ñòð. 47 |

uxxx ? u2 ux ? ux ?x uy + ?x uyy . (27)

ut =

4 8 4 4

In [16] we show also that this hierarchy actually has two Virasoro algebras and two

graded Lie algebras.

We also hope to show elsewhere the connections between the rather general al-

gebraic structure established in this paper and the speci?c representation of the W?

and W1+? algebras developed in connection with two-dimensional quantum gravity

as described in in Refs. [18, 19]. (In [18, 19], these two in?nite dimensional algebras

were developed for the ordinary KP hierarchy and included the algebra, containing

the centreless Virasoro algebra, of Ref. [5].) In this connection, we note already that

if, for example, E(Ri ) = span{Aim | m ? 1}, i ? 0, and we impose

i+j?2

al (i ? 1, j ? 1, m ? 1, n ? 1)Al+1,m+n?1 ,

[Aim , Ajn ] =

l=min(i?1,j?1)

where the coe?cients al are de?ned by

i+j

di+1 j+n+1 dj+1 dl+1

i+m+1 l+m+n+1

x ,x = al (i, j, m, n)x ,

dxi+1 dxj+1 dxl+1

l=min(i,j)

?

then the E(R) = E(Ri ) is a sub-algebra of the W1+? algebra of Refs. [18, 19]

i=0

by the identi?cation Aim = ?i?1,m?1 ; here the ?i?1,m?1 are the elements forming the

di

W1+? algebra [18, 19] and they may be realized by xi+m?1 dxi .

Acknowledgements. One of the authors (WXM) would like very much to thank

the Alexander von Humboldt Foundation for the ?nancial support, which made his

Time-dependent symmetries of variable-coe?cient evolution equations 209

visit to UMIST possible. He is also grateful to Prof. B. Fuchssteiner for his kind and

stimulating discussions.

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2. Fuchssteiner B., Mastersymmetries, higher order time-dependent symmetries and conserved

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3. Chen H.H., Lee Y.C., A new hierarchy of symmetries for the integrable nonlinear evolution

equations, in Advances in Nonlinear Waves, Vol. II, Editor L. Debnath, Research Notes in

Mathematics, V.111, Boston, Pitman, 1985, 233–239.

4. Ma W.X., Fuchssteiner B., Algebraic structure of discrete zero curvature equations and master

symmetries of discrete evolution equations, Preprint, 1996.

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Kadomtsev–Petviashvili equations in 2 + 1 dimensions, J. Phys. A: Math. Gen., 1988, 21,

L443–L449.

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of nonlinear mathematical physics, Dordrecht, Kluwer, 1993.

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8. Ma W.X., K-symmetries and ? -symmetries of evolution equations and their Lie algebras,

J. Phys. A: Math. Gen., 1990, 23, 2707–2716.

9. Oevel W., Fuchssteiner B., Zhang H.W., Mastersymmetries and mutil-Hamiltonian formulations

for some integrable lattice systems, Prog. Theor. Phys., 1989, 81, 294–308.

10. Ma W.X., Lax representations and Lax operator algebras of isospectral and nonisospectral

hierarchies of evolution equations, J. Math. Phys., 1992, 33, 2464–2476.

11. Tu G.Z., A trace identity and its applications to the theory of discrete integrable systems,

J. Phys. A: Math. Gen., 1990, 23, 3903–3922.

12. Ma W.X., The algebraic structures of isospectral Lax operators and applications to integrable

equations, J. Phys. A: Math. Gen., 1992, 25, 5329–5343.

13. Kac V.G., In?nite dimensional Lie algebras, 3rd ed., Cambridge, Cambridge University Press,

1990.

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Kadomtsev–Petviashvili equations, Phys. Lett. A, 1982, 88, 323–327.

15. Chen H.H., Lee Y.C., Lin J.E., On a new hierarchy of symmetries for the Kadomtsev-Petviashvili

equation, Physica D, 1983, 9, 439–445.

16. Ma W.X., Bullough R.K., Caudrey P.J., Graded symmetry algebras of time-dependent evolution

equations and application to the modi?ed KP equations, Dedicated to W.I. Fushchych2 on his

60th birthday, J. Nonlinear Math. Phys., 1997, 4, 293–309.

17. Caudrey P.J., Some thoughts on integrating non-integrable systems, in Nonlinear Physics:

Theory and Experiment, Editors E. Al?nito, M. Boiti, L. Martina, F. Pempinelli, Singapore,

World Scienti?c, 1996, 60–66.

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in 1834–1995, Acta Appl. Math., 1995, 39, 193–228 (and references therein).

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quantum gravity: a ‘map’, J. Nonlinear Math. Phys., 1996, 3, 245–259 (and references therein).

2 With sadness we report the death of Professor Wilhelm I. Fushchych on Monday 7 April 1997

after a short illness. The remaining three authors of this paper, Wen-Xiu Ma, Robin Bullough and

Philip Caudrey, dedicate this paper to his memory. Appreciations of W.I. Fushchych will appear in

J. Nonlinear Math. Phys. of which he was Editor in Chief.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 210–219.

On new representations of Galilei groups

R.Z. ZHDANOV, W.I. FUSHCHYCH

We have constructed new realizations of the Galilei group and its natural extensions

by Lie vector ?elds. These realizations together with the ones obtained by Fushchych

& Cherniha (Ukr. Math. J., 1989, 41, ¹ 10, 1161; ¹ 12, 1456) and Rideau & Win-

ternitz (J. Math. Phys., 1993, 34, 558) give a complete description of inequivalent

representations of the Galilei, extended Galilei, and generalized Galilei groups in the

class of Lie vector ?elds with two independent and two dependent variables.

1. Introduction

As is well known, the problem of classi?cation of linear and nonlinear partial di?eren-

tial equations (PDEs) admitting a given Lie transformation group G is closely con-

nected to the one of describing inequivalent representations of its Lie algebra AG in

the class of Lie vector ?elds (LVFs) [1–3]. Given a representation of the Lie algebra

AG, one can, in principle, construct all PDEs admitting the group G by means of the

in?nitesimal Lie method [1, 2, 4].

In the present paper we study representations of the Lie algebra of the Galilei

group G(1, 1) (which will be called in the sequel the Galilei algebra AG(1, 1)) and its

natural extensions in the class of LVFs

Q = ? 1 (t, x, u, v)?t + ? 2 (t, x, u, v)?x + ? 1 (t, x, u, v)?u + ? 2 (t, x, u, v)?v , (1)

where t, x and u, v are considered as independent and dependent variables, corres-

pondingly, and ? 1 , . . . , ? 2 are some su?ciently smooth real-valued functions.

Representations of the Galilei group with basis generators (1) are realized on the

set of solutions of the linear and nonlinear (1 + 1)-dimensional heat, Schr?dinger,o

Hamilton–Jacobi, Burgers and KdV equations to mention only a few PDEs (for more

details, see [4]).

We say that operators P0 , P1 , M , G, D, A of the form (1) realize a representati-

on of the generalized Galilei algebra AG2 (1, 1) (called also the Schr?dinger algebra

o

ASch(1, 1)) if

• they are linearly independent,

• they satisfy the following commutation relations:

[P0 , P1 ] = 0, [P0 , M ] = 0, [P1 , M ] = 0,

1

[P0 , G] = P1 , [P1 , G] = 2 M, [P0 , D] = 2P0 ,

[P1 , D] = P1 , [P0 , A] = D, [P1 , A] = G,

(2)

[M, G] = 0, [M, D] = 0, [M, A] = 0,

[M, G] = 0, [M, D] = 0, [M, A] = 0,

[G, D] = ?G, [G, A] = 0, [D, A] = 2A.

J. Nonlinear Math. Phys., 1997, 4, ¹ 3–4, P. 426–435.

On new representations of Galilei groups 211

In the above formulae, [Q1 , Q2 ] ? Q1 Q2 ? Q2 Q1 is the commutator.

The subalgebra of the above algebra spanned by the operators P0 , P1 , M, G, is

the Galilei algebra. The Lie algebra having the basis elements P0 , P1 , M , G, D is

called the extended Galilei algebra AG1 (1, 1).

It is straightforward to verify that relations (2) are not altered by an arbitrary

invertible transformation of the independent and dependent variables

t > t = f1 (t, x, u, v), x > x = f2 (t, x, u, v),

(3)

u > u = g1 (t, x, u, v), v > v = g2 (t, x, u, v),

where f1 , . . . , g2 are su?ciently smooth functions. Invertible transformations of the

form (3) form a group (called di?eomorphism group) which establishes a natural

equivalence relation on the set of all possible representations of the algebra AG(1, 1).

Two representations of the Galilei algebra are called equivalent if the corresponding

basis operators can be transformed one into another by a change of variables (3).

In the papers by Fushchych and Cherniha [5, 6] di?erent linear representations

of the Galilei group and of its generalizations were used to classify Galilei-invariant

nonlinear PDEs in n dimensions with an arbitrary N ? N (see also [7]). The next

paper in this direction was the one by Rideau and Winternitz [8]. It gives a description

of inequivalent representations of the algebras AG(1, 1), AG1 (1, 1), AG2 (1, 1) under

supposition that commuting operators P0 , P1 , M can be reduced to the form

(4)

P 0 = ?t , P1 = ? x , M = ?u

by transformation (3).

The results of [8] can be summarized as follows. The basis elements P0 , P1 , M are

given formulae (4) and the remaining basis elements are adduced below

1. Inequivalent representations of the Galilei algebra

G = t?x + 1 x?u + f (v)?t ,

(a) 2

(5)

G = t?x + 1 x?u + v?v .

(b) 2

2. Inequivalent representations of the extended Galilei algebra

1

(a) G = t?x + 2 x?u , D = 2t?t + x?x + f (v)?u ,

D = 2t?t + x?x ? 1 v?v ,

1

(b) G = t?x + 2 x?u , 2

(6)

1

(c) G = t?x + 2 x?u + v?t , D = 2t?t + x?x + 3v?v ,

?v , D = 2t?t + x?x + ??u ? v?v .

1

(d) G = t?x + 2 x?u +

3. Inequivalent representations of the generalized Galilei algebra

G = t?x + 1 x?u , D = 2t?t + x?x + f (v)?u ,

(a) 2

A = t ?x + tx?x + 1 x2 + f (v)t ?u ,

2

4

1

(b) G = t?x + 2 x?u , D = 2t?t + x?x + 2v?v ,

(7)

A = t2 ?x + tx?x + 1 x2 + ?v ?u + (2t + ?v)v?v ,

4

G = t?x + 2 x?u + ?v , D = 2t?t + x?x + ??u ? v?v ,

1

(c)

A = t2 ?t + tx?x + 1 x2 + ?t ?u + (x ? tv)?v .

4

212 R.Z. Zhdanov, W.I. Fushchych

Here

?,

f (v) =

v,

? is an arbitrary constant and ? = 0, 1.

Remark 1. Representation (7b) with (? = 0, ? = 0) were obtained for the ?rst time

in [5, 6].

Remark 2. The forms of basis operators of the extended Galilei and generalized

Galilei algebras are slightly simpli?ed as compared to those given in [8]. For example,

the operators D, A from (7c) read as

1

?

D = 2t?t + x?x + ??u ? (1 + 2 ln v )??v ,

?v ?

2

12 1

? x + ?t ?u + x ? t(1 + 2 ln v ) v ?v .

A = t2 ?t + tx?x + ? ??

4 2

??

It is readily seen that the operators {D, A} and {D, A} are related to each other

1

by the transformation v = ln(?e 2 ).

v

Generally speaking, basis elements P0 , P1 , M have not to be reducible to the

form (4). The requirement of reducibility imposes an additional constraint on the

choice of basis elements of the algebras AG(1, 1), AG1 (1, 1), AG2 (1, 1), thus narrowing

the set of all possible inequivalent representations. This is the reason why formulae

(5)–(7) give no complete description of representations of the Galilei, extended Galilei,

and generalized Galilei algebras. As established in the present paper, there are ?ve

more classes of representations of AG(1, 1), six more classes of representations of

AG1 (1, 1) and one new representation of the generalized Galilei algebra AG2 (1, 1).

2. Principal results

Before formulating the principal assertion we prove an auxiliary lemma.

Lemma 1. Let P0 , P1 , M be mutually commuting linearly independent operators of

the form (1). Then there exists transformation (3) reducing these operators to one of

the forms

(8)

P 0 = ?t , P1 = ? x , M = ?u ;

(9)

P 0 = ?t , P1 = ? x , M = ?(u, v)?t + ?(u, v)?x ;

(10)

P 0 = ?t , P1 = x?t , M = 2?u ;

(11)

P 0 = ?t , P1 = x?t , M = ?(x)?t ;

(12)

P 0 = ?t , P1 = x?t , M = 2u?t ,

where ?, ?, ? are arbitrary smooth functions of the corresponding arguments.

Proof. Let R be a 2 ? 4 matrix whose entries are coe?cients of the operators P0 , P1 .

Case 1. rank R = 2. It is a common knowledge that any nonzero operator Q of

the form (1) having smooth coe?cients can be transformed by the change of variables

On new representations of Galilei groups 213

(3) to become Q = ?t (see, e.g. [1]). Consequently, without loosing generality, we

can suppose that the relation P0 = ?t holds (hereafter we skip the primes). As the

operator P1 commutes with P0 , its coe?cients do not depend on t, i.e.,

ñòð. 47 |