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(q 5 )x = q 1 p1,1 u + 3q 1 ww1 + •1 w + ψu ’ •p1,1 w,
2 2 2

(q 5 )t = 3q 1 p1,1 u ’ 6q 1 p1,1 uw2 ’ q 1 p1,1 u2 + 3q 1 p1,1 w1 ’ 18q 1 w3 w1
2 2
2 2 2 2 2 2
’ 3q 1 ww3 ’ •3 w ’ ψ2 u + •2 p1,1 w + •2 w1 ’ ψ1 p1,1 u + ψ1 u1
2

’ •1 p1,1 w1 + 2•1 uw ’ 6•1 w3 ’ •1 w2 ’ 3ψψ1 q 1 p1,1 ’ 9ψ•1 q 1 w
2 2
2 2
+ ψp1,1 u1 ’ 3ψp1,1 ww1 + 4ψu ’ 6ψuw ’ ψu2 + 6ψww2
+ 9•ψ1 q 1 w ’ 3••1 q 1 p1,1 + 12•ψq 1 p1,1 w + 3•ψ•1
2 2 2

’ 3•p1,1 uw + 6•p1,1 w + •p1,1 w2 ’ 4•uw1 + 4•u1 w ’ 12•w 2 w1 ;
3


(q 5 )x = ’q 1 p1,1 u + q 1 u1 ’ 3q 1 ww1 + ψ1 w ’ ψp1,1 w,
2 2 2 2

(q 5 )t = ’3q 1 p1,1 u + 6q 1 p1,1 uw2 + q 1 p1,1 u2 ’ 3q 1 p1,1 w1 + 6q 1 uu1
2 2
2 2 2 2 2 2
2 3
’ 12q 1 uww1 ’ 6q 1 u1 w ’ q 1 u3 + 18q 1 w w1 + 3q 1 ww3 + 6q 1 w1 w2
2 2 2 2 2 2
3
’ ψ3 w + ψ2 p1,1 w + ψ2 w1 ’ ψ1 p1,1 w1 + 2ψ1 uw ’ 6ψ1 w ’ ψ1 w2
+ •1 p1,1 u ’ •1 u1 ’ 3ψψ2 q 1 + 3ψψ1 q 1 p1,1 ’ 3ψ•1 q 1 w ’ 3ψp1,1 uw
2 2 2

+ 6ψp1,1 w3 + ψp1,1 w2 ’ ψuw1 + 4ψu1 w ’ 12ψw 2 w1 ’ 3••2 q 1
2
+ 3•ψ1 q 1 w + 3••1 q 1 p1,1 ’ 12•ψq 1 p1,1 w + 12•ψq 1 w1 + 3•ψψ1
2 2 2 2
2
’ •p1,1 u1 + 3•p1,1 ww1 ’ •u + •u2 ’ 6•ww2 . (7.49)

Thus the entire nonlocal setting comprises the following 14 nonlocal vari-
ables:

p0,1 , p0,2 of degree
0,
p1,1 , p1,2 , p1,3 , p1,4 of degree
1,
p2,1 of degree
2,
p3,1 of degree
3,
1
q1 , q 1 of degree ,
2
2 2

3
q3 , q 3 of degree ,
2
2 2

5
q5 , q 5 of degree . (7.50)
2
2 2


In the next subsections the augmented system of equations associated
to the local and the nonlocal variables denoted above will be considered in
computing higher and nonlocal symmetries and the recursion operator.
4.1.2. Higher and nonlocal symmetries. In this subsection, we present
results for higher and nonlocal symmetries for the N = 2 supersymmetric
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 329

extension of KdV equation (7.42),
‚ ‚ ‚ ‚
Y =Yu +Yw +Y• +Yψ + ...
‚u ‚w ‚• ‚ψ
We obtained the following odd symmetries, just giving here the components
of their generating functions,
u u
Y 1 ,1 = ψ1 , Y 1 ,2 = •1 ,
2 2
w u
= ’•,
Y 1 ,1 Y 1 ,2 = •1 ,
2 2
• •
= ’w1 ,
Y 1 ,1 Y 1 ,2 = u,
2 2
ψ ψ
Y 1 ,1 = u; Y 1 ,2 = w1 (7.51)
2 2

and
u
Y 3 ,1 = 2q 1 u1 ’ •2 + 3ψ1 w ’ •1 p1,1 + 3ψw1 + •u,
2
2
w
= 2q 1 w1 + ψ1 ’ ψp1,1 + •w,
Y 3 ,1
2
2

= ’2•1 q 1 ’ p1,1 u + u1 ’ 3ww1 ,
Y 3 ,1
2
2
ψ
Y 3 ,1 = ’2ψ1 q 1 + 2•ψ ’ p1,1 w1 ’ uw ’ w2 ;
2
2

u
Y 3 ,2 = 2q 1 u1 ’ ψ2 ’ ψ1 p1,1 ’ 3•1 w + ψu ’ 3•w1 ,
2
2
w
= 2q 1 w1 ’ •1 + ψw + •p1,1 ,
Y 3 ,2
2
2

= ’2•1 q 1 ’ 2•ψ + p1,1 w1 + uw + w2 ,
Y 3 ,2
2
2
ψ
Y 3 ,2 = ’2ψ1 q 1 ’ p1,1 u + u1 ’ 3ww1 . (7.52)
2
2

We also obtained the following even symmetries:
u
Y0,1 = 0,
w
Y0,1 = 0,

Y0,1 = ψ,
ψ
Y0,1 = ’•;
u
Y1,1 = u1 ,
w
Y1,1 = w1 ,

Y1,1 = •1 ,
ψ
Y1,1 = ψ1 ;
u
Y1,2 = •1 q 1 + 2ww1 ,
2
w
Y1,2 = ψq 1 + w1 ,
2

= ’q 1 u + •1 ’ ψw,
Y1,2
2
330 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

ψ
Y1,2 = ’q 1 w1 ’ •w;
2

u
Y1,3 = ψ1 q 1 ’ •1 q 1 ,
2 2
w
= ’ψq 1 ’ •q 1 ,
Y1,3
2 2

= q 1 w1 + q 1 u ’ •1 + 2ψw,
Y1,3
2 2
ψ
Y1,3 = ’q 1 u + q 1 w1 + ψ1 + 2•w;
2 2

u
Y1,4 = ψ1 q 1 + •1 q 1 ,
2 2
w
= ψq 1 ’ •q 1 ,
Y1,4
2 2

= ’q 1 u + q 1 w1 + ψ1 + 2•w,
Y1,4
2 2
ψ
= ’q 1 w1 ’ q 1 u + •1 ’ 2ψw.
Y1,4 (7.53)
2 2

Moreover there is a symmetry of degree 2 with the generating function
u
Y2,1 = 2q 1 q 1 u1 + ψ2 q 1 ’ •2 q 1 ’ ψ1 q 3 + 3ψ1 q 1 w ’ •1 q 3 + 3•1 q 1 w
2 2 2 2 2 2 2 2
+ 3ψq 1 w1 ’ ψq 1 u + •q 1 u + 3•q 1 w1 + •1 ψ + •ψ1 ,
2 2 2 2
w
= 2q 1 q 1 w1 + ψ1 q 1 + •1 q 1 ’ ψq 3 ’ ψq 1 w + •q 3 + •q 1 w,
Y2,1
2 2 2 2 2 2 2 2

= ’q 3 w1 + q 3 u ’ q 1 u1 + 3q 1 ww1 + q 1 uw + q 1 w2 + ψ2 + 2•1 q 1 q 1
Y2,1
2 2 2 2 2 2 2 2

’ 2ψu + 4ψw 2 ’ 2•ψq 1 + 2•w1 ,
2
ψ
= q 3 u + q 3 w1 + q 1 uw + q 1 w2 + q 1 u1 ’ 3q 1 ww1 ’ •2 + 2ψ1 q 1 q 1
Y2,1
2 2 2 2 2 2 2 2
2
+ 2ψw1 ’ 2•ψq 1 + 2•u ’ 4•w . (7.54)
2

4.1.3. Recursion operator. Here we present the recursion operator R
for symmetries for this case obtained as a higher symmetry in the Cartan
covering of the augmented system of equations (7.50). The result is
‚ ‚ ‚ ‚
R = Ru + Rw + R• + Rψ + ..., (7.55)
‚u ‚w ‚• ‚ψ
where the components Ru , Rw , R• , Rψ are given by
Ru = ωu2 + ωu (’4u + 4w 2 )
+ ωw1 (’4w1 ) + ωw (8uw ’ 2w2 ’ 6•ψ)
+ ω•1 (’2•) + ω• (•1 ’ 8ψw) + ωψ1 (’2ψ) + ωψ (ψ1 + 8•w)
+ ωq 1 (•2 ’ 3ψ1 w ’ 3ψw1 ’ •u ’ q 1 u1 )
2
2

+ ωq 1 (ψ2 + 3•1 w + 3•w1 ’ ψu ’ q 1 u1 )
2
2

+ ωq 3 (ψ1 ) + ωq 3 (’•1 ) + ωp1,4 (2u1 ) + ωp1,2 (u1 )
2 2

+ ωp1,1 (’2u1 + 4ww1 + •1 q 1 + ψ1 q 1 ),
2 2
2
Rw = ωw2 + ωw (4w ) + ω• (’2ψ) + ωψ (2•)
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 331

+ ωq 1 (’ψ1 ’ •w ’ q 1 w1 ) + ωq 1 (•1 ’ ψw ’ q 1 w1 )
2 2
2 2

+ ωq 3 (’•) + ωq 3 (’ψ) + ωp1,4 (2w1 ) + ωp1,2 (w1 )
2 2

+ ωp1,1 (ψq 1 ’ •q 1 ),
2 2

R• = ωu (’2•) + ωw1 (’2ψ) + ωw (’ψ1 + 8•w)
+ ω•2 + ω• (’2u + 4w 2 ) + ωψ (’2w1 )
+ ωq 1 (’u1 + 3ww1 + •1 q 1 )
2
2

+ ωq 1 (’uw ’ w2 + 2•ψ + •1 q 1 )
2
2

+ ωq 3 (’w1 ) + ωq 3 (’u) + ωp1,4 (2•1 ) + ωp1,2 (•1 )
2 2

+ ωp1,1 (’•1 ’ q 1 u + q 1 w1 ),
2 2

Rψ = ωu (’2ψ) + ωw1 (2•) + ωw (•1 + 8ψw)
+ ω• (2w1 ) + ωψ2 + ωψ (’2u + 4w 2 )
+ ωq 1 (uw + w2 ’ 2•ψ + ψ1 q 1 )
2
2

+ ωq 1 (’u1 + 3ww1 + ψ1 q 1 )
2
2

+ ωq 3 (u) + ωq 3 (’w1 ) + ωp1,4 (2ψ1 ) + ωp1,2 (ψ1 )
2 2

+ ωp1,1 (’ψ1 ’ q 1 w1 ’ q 1 u). (7.56)
2 2

It should be noted that the components are given in the right-module struc-
ture (see Chapter 6).

4.2. Case a = 4. In this subsection we discuss the case a = 4, which
does lead to the following system of partial di¬erential equations:
ut = ’u3 + 6uu1 ’ 3••2 ’ 3ψψ2 ’ 6ww3 ’ 12w1 w2 + 24uww1 + 12u1 w2
+ 24ψ•1 w ’ 24•ψ1 w ’ 24•ψw1 ,
•t = ’•3 + 3•u1 + 3•1 u ’ 6ψ2 w ’ 9ψ1 w1 ’ 3ψw2 + 12•1 w2 + 24•ww1 ,
ψt = ’ψ3 + 3ψu1 + 3ψ1 u + 6•2 w + 9•1 w1 + 3•w2 + 12ψ1 w2 + 24ψww1 ,
wt = ’w3 + 12w 2 w1 + 6u1 w + 6uw1 + 3ψ•1 ’ 3•ψ1 . (7.57)
The results obtained in this case for conservation laws, higher symmetries
and deformations or recursion operator will be presented in subsequent sub-
sections.
4.2.1. Conservation laws. For the even conservation laws and the asso-
ciated even nonlocal variables we obtained the following results.
1. Nonlocal variables p0,1 and p0,2 of degree 0 are
(p0,1 )x = w,
(p0,1 )t = ’3•ψ + 6uw + 4w 3 ’ w2 ;

(p0,2 )x = p1,1 ,
332 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

(p0,2 )t = ’24p3,1 ’ u1 ’ 3ww1 . (7.58)
2. Nonlocal variables p1,1 and p1,2 of degree 1 are de¬ned by
(p1,1 )x = u,
(p1,1 )t = ’3ψψ1 ’ 3••1 ’ 24•ψw + 3u2 + 12uw 2 ’ u2 ’ 6ww2 ’ 3w1 ;
2


(p1,2 )x = ψq 1 + •q 1 ,
2 2

(p1,2 )t = ’ψ2 q 1 ’ •2 q 1 ’ 6ψ1 q 1 w + 6•1 q 1 w ’ 3ψq 1 w1 + 3ψq 1 u
2 2 2 2 2 2
2 2
+ 12ψq 1 w ’ 2ψψ1 + 3•q 1 u + 12•q 1 w + 3•q 1 w1
2 2 2 2

’ 2••1 ’ 12•ψw. (7.59)
3. Nonlocal variables p2,1 and p2,2 of degree 2 are
(p2,1 )x = •ψ ’ uw,
(p2,1 )t = •1 ψ1 + ψ•2 + 9ψψ1 w ’ •ψ2 + 9••1 w + 6•ψu + 36•ψw 2
’ 6u2 w ’ 12uw 3 + uw2 ’ u1 w1 + u2 w + 6w2 w2 ;
1
(p2,2 )x = (’q 1 q 1 u ’ ψq 1 w ’ •q 1 w + uw),
3 22 2 2

1
(p2,2 )t = (’3q 1 q 1 u2 ’ 12q 1 q 1 uw2 + q 1 q 1 u2 + 6q 1 q 1 ww2 + 3q 1 q 1 w1
2
3 22 22 22 22 22

+ ψ2 q 1 w + •2 q 1 w + ψ1 q 1 u + 6ψ1 q 1 w2 ’ ψ1 q 1 w1 ’ •1 q 1 w1
2 2 2 2 2 2
2
’ •1 q 1 u ’ 6•1 q 1 w ’ ψq 1 u1 ’ 3ψq 1 ww1 ’ 9ψq 1 uw
2 2 2 2 2

’ 12ψq 1 w + ψq 1 w2 + 3ψψ1 q 1 q 1 ’ ψψ1 w ’ 9•q 1 uw ’ 12•q 1 w3
3
2 2 2 2 2 2
+ •q 1 w2 + •q 1 u1 + 3•q 1 ww1 + 3••1 q 1 q 1 ’ ••1 w + 24•ψq 1 q 1 w
2 2 2 2 2 2 2

’ 2•ψu ’ 12•ψw 2 + 6u2 w + 12uw 3 ’ uw2 + u1 w1
’ u2 w ’ 6w2 w2 ). (7.60)
4. Finally, the variables p3,1 and p3,2 of degree 3 are de¬ned by
1
(p3,1 )x = (ψψ1 + ••1 + 8•ψw ’ u2 ’ 4uw2 + ww2 ),
8
1
(p3,1 )t = (2ψ1 ψ2 + 2•1 •2 + 14•1 ψ1 w ’ ψψ3 + 17ψ•2 w + 9ψψ1 u
8
+ 72ψψ1 w2 ’ 2ψ•1 w1 ’ ••3 ’ 17•ψ2 w + 2•ψ1 w1 + 9••1 u
+ 72••1 w2 + 96•ψuw + 192•ψw 3 ’ 14•ψw2 ’ 4u3 ’ 48u2 w2
+ 2uu2 ’ 48uw 4 + 26uww2 + 2uw1 ’ u2 ’ 2u1 ww1 + 10u2 w2
2
1
+ 36w 3 w2 + 12w 2 w1 ’ ww4 + w1 w3 ’ w2 );
2 2


1
(27q 1 q 3 u ’ 27q 1 q 3 w1 ’ 45q 1 q 3 w1 + 27q 1 q 3 u ’ 8q 1 q 1 p1,1 w1
(p3,2 )x =
27 22 22 22 22 22

+ 6q 1 q 1 uw ’ 10q 1 q 1 w2 ’ 9ψ2 q 1 ’ 9•2 q 1 ’ 186ψ1 q 3 + 16ψ1 q 1 w
2 2 2 2 2 2 2 2
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 333

+ 52ψ1 q 1 p1,1 + 36ψ1 q 1 p1,2 + 18•1 q 3 ’ 24ψq 5 ’ 72ψq 3 w
2 2 2 2 2

’ 48ψq 1 p2,1 + 288•q 5 ). (7.61)
2 2

For the odd conservation laws and the associated odd nonlocal variables
we derived the following results.
1. At degree 1/2, we have the variables q 1 and q 1 de¬ned by the relations
2 2

(q 1 )x = ψ,
2

(q 1 )t = ’ψ2 + 6•1 w + 3ψu + 12ψw 2 + 3•w1 ;
2


(q 1 )x = •,
2

(q 1 )t = ’•2 ’ 6ψ1 w ’ 3ψw1 + 3•u + 12•w 2 . (7.62)
2

2. At degree 3/2, the variables are q 3 and q 3 :

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