p1 • dx. (6.124)

2

’∞

So prolongation of Z1 towards the nonlocal variable q 1 , or equivalently,

2

computation of the ‚/‚q 1 -component of the vector ¬eld Z1 , requires formal

2

introduction of a new odd nonlocal variable q 3 de¬ned by

2

x

q3 = p1 • dx, (6.125)

2

’∞

where

(q 3 )x = p1 •,

2

(q 3 )t = p1 (’•2 + 3u•) ’ u1 • + u•1 , (6.126)

2

while the compatibility condition on (6.126) is satis¬ed; so q 3 is a new odd

2

potential, Q 3 being the new odd conserved quantity.

2

The vector ¬eld Z1 is now given in the augmented setting (6.120) by

‚ ‚ ‚ ‚

+ (q 1 u ’ •1 ) + (q 1 p1 ’ • ’ q 3 )

= q 1 •1 + q1 • .

Z1

‚u ‚• 2 ‚q 1 ‚p1

2 2 2 2

2

(6.127)

The system of graded partial di¬erential equations under consideration is

now the once more augmented system (6.120):

ut = ’u3 + 6u1 u ’ 3••2 ,

•t = ’•3 + 3•1 u + 3•u1 ,

(q 1 )x = •,

2

(q 1 )t = ’•2 + 3•u,

2

6. SUPERSYMMETRIC KDV EQUATION 287

(p1 )x = u,

(p1 )t = ’u2 + 3u2 ’ 3••1 ,

(q 3 )x = p1 •,

2

(q 3 )t = p1 (’•2 + 3u•) ’ u1 • + u•1 . (6.128)

2

The prolongation of the vector ¬eld towards the nonlocal variables q 1 ,

Y3

2

2

p1 is now constructed from the respective equations for (q 1 )x and (p1 )x ,

2

(6.128) resulting in

q1 1

Y 3 2 = 2q 1 • ’ p2 + u,

21

2

2

p

Y 3 1 = 2q 1 u ’ p1 • ’ •1 . (6.129)

2

2

Computation of the graded Lie bracket [ Z1 , Y3 ] leads to

2

1 ‚

= (’2q 3 u1 + p2 •1 + p1 (•2 ’ u•) ’ 4u1 • ’ 3u•1 + •3 )

Y5 1

2 ‚u

2

2

1 ‚

+ (’2q 3 •1 + p2 u ’ p1 u1 + u2 ’ 2u2 + ••1 )

21 ‚•

2

1 ‚

+ (’2q 3 u + p2 • + p1 •1 ’ 4u• + •2 )

21 ‚p1

2

1 1 ‚

+ (’2q 3 p1 • + p4 ’ p2 u + p1 u1 ’ u2 ’ ••1 ) + · · · , (6.130)

81 1

2 ‚q 3

2

2

whereas the ‚/‚p1 - and ‚/‚q 3 -components of are obtained by the in-

Y5

2 2

variance of the associated di¬erential equations for these variables in (6.128).

In order to obtain the ‚/‚q 1 -component of Y 5 , we have to require the

2 2

invariance of the equation (q 1 )x ’ • = 0, which results in the following

2

condition

q1 1

Dx (Y 5 2 ) = ’2q 3 •1 + p2 u ’ p1 u1 + u2 ’ 2u2 + ••1 , (6.131)

21

2

2

from which we have

x

q1 13

(u2 + 2(p1 •)• + ••1 ’ 2u2 ) dx

Y 5 = p1 ’ p1 u + u1 ’ 2q 3 • +

2

6 2

’∞

2

x

13

(u2 ’ ••1 ) dx.

= p1 ’ p1 u + u1 ’ 2q 3 • ’ (6.132)

6 2

’∞

So expression (6.132) requires in a natural way the introduction of the even

nonlocal variable p3 , de¬ned by

x

(u2 ’ ••1 ) dx,

p3 = (6.133)

’∞

where

(p3 )x = u2 ’ ••1 ,

288 6. SUPER AND GRADED THEORIES

(p3 )t = 4u3 ’ 2u2 u + u2 ’ 9u••1 + ••3 ’ 2•1 •2 . (6.134)

1

Here p3 is a well-known potential, P3 being the associated conserved quan-

tity.

Finally, the commutator Y 7 = [ Z1 , Y 5 ] requires the prolongation

2 2

of the vector ¬eld Z1 towards the nonlocal variable q 3 , obtained by the

2

invariance of the condition (q 3 )x ’ p1 • = 0 by Z1 , so

2

q3

= (q 1 •)• + p1 (q 1 u ’ •1 ).

Dx (Z1 2 ) = Z1 (p1 •) (6.135)

2 2

Integration of (6.135) leads to

x

q3 1 1

= p2 q 1 ’ p 1 • ’ ( p2 • ’ u•) dx.

2

Z1 (6.136)

212 1

’∞ 2

The new odd nonlocal variable q 5 is, due to (6.136), formally de¬ned by

2

x

1

( p2 • ’ u•) dx.

q5 = (6.137)

1

’∞ 2

2

Here q 5 is a nonlocal odd potential of the supersymmetric KdV equation

2

(6.104),

1

(q 5 )x = p2 • ’ u•,

21

2

1

(q 5 )t = p2 (’•2 + 3u•) + p1 (’u1 • + u•1 ) + u2 • ’ u1 •1 ’ 4u2 • + u•2 .

21

2

(6.138)

Proceeding in this way, we obtain a hierarchy of nonlocal higher supersym-

metries by induction,

n ∈ N.

=[ Z1 , Yn’ 1 ], (6.139)

Yn+ 1

2 2

The higher even potentials p1 , p3 , . . . arise in a natural way in the prolon-

gation of the vector ¬elds Y2n+ 1 towards the nonlocal variable q 1 , whereas

2

2

the higher nonlocal odd potentials q 1 , q 3 , q 5 , . . . are obtained in the pro-

2 2 2

longation of the recursion symmetry Z1 .

To obtain the graded Lie algebra structure of symmetries we calculate

the graded Lie bracket of the vector ¬elds derived so far. The result is

remarkable and fascinating:

[ Y1 , Y1 ] =2 X1 ,

2 2

[ Y3 , Y3 ] =2 X3 ,

2 2

[ Y5 , Y5 ] =2 X5 , (6.140)

2 2

so the “squares” of the supersymmetries Y1 , Y3 , are just the “clas-

Y5

2 2 2

sical” symmetries 2 X1 , 2 X3 , 2 obtained previously (see (6.101)). The

X5

6. SUPERSYMMETRIC KDV EQUATION 289

other commutators are

[ Y1 , Y3 ] = 0,

2 2

= ’2

[ Y1 , Y5 ] X3 ,

2 2

[ Y3 , Y5 ] = 0,

2 2

[ X1 , X3 ] =[ X1 , X5 ] =[ X3 , X5 ] = 0,

[ Z1 , X1 ] =[ Z1 , X3 ] =[ Z1 , X5 ] = 0,

[ Yn+ 1 , X2m+1 ] = 0, (6.141)

2

where n = 0, 1, 2, m = 0, 1, 2. We conjecture that in this way we obtain

an in¬nite hierarchy of nonlocal odd symmetries Yn+ 1 , n ∈ N, and an in¬-

2

nite hierarchy of ordinary even higher symmetries X2n+1 , n ∈ N, while the

even and odd nonlocal variables p2n+1 , qn+ 1 and the associated conserved

2

quantities P2n+1 , Qn+ 1 are obtained by the prolongation of the vector ¬elds

2

Yn+ 1 and Z1 respectively.

2

We ¬nish this section with a lemma concerning the Lie algebra structure

of the symmetries.

Lemma 6.36. Let X2n+1 , n ∈ N, be de¬ned by

1

X2n+1 = [Yn+ 1 , Yn+ 1 ], (6.142)

2 2 2

and assume that

n ∈ N.

[Z1 , X2n+1 ] = 0, (6.143)

Then

(’1)m’n 2Xn+m+1 m ’ n is even,

1. [Yn+ 1 , Ym+ 1 ] =

m ’ n is odd.

0

2 2

2. [Yn+ 1 , X2m+1 ] = 0, n, m ∈ N.

2

3. [X2n+1 , X2m+1 ] = 0, n, m ∈ N.

Proof. The proof of (1) is by induction on k = m ’ n. First consider

the cases k = 1 and k = 2:

0 = [Z1 , [Yn+ 1 , Yn+ 1 ]] = [Yn+1+ 1 , Yn+ 1 ] + [Yn+ 1 , Yn+1+ 1 ]

2 2 2 2 2 2

= 2[Yn+ 1 , Yn+1+ 1 ],

2 2

0 = [Z1 , [Yn+ 1 , Yn+1+ 1 ]] = [Yn+1+ 1 , Yn+1+ 1 ] + [Yn+ 1 , Yn+2+ 1 ], (6.144)

2 2 2 2 2 2

so

[Yn+ 1 , Yn+2+ 1 ] = ’2X2n+3 . (6.145)

2 2

For general k, the result is obtained from the identity

0 = [Z1 , [Yn+ 1 , Yn+k+ 1 ]] = [Yn+1+ 1 , Yn+k+ 1 ] + [Yn+ 1 , Yn+k+1+ 1 ], (6.146)

2 2 2 2 2 2

290 6. SUPER AND GRADED THEORIES

i.e.,

[Yn+ 1 , Yn+k+1+ 1 ] = ’[Yn+1+ 1 , Yn+k+ 1 ]. (6.147)

2 2 2 2

The proof of (3) is a consequence of (2) by

[X2n+1 , X2m+1 ] = [[Yn+ 1 , Yn+ 1 ], X2m+1 ] = 2[Yn+ 1 , [Yn+ 1 , X2m+1 ]] = 0.

2 2 2 2

(6.148)

So we are left with the proof of statement (2), the proof of which is by

induction too. Let us prove the following statement:

E(n) : for all i ¤ n, j ¤ n one has [Yi+ 1 , X2j+1 ] = 0.

2

One can see that E(0) is true for obvious reasons: [Y 1 , X1 ] = 0. The

2

induction step is in three parts,

(b1): [Yn+1+ 1 , X2n+3 ] = 0;

2

(b2): [Yn+1+ 1 , X2j+1 ] = 0, j ¤ n;

2

(b3): [Yi+ 1 , X2n+3 ] = 0, i ¤ n.

2

The proof of (b1) is obvious by means of the de¬nition of X2n+3 .

The proof of (b2) follows from

[Yn+1+ 1 , X2j+1 ]] = [[Z1 , Yn+ 1 ], X2j+1 ]

2 2

= [Z1 , [Yn+ 1 , X2j+1 ]] + [[Z1 , X2j+1 ], Yn+ 1 ] = 0, (6.149)

2 2

while both terms in the right-hand side are equal to zero by assumption and

(6.144) respectively.

Finally, the proof of (b3) follows from

1

[Yi+ 1 , X2n+3 ] = [Yi+ 1 , [Yn+1+ 1 , Yn+1+ 1 ]] = [[Yi+ 1 , Yn+1+ 1 ], Yn+1+ 1 ] = 0,

2

2 2 2 2 2 2 2

(6.150)

by statement 1 of Lemma 6.36, which completes the proof of this lemma.

7. Supersymmetric mKdV equation

Since constructions and computations in this section are completely sim-

ilar to those carried through in the previous section, we shall here present

just the results for the supersymmetric mKdV equation (6.151)

vt = ’v3 + 6v 2 v1 ’ 3•(v•1 )1 ,

•t = ’•3 + 3v(v•)1 . (6.151)

Note that the supersymmetric mKdV equation (6.151) is graded

deg(x) = ’1, deg(t) = ’3,

1

deg(v) = 1, deg(•) = . (6.152)

2

7. SUPERSYMMETRIC MKDV EQUATION 291

The supersymmetry Y of (6.151) is given by

1

2

‚ ‚

Y = •1 +v . (6.153)

1

‚v ‚•

2

The associated nonlocal variable q 1 and the conserved quantity Q 1 are given

2 2

by

∞

x

q1 = (v•) dx, Q1 = (v•) dx, (6.154)

2 2

’∞ ’∞

where

(q 1 )x = v•,

2

(q 1 )t = ’v2 • + v1 •1 ’ v•2 + 3v 3 •. (6.155)

2

The nonlocal symmetry Z 1 is given by

‚ ‚

+ (q 1 v ’ •1 ) .

Z 1 = (q 1 •1 ) (6.156)

‚v ‚•

2 2

We now present the even nonlocal variables p1 , p3 and the odd nonlocal

variables q 1 , q 3 , q 5 , where

2 2 2

x

(v 2 ’ ••1 ) dx,

p1 =

’∞

x

(’v 4 ’ v1 + 3••1 v 2 + •1 •2 ) dx,

2

p3 =

’∞

x

q1 = (v•) dx,

2

’∞

x

q3 = (p1 v• + v•1 ) dx,

2

’∞

x

(’p2 v• ’ 2p1 v•1 + v 3 • ’ 2v•2 ) dx.

q5 = (6.157)

1

2

’∞

The x-derivatives of these nonlocal variables are just the integrands in

(6.157), while the t-derivatives are given by

(p1 )t = 3v 4 + v1 ’ 2vv2 ’ 9v 2 ••1 + ••3 ’ 2•1 •2 ,

2

(p3 )t = ’4v 6 + 4v2 v 3 ’ v2 + 2v1 v3 ’ 12v 2 v1 + 21v 4 ••1

2 2

’ 9vv2 ••1 + 3v1 ••1 + 12vv1 ••2 ’ 3v 2 ••3

2

+ 9v 2 •1 •2 ’ •1 •4 + 2•2 •3 ,