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+ 7 cos
2
A2 = C1 |cx|?1 e2 ,
A1 = 0,
1 1
A0 = A3 = e?? C1 cos
(3) eC3 ? + C2 e2 + sin eC3 ? + C2 e3 ,
2 2
A1 = [(bx)2 + (cx)2 ]?1 (bx)C3 e1 ? 2?C1 (cx)e?? ?
1 1
? cos eC3 ? + C2 e2 + sin eC3 ? + C2 e3 ,
2 2
A2 = [(bx)2 + (cx)2 ]?1 (cx)C3 e1 + 2?C1 (bx)e?? ?
1 1
? cos eC3 ? + C2 e2 + sin eC3 ? + C2 e3 ,
2 2
? = kx ? log[(bx)2 + (cx)2 ] + 2? arctan cx(bx)?1 ,
200 V.I. Lahno, W.I. Fushchych

(4) A0 = A3 = C4 e3 ,
A1 = [(bx)2 + (cx)2 ]?1 C1 (bx)(sin(eC4 ? + C2 )e1 + cos(eC4 ? + C2 )e2 +
+ (C3 ? 2C4 ?)e3 ) ? 2C4 (cx)e3 ,
A2 = [(bx)2 + (cx)2 ]?1 C1 (cx)(sin(eC4 ? + C2 )e1 + cos(eC4 ? + C2 )e2 +
? = kx + 2 arctan(cx(bx)?1 ).
+ (C3 ? 2C4 ?)e3 ) + 2C4 (bx)e3 ,

The values of 1 and 7 are given in Table 1, ? is given in the list of subalgebras, and
?0 , ?3 , C1 , C2 , C3 , and C4 are arbitrary real constants.

Conclusions
?
In this paper, we have investigated the structure of P (1, 3)-invariant ansatze for the
vector potential of the Yang–Mills ?eld. The linear form we obtained for the ansatze
is reduced to a covariant form, which allows us to simplify considerably the procedure
for the symmetry reduction of SDYME (1) to systems of ODE. We have demonstrated
the possibility of constructing real solutions of SDYME (1).
Let us note that ansatz (11) can also be used for symmetry reduction in the
Minkowski space R(1, 3).

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9. Schwarz F., Lett. Math. Phys., 1982, 6, 355.
10. Olver P., Applications of Lie groups to di?erential equations, New York, Springer, 1993.
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12. Fushchych W.I., Shtelen W.M., Lett. Nuovo Cimento, 1983, 38, 37.
13. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equation of
nonlinear mathematical physics, Dordrecht, Kluwer, 1993.
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15. Zhdanov R.Z., Lahno V.I., Fushchych W.I., Ukr. Math. J., 1995, 47, 456.
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Amsterdam, North-Holland, 1989.
W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 201–209.

Time-dependent symmetries
of variable-coe?cient evolution equations
W.X. MA, R.K. BULLOUGH, P.J. CAUDREY, W.I. FUSHCHYCH
Polynomial-in-time dependent symmetries are analysed for polynomial-in-time de-
pendent evolution equations. Graded Lie algebras, especially Virasoro algebras, are
used to construct nonlinear variable-coe?cient evolution equations, both in 1 + 1
dimensions and in 2 + 1 dimensions, which possess higher-degree polynomial-in-time
dependent symmetries. The theory also provides a kind of new realisation of graded
Lie algebras. Some illustrative examples are given.

It is well known that the usual family of KdV equations has polynomial-in-time
dependent symmetries (ptd-symmetries) which are only of the ?rst-degree. This is
because only master symmetries of ?rst degree are so far found. Moreover there are
usually1 no higher-degree ptd-symmetries for time-independent integrable equations
in 1 + 1 dimensions; but this may not be so in 2 + 1 dimensions.
However a form of special graded Lie algebras, namely centreless Virasoro sym-
metry algebras is apparently common to all time-independent integrable equations
in whatever dimensions both in the continuous case and in the discrete case. This
feature would therefore seem to be an important one in the discussion of integrability
and integrable nonlinear equations. For the higher dimensional integrable equations,
there may also exist still more general graded symmetry Lie algebras.
The purpose of the present paper is to discuss ptd-symmetries for evolution equa-
tions with polynomial-in-time dependent coe?cients (conveniently expressed in terms
of monomials in t as in equation (4) below). We provide a purely algebraic structure for
constructing such integrable equations with these forms of symmetries. This way we
show there do exist integrable equations in 1+1 dimensions which possess these forms
of symmetries and we construct actual examples. Graded Lie algebras, and especially
centreless Virasoro algebras, are used for these constructions. In consequence new
features are extracted from the graded Lie algebras which provide new realisations of
these algebras and most particularly of the centreless Virasoro algebras.
We ?rst de?ne a symmetry for an evolution equation, linear and nonlinear [1–5].
For a given evolution equation ut = K(u), a vector ?eld ?(u) is called its symmetry
if ?(u) satis?es its linearized equation
d?(u) ??
(1)
= K [?], i.e. = [K, ?],
dt ?t
where the prime and [·, ·] denote the Gateaux derivative and the Lie product
?
[K, ?] = K [?] ? ? [K], (2)
K [S] = K(u + ?S)|?=0 ,
??
J. Phys. A: Math. Gen., 1997, 30, ¹ 14, P. 5141–5149.
1 TheBenjamin–Ono equation is a counter-example.
202 W.X. Ma, R.K. Bullough, P.J. Caudrey, W.I. Fushchych

respectively. Of course, a symmetry ? may also depend explicitly on the time variab-
le t. For example, ? may be of polynomial type in t, i.e.
n
tj tn
Sj (u) = S0 + tS1 + · · · + Sn , (3)
?(t, u) =
j! n!
j=0

where the vector ?elds Sj (u), 0 ? j ? n, do not depend explicitly on the time
variable t.
If we consider a variable-coe?cient evolution equation ut = K(t, u) of the form
m
ti tm
Ti (u) = T0 + tT1 + · · · + (4)
ut = K(t, u) = Tm ,
i! m!
i=0

where the vector ?elds Ti (u), 0 ? i ? m, do not depend explicitly on the time
variable t, either, then a precise result may be obtained which states (3) is a symmetry
of (4). At this stage, we can have
n n?1 k
ti?1
?? t
= Si (u) = Sk+1 (u),
(i ? 1)!
?t k!
? ?
i=0 k=0
m n m+n k
ti tj t k
[K, ?] = ? Sj (u)? =
Ti (u), [Ti , Sj ].
i! j! k! i
i=0 j=0 k=0 i+j =k
0?i?m
0?j?n

Therefore a simple comparison of each power of t in (1) leads to
k
0 ? k ? n ? 1, (5)
Sk+1 = [Ti , Sj ],
i
i+j =k
0?i?m
0?j?n

k
n ? k ? m + n. (6)
[Ti , Sj ] = 0,
i
i+j =k
0?i?m
0?j?n

These equalities in (5) and (6) constitute a necessary and su?cient condition to state
that (3) is a symmetry of (4). If we look at them a little more, it may be seen that
S1 = [T0 , S0 ],
S2 = [T0 , S1 ] + [T1 , S0 ],
························
[T1 , Sn?2 ] + · · · +
n?1 n?1 n?1
Sn = [T0 , Sn?1 ] + [Tn?1 , S0 ],
0 1 n?1

where Ti = 0, i ? m+1, and so a higher-degree ptd-symmetry ?(t, u) de?ned by (3) is
determined completely by a vector ?eld S0 . However this vector ?eld S0 needs to satis-
fy (6). This kind of vector ?eld S0 is a generalisation of the master symmetries de?ned
in [2] which here we still call a master symmetry of degree n for the more general
evolution equation, equation (4). We conclude the discussion above as a theorem.
Time-dependent symmetries of variable-coe?cient evolution equations 203

Theorem 1. Let ? be a vector ?eld not depending explicitly on the time variable t.
De?ne
k
k
k ? 0, (7)
S0 (?) = ?, Sk+1 (?) = [Tj , Sk?j (?)],
j
j=0

where we assume Ti = 0, i ? m + 1. If there exists n ? N so that Sj (?) = 0, j ? n + 1,
then
n
tj
(8)
?(?) = Sj (?)
j!
j=0

is a polynomial-in-time dependent symmetry of the evolution equation (4).
We shall go on to construct variable-coe?cient integrable equations which possess
higher-degree ptd-symmetries as de?ned by (3). We need to start from the centreless
Virasoro algebra
[Kl1 , Kl2 ] = 0, l1 , l2 ? 0,
[Kl1 , ?l2 ] = (l1 + ?)Kl1 +l2 , l1 , l2 ? 0, (9)
[?l1 , ?l2 ] = (l1 ? l2 )?l1 +l2 , l1 , l2 ? 0
in which the vector ?elds Kl1 = Kl1 (u), ?l2 = ?l2 (u), l1 , l2 ? 0, do not depend
explicitly on the time variable t and ? is a ?xed constant. Although the vector ?elds ?l ,
l ? 0, are not symmetries of any equations that we want to discuss, an algebra
isomorphic to this kind of Lie algebra commonly arises as a symmetry algebra for
many well-known continuous and discrete integrable equations [3–5]. In equation (9),
the vector ?elds ?l , l ? 0, may provide the generators of Galilean invariance [6] and
invariance under scale transformations for any standard equation ut = Kk (u). Let us
choose a set of speci?c vector ?elds
0 ? j ? m, (10)
Tj = Kij ,
which yields the following variable-coe?cient evolution equation
t2 tm
ut = Ki0 + tKi1 + Ki2 + · · · + (11)
Ki .
m! m
2!
This equation still has a hierarchy of time-independent symmetries Kl , l ? 0, and
therefore it is integrable in the sense of symmetries [7]. What is more, it will inherit
many integrable properties of ut = Kl , l ? 0. For example, if ut = Kl , l ? 0, have
Hamiltonian structures of the form
?Hl
l ? 0,
u t = Kl = J ,
?u
where J is a symplectic operator and Hl , l ? 0, do not depend explicitly on t,
then the Hl are still conserved densities of equation (11) and equation (11) is then
completely integrable in the commonly used sense for pdes. In what follows, we need
to prove that ?l is a master symmetry (as explained above) of degree m + 1 of equa-
tion (11). In fact, according to (7), we have
0 ? k ? m,
S0 (?l ) = ?l , Sk+1 (?l ) = [Tk , S0 (?l )] = [Kik , ?l ] = (ik + ?)Kik +l ,
204 W.X. Ma, R.K. Bullough, P.J. Caudrey, W.I. Fushchych

and further we can prove that Sj (?l ) = 0 when j ? m + 2, which shows that ?l is a
master symmetry of degree m + 1 of equation (11). Therefore we obtain a hierarchy
of ptd-symmetries of the form
m+1 j m+1
t ij?1 + ? j
?l (t, u) = Sj (?l ) = t Kij?1 +l + ?l =
j! j!
j=0 j=1
(12)
m
ij + ? j+1
l ? 0,
= t Kij +l + ?l ,
(j + 1)!
j=0

for the variable-coe?cient and integrable equation (11). Moreover these higher-degree
ptd-symmetries together with time-independent symmetries Kl , l ? 0, constitute the
same centreless Virasoro algebra as (9), namely
[Kl1 , Kl2 ] = 0, l1 , l2 ? 0,
[Kl1 , ?l2 ] = (l1 + ?)Kl1 +l2 , l1 , l2 ? 0, (13)
[?l1 , ?l2 ] = (l1 ? l2 )?l1 +l2 , l1 , l2 ? 0.
For example, we can calculate that
? ?
m m
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