ñòð. 46 |

[?l1 , ?l2 ] = ? t Kij +l2 + ?l2 ? =

t Kij +l1 + ?l1 ,

(j + 1)! (j + 1)!

j=0 j=0

? ?? ?

m m

ij + ? j+1 ij + ? j+1

=? t Kij +l1 , ?l2 ? + ??l1 , t Kij +l2 ? + [?l1 , ?l2 ] =

(j + 1)! (j + 1)!

j=0 j=0

m

(l1 ? l2 )(ij + ?) j+1

t Kij +l1 +l2 + (l1 ? l2 )?l1 +l2 = (l1 ? l2 )?l1 +l2 .

=

(j + 1)!

j=0

The algebra (13) also gives us a new realisation of centreless Virasoro algebras. By

now we may very much see that there exist higher-degree ptd-symmetries for some

evolution equations in 1 + 1 dimensions. Moreover our derivation does not refer to

any particular choices of dimensions and space variables. Hence the evolution equa-

tion (11) may be not only both continuous and discrete, but also both 1 + 1 and 2 + 1

dimensional.

Actually there are many integrable equations which possess a centreless Virasoro

algebra (9) (see [3–5, 8, 9] for example). Among the most famous examples are the

KdV hierarchy in the continuous case and the Toda lattice hierarchy in the discrete

case. Through the theory above, we can say that a KdV-type equation

(14)

ut = tK0 + K1 = tux + uxxx + 6uux

possesses a hierarchy of second-degree time-polynomial-dependent symmetries

3 1

l ? 0,

tKl+1 + t2 Kl + ?l , (15)

?l =

2 4

where the vector ?elds Kl , ?l , l ? 0, are de?ned by

1

? = ? 2 + 4u + 2ux ? ?1 , l ? 0.

Kl = ?l ux , ?l = ?l (u + xux ),

2

Time-dependent symmetries of variable-coe?cient evolution equations 205

They constitute a centreless Virasoro algebra (9) with ? = 1 [8, 10] and thus so do

2

the symmetries Kl , ?l , l ? 0. We can also conclude that a Toda-type lattice equation

p(n) t2

(u(n))t = = K0 + tK1 + K0 =

2!

v(n) t

v(n) ? v(n ? 1)

1

1 + t2 (16)

= +

v(n)(p(n) ? p(n ? 1))

2

p(n)(v(n) ? v(n ? 1)) + v(n)(p(n + 1) ? p(n ? 1))

+t

v(n)(v(n ? 1) ? v(n + 1)) + v(n)(p(n)2 ? p(n ? 1)2 )

possesses a hierarchy of third-degree time-polynomial-dependent symmetries

1

?l = tKl + t2 Kl+1 + t3 Kl + ?l , l ? 0, (17)

6

where the corresponding vector ?elds are de?ned by

v ? v (1)

l ? 0,

l

Kl = ? K0 , K0 = ,

v(p ? p(?1) )

p

l ? 0,

?l = ?l ?0 , ?0 = ,

2v

in which the hereditary operator ? is de?ned by

(v (1) E 2 ? v)(E ? 1)?1 v ?1

p

?= .

v(E ?1 + 1) v(pE ? p(?1) )(E ? 1)?1 v ?1

Here we have used a normal shift operator E: (Eu)(n) = u(n + 1) and u(m) = E m u.

These discrete vector ?elds Kl , l ? 0, (see [11] for more information) together with

the discrete vector ?elds ?l , l ? 0, constitute a centreless Virasoro algebra (9) with

? = 1 [4] and the symmetry Lie algebra of ?l , l ? 0 and Kl , l ? 0, has the same

commutation relations as that Virasoro algebra.

More generally, we can consider further algebraic structures by starting from

a more general graded Lie algebra. In keeping with the notation in [12], let us write

a graded Lie algebra consisting of vector ?elds not depending explicitly on the time

variable t as follows:

?

[E(Ri ), E(Rj )] ? E(Ri+j?1 ), i, j ? 0, (18)

E(R) = E(Ri ),

i=0

where E(R?1 ) = 0. Note that such a graded Lie algebra is called a master Lie algebra

in [12] since it is actually not a graded Lie algebra as de?ned in [13]. However we still

call it a graded Lie algebra because it is very similar. Choose

Ti = Ki ? E(R0 ), 0 ? i ? m, (19)

and consider a variable-coe?cient evolution equation

m

ti t2 tm

Ti = K0 + tK1 + K2 + · · · + (20)

ut = Km .

i! 2! m!

i=0

206 W.X. Ma, R.K. Bullough, P.J. Caudrey, W.I. Fushchych

Before we state the main result, we derive two properties of the generating vector

?elds Sj , j ? 0.

Lemma 1. Assume that Ti , 0 ? i ? m, are de?ned by (19), and let l ? 0 and

?l ? E(Rl ). Then the vector ?elds Sj (?l ), j ? 0, de?ned by (7) satisfy the following

property

l??

S(??1)(m+1)+? (?l ) ? 1 ? ? ? l, 1 ? ? ? m + 1, (21)

E(Ri ),

i=0

j ? l(m + 1) + 1. (22)

Sj (?l ) = 0,

Proof. Note the de?nition (7) of Sj (?l ), j ? 0, and Ti = Ki , 0 ? i ? m. We can

calculate that

m

?(m + 1) + ?

S?(m+1)+?+1 (?l ) = K? , S?(m+1)+??? (?l ) =

?

?=0

??1

?(m + 1) + ?

= K? , S?(m+1)+??? (?l ) +

?

?=0

m

?(m + 1) + ?

K? , S(??1)(m+1)+[(m+1)?(???)] (?l ) ?

+

?

?=?

l?(?+2) l?(?+1) l?(?+1)

? E(Ri ) + E(Ri ) = E(Ri ),

i=0 i=0 i=0

where in the last but one step we have used the induction assumption. This result

shows that (21) is true by mathematical induction. The proof of (22) is the same so

that the proof of the Lemma is complete.

Lemma 2. Assume that Ti , 0 ? i ? m, are de?ned by (19), and let l1 , l2 ? 0 and

?l1 ? E(Rl1 ), ?l2 ? E(Rl2 ). Then we have

k

k ? 0, (23)

Sk ([?l1 , ?l2 ]) = [Si (?l1 ), Sj (?l2 )],

i

i+j=k

where the Sj (?), j ? 0, are de?ned by (7).

Proof. We use mathematical induction to prove the required result. Noting that

Ti = Ki , 0 ? i ? m, we can calculate that

k

Sk+1 ([?l1 , ?l2 ]) = Ki , Sj ([?l1 , ?l2 ]) =

i

i+j=k

? ?

k? j (by the induction

[S? (?l1 ), S? (?l2 )]?

= Ki , =

assumption)

i ?

i+j=k ?+?=j

k j

= [Ki , [S? (?l1 ), S? (?l2 )]] =

i ?

i+j=k ?+?=j

k!

= [Ki , [S? (?l1 ), S? (?l2 )]] =

i!?!?!

i+?+?=k

Time-dependent symmetries of variable-coe?cient evolution equations 207

k!

{[[Ki , S? (?l1 )] , S? (?l2 )] + [S? (?l1 ), [Ki , S? (?l2 )]]} =

=

i!?!?!

i+?+?=k

? ?

k? j

[Ki , S? (?l1 )] , S? (?l2 )? +

=

j i

i+?=j

j+?=k

? ?

k? j

[Ki , S? (?l2 )]? =

+ S? (?l1 ),

j i

?+j=k i+?=j

k k

= [Sj+1 (?l1 ), S? (?l2 )] + [S? (?l1 ), Sj+1 (?l2 )] =

j j

j+?=k ?+j=k

k+1

k ? 0,

= [Si (?l1 ), Sj (?l2 )] ,

i

i+j=k+1

and this yields the key step in the mathematical induction. On the other hand, we

easily see that

S0 ([?l1 , ?l2 ]) = [?l1 , ?l2 ] = [S0 (?l1 ), S0 (?l2 )].

Therefore mathematical induction gives the proof of the equality (23).

Theorem 2. Assume that Ti , 0 ? i ? m, are de?ned by (19), and let l ? 0 and

?l ? E(Rl ). Then the vector ?eld

l(m+1) j

t

(24)

?(?l ) = Sj (?l ),

j!

j=0

where the Sj (?l ), 0 ? j ? l(m+1), are de?ned by (7), is a time-independent symmetry

of (20) when l = 0 and a polynomial-in-time dependent symmetry of (20) when l > 0.

Furthermore we have

[?(?l1 ), ?(?l2 )] = ?([?l1 , ?l1 ]), ?l1 ? E(Rl1 ), ?l2 ? E(Rl2 ), l1 , l2 ? 0, (25)

and thus all symmetries ?(?l ) with ?l ? E(Rl ), l ? 0, constitute the same graded

Lie algebra as (18) and the map ? : ?l > ?(?l ) is a Lie homomorphism between two

graded Lie algebras E(R) and ?(E(R)).

Proof. By Lemma 1, we can observe that ?(?l ) de?ned by (24) is a symmetry of

(20). We go on to prove (25). Assume that ?l1 ? E(Rl1 ), ?l2 ? E(Rl2 ), l1 , l2 ? 0. By

Lemmas 1 and 2, we can make the following calculation

? ?

l1 (m+1) i l2 (m+1) j

t t

[?(?l1 ), ?(?l2 )] = ? Sj (?l2 )? =

Si (?l1 ),

i! j!

i=0 j=0

(l1 +l2 ?1)(m+1) k

t k

= [Si (?l1 ), Sj (?l2 )] (by Lemma 1) =

k! i

k=0 i+j=k

(l1 +l2 ?1)(m+1) k

t

= Sk ([?l1 , ?l2 ]) (by Lemma 2) =

k!

k=0

= ?([?l1 , ?l2 ]).

The rest is then obvious and the required result is obtained.

208 W.X. Ma, R.K. Bullough, P.J. Caudrey, W.I. Fushchych

A graded Lie algebra has been exhibited for the time-independent KP hierar-

chy [14] in [2, 12], and it includes a centreless Virasoro algebra [5, 15]. The ordinary

time-independent KP equation being considered here is the following

?1

ut = ?x uyy ? uxxx ? 6uux . (26)

From this we may now go on to generate the corresponding graded Lie algebra of ptd-

symmetries for a resulting new set of variable-coe?cient KP equations, but in this

connection the reader must be referred to the comparable analysis in [16] mentioned

below.

The idea of using graded Lie algebras as described in this paper is rather similar

to the thinking used to extend the inverse scattering transform from 1 + 1 to higher

dimensions [17]. Moreover the resulting symmetry algebra consisting of the ?(?l ),

l ? 0, provides a new realisation of a graded Lie algebra (18). The theory also shows us

that more information can be extracted from graded Lie algebras, which is itself very

interesting. What is more, we have shown here that there do exist various integrable

equations in 1 + 1 dimensions, such as KdV-type equations, possessing higher-degree

polynomial-in-time dependent symmetries. We report a graded Lie algebra of ptd

symmetries for a corresponding new set of variable coe?cient modi?ed KP equations

in a second article [16]. In [16] we display this modi?ed KP hierarchy explicitly, the

time independent modi?ed KP equation being, in comparison with (26), the equation

1 3 3 3 ?1

ñòð. 46 |