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2
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 ,

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