier than those ofŽ. 74 ; when one is at the limit of a program’s capability, this can make a great difference in the feasibility or otherwise of a calculation. with the Lie Brackets

wxG12,GsG 2

wxG13,GsG 3

wxG23,Gs077Ž. which is the same algebra as for the first integrals of YЉ s 0 as was mentioned in the Introduction. Consequently there exists a transformation Ž.Ž.point in this case to transform 74 to YЉ s 0. It iswx 22 x 1 1 2 X sy2x q , Ysxy .7Ž.8 yy

The solution of the nonlinear equationŽ. 74 follows immediately from that of YЉ s 0.

ACKNOWLEDGMENTS

G. P. F. thanks the Research Committee of the University of the Aegean for its continuing support. K. S. G. and P. G. L. L. thank the Foundation for Research Development of South Africa and the University of Natal for their continuing support.

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