ñòð. 45 |

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).

1. Actor A., Rev. Mod. Phys., 1979, 51, 461.

2. Prasad M.K., Physica D, 1980, 1, 167.

3. Chakravarty S., Ablowitz M.J., Clarkson P.A., Phys. Rev. Lett., 1990, 65, 1085.

4. Chakravarty S., Kent S.L., Newman E.T., J. Math. Phys., 1995, 36, 763.

5. Tafel J., J. Math. Phys., 1993, 34, 1892.

6. Kovalyov M., Legar? M., Gagnon L., J. Math. Phys., 1993, 34, 3245.

e

7. Ivanova T.A., Popov A.D., Phys. Lett. A, 1995, 205, 158.

8. Legar? M., Popov A.D., Phys. Lett. A, 1995, 198, 195.

e

9. Schwarz F., Lett. Math. Phys., 1982, 6, 355.

10. Olver P., Applications of Lie groups to di?erential equations, New York, Springer, 1993.

11. Ovsyiannikov L.V., Group analysis of di?erential equations, New York, Academic Press, 1982.

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.

14. Lahno V., Zhdanov R., Fushchych W., J. Nonlinear Math. Phys., 1995, 2, 51.

15. Zhdanov R.Z., Lahno V.I., Fushchych W.I., Ukr. Math. J., 1995, 47, 456.

16. Zhdanov R.Z., Fushchych W.I., J. Phys. A, 1995, 28, 6253.

17. Patera J., Winternitz P., Zassenhaus H., J. Math. Phys., 1975, 16, 1615.

18. Fushchych W.I., Barannik L.F., Barannik A.F., Subgroup analysis of Galilei and Poincar?

e

groups and reduction of nonlinear equations, Kiev, Naukova Dumka, 1991 (in Russian).

19. Fushchych W.I., Zhdanov R.Z., Symmetries and exact solutions of nonlinear spinor equations,

Amsterdam, North-Holland, 1989.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 201–209.

Time-dependent symmetries

of variable-coe?cient evolution equations

and graded Lie algebras

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

ñòð. 45 |