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?1
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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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.
6. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of equations
of nonlinear mathematical physics, Dordrecht, Kluwer, 1993.
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for some integrable lattice systems, Prog. Theor. Phys., 1989, 81, 294–308.
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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.
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equation, Physica D, 1983, 9, 439–445.
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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, 10, 1161; 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, 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.,

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