<< . .

. 51
( : 58)



. . >>

2 2

1
(q 3 )x = (q 1 u + •w),
32
2

1
(q 3 )t = (3q 1 u2 + 12q 1 uw2 ’ q 1 u2 ’ 6q 1 ww2 ’ 3q 1 w1 ’ •2 w ’ ψ1 u
2
3
2 2 2 2 2 2

’ 6ψ1 w2 + •1 w1 ’ 3ψψ1 q 1 + ψu1 + 3ψww1 ’ 3••1 q 1 ’ 24•ψq 1 w
2 2 2
3
+ 9•uw + 12•w ’ •w2 );
1
(q 3 )x = (q 1 u ’ ψw),
32
2

1
(q 3 )t = (3q 1 u2 + 12q 1 uw2 ’ q 1 u2 ’ 6q 1 ww2 ’ 3q 1 w1 + ψ2 w ’ ψ1 w1
2
3
2 2 2 2 2 2
2 3
’ •1 u ’ 6•1 w ’ 3ψψ1 q 1 ’ 9ψuw ’ 12ψw + ψw2 ’ 3••1 q 1
2 2

’ 24•ψq 1 w + •u1 + 3•ww1 ). (7.63)
2

3. Finally, at degree 5/2 we have q 5 and q 5 which are de¬ned by the
2 2
relations, i.e.,
1
(q 5 )x = (2q 1 p1,1 u ’ 2ψ1 w ’ 2ψp1,1 w + 4ψp2,1 + 2•u + 3•w 2 ),
24
2 2

1
(q 5 )t = (6q 1 p1,1 u2 + 24q 1 p1,1 uw2 ’ 2q 1 p1,1 u2 ’ 12q 1 p1,1 ww2
24
2 2 2 2 2
2
’ 6q 1 p1,1 w1 + 2ψ3 w + 2ψ2 p1,1 w ’ 4ψ2 p2,1 ’ 2ψ2 w1 ’ 2•2 u
2

’ 15•2 w2 ’ 2ψ1 p1,1 w1 ’ 24ψ1 uw ’ 42ψ1 w3 + 2ψ1 w2 ’ 2•1 p1,1 u
’ 12•1 p1,1 w2 + 24•1 p2,1 w + 2•1 u1 ’ 6ψψ1 q 1 p1,1 ’ 18ψp1,1 uw
2

’ 24ψp1,1 w3 + 2ψp1,1 w2 + 12ψp2,1 u + 48ψp2,1 w2 ’ 4ψuw1
’ 21ψw 2 w1 ’ 6••1 q 1 p1,1 ’ 48•ψq 1 p1,1 w + 2•ψψ1 + 2•p1,1 u1
2 2

+ 6•p1,1 ww1 + 12•p2,1 w1 + 8•u + 69•uw 2
2
334 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

’ 2•u2 + 36•w 4 ’ 24•ww2 );
1
(q 5 )x = (’4q 1 p1,1 w1 ’ 2q 1 p1,1 u + 2ψ1 p1,1 + 4•1 q 1 q 1 ’ 2•1 w
6
2 2 2 22

’ 3ψw 2 ’ 6•p1,1 w). (7.64)
We omitted explicit expressions for (p3,2 )t and (q 5 )t in (7.61) and (7.64)
2
because they are too massive.
Thus, we obtained the following 14 nonlocal variables:
p0,1 , p0,2 of degree
0,
p1,1 , p1,2 of degree
1,
p2,1 , p2,2 of degree
2,
p3,1 , p3,2 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 . (7.65)
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.
4.2.2. Higher and nonlocal symmetries. In this subsection we present
results for higher and nonlocal symmetries for the N = 2 supersymmetric
extension of the KdV equation (7.57) in the case a = 4,
‚ ‚ ‚ ‚
Y =Yu +Yw +Y• +Yψ + ...
‚u ‚w ‚• ‚ψ
We obtained the following odd symmetries. The components of their gener-
ating functions are given below:
u u
Y 1 ,1 = ψ1 , Y 1 ,2 = •1 ,
2 2
w w
= ’•,
Y 1 ,1 Y 1 ,2 = ψ,
2 2
• •
= ’w1 ,
Y 1 ,1 Y 1 ,2 = u,
2 2
ψ ψ
Y 1 ,1 = u; Y 1 ,2 = w1 (7.66)
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 ’ 3•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 ’ 4•ψ + p1,1 w1 + 3uw ’ w2 ;
2
2
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 335

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

= ’2•1 q 1 ’ 4•ψ + p1,1 w1 + 3uw ’ w2 ,
Y 3 ,2
2
2
ψ
Y 3 ,2 = ’2ψ1 q 1 ’ p1,1 u + u1 + 3ww1 . (7.67)
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 = ψ1 q 1 + •1 q 1 ,
2 2
w
= ψq 1 ’ •q 1 ,
Y1,1
2 2

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

u
Y1,2 = u1 ,
w
Y1,2 = w1 ,

Y1,2 = •1 ,
ψ
Y1,2 = ψ1 . (7.68)
4.2.3. Recursion operator. Here we present the recursion operator R for
symmetries for the case a = 4 obtained as a higher symmetry in the Cartan
covering of the augmented system of equations (7.65). This operator is of
the form
‚ ‚ ‚ ‚
R = Ru + Rw + R• + Rψ + ..., (7.69)
‚u ‚w ‚• ‚ψ
where the components Ru , Rw , R• , Rψ are given by
Ru = ωu2 + ωu (’4u ’ 4w 2 ) + ωw2 (4w)
+ ωw1 (6w1 ) + ωw (’16uw + 6w2 + 18•ψ)
+ ω•1 (’2•) + ω• (•1 + 12ψw) + ωψ1 (’2ψ) + ωψ (ψ1 ’ 12•w)
+ ωq 1 (ψ2 ’ 3•1 w ’ 3•w1 ’ ψu ’ q 1 u1 )
2
2

+ ωq 1 (•2 + 3ψ1 w + 3ψw1 ’ •u ’ q 1 u1 )
2
2

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

+ ωp1,1 (’2u1 + ψ1 q 1 + •1 q 1 )
2 2

+ ωp0,1 (2w3 ’ 8uw1 ’ 8u1 w + 8•1 ψ + 8•ψ1 ),
336 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

Rw = ωu (’4w) + ωw2 + ωw (’4u ’ 4w 2 ) + ω• (2ψ) + ωψ (’2•)
+ ωq 1 (’•1 ’ 3ψw ’ q 1 w1 ) + ωq 1 (ψ1 ’ 3•w ’ q 1 w1 )
2 2
2 2

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

+ ωp1,1 (’2w1 + ψq 1 ’ •q 1 ) + ωp0,1 (’2u1 ’ 8ww1 ),
2 2


R• = ωu (’2•) + ωw1 (2ψ) + ωw (5ψ1 ’ 12•w)
+ ω•2 + ω• (’2u ’ 4w 2 ) + ωψ1 (4w) + ωψ (4w1 )
+ ωq 1 (w2 ’ 3uw + 4•ψ + •1 q 1 ) + ωq 1 (’u1 ’ 3ww1 + •1 q 1 )
2 2
2 2

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

+ ωp1,1 (’•1 + 2ψw + q 1 w1 ’ q 1 u)
2 2

+ ωp0,1 (2ψ2 ’ 8•1 w ’ 8•w1 ) + ωp2,1 (’2ψ),

Rψ = ωu (’2ψ) + ωw1 (’2•) + ωw (’5•1 ’ 12ψw)
+ ω•1 (’4w) + ω• (’4w1 ) + ωψ2 + ωψ (’2u ’ 4w 2 )
+ ωq 1 (’u1 ’ 3ww1 + ψ1 q 1 )
2
2

+ ωq 1 (3uw ’ w2 ’ 4•ψ + ψ1 q 1 )
2
2

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

+ ωp1,1 (’ψ1 ’ 2•w ’ q 1 u ’ q 1 w1 ) + ωp0,1 (’2•2 ’ 8ψ1 w ’ 8ψw1 )
2 2

+ ωp2,1 (2•). (7.70)

It should be noted that the components are again given here in the right-
module structure (see Chapter 6).

Remark 7.2. Personal communication with Prof. A. Sorin informed us
about existence of a deformation, or recursion operator of order 1 in this
speci¬c case, a fact which might be indicated by the structure of the existing
nonlocal variables. The result is given by

R1 = ωu (2w) ’ ωw2 + ωw (4u) + ω• (’3ψ) + ωψ (3•)

+ ωq 1 (•1 ) + ωq 1 (’ψ1 ) + ωp0,1 (2u1 )
‚u
2 2

+ ωu + ωw (2w) + ωq 1 (ψ)
2

+ ωq 1 (•) + ωp0,1 (2w1 )
‚w
2

+ ωw (3•) ’ ωψ1 + ω• (2w) + ωq 1 (u)
2

+ ωq 1 (w1 ) + ωp0,1 (2•1 ) + ωp1,1 (’ψ)
‚•
2
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 337


+ ωw (3ψ) + ω•1 + ωψ (2w) + ωq 1 (w1 )
2

+ ωq 1 (’u) + ωp0,1 (2ψ1 ) + ωp1,1 (•) .
‚ψ
2

4.3. Case a = 1. In this section we discuss the case a = 1, which does
lead to the following system of partial di¬erential equations:
ut = ’u3 + 6uu1 ’ 3••2 ’ 3ψψ2 ’ 3ww3 ’ 3w1 w2 + 3u1 w2 + 6uww1
+ 6ψ•1 w ’ 6•ψ1 w ’ 6•ψw1 ,
•t = ’•3 + 3•u1 + 3•1 u ’ 3ψ2 w ’ 3ψ1 w1 + 3•1 w2 + 6•ww1 ,
ψt = ’ψ3 + 3ψu1 + 3ψ1 u + 3•2 w + 3•1 w1 + 3ψ1 w2 + 6ψww1 ,
wt = ’w3 + 3w2 w1 + 3uw1 + 3u1 w. (7.71)
The results obtained in this case for conservation laws, higher symmetries
and recursion symmetries will be presented in subsequent subsections.
4.3.1. 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 = 3uw + w 3 ’ w2 ;

(p0,2 )x = p1 ,
(p0,2 )t = ’6p3 ’ u1 . (7.72)
2. Nonlocal variables p1,1 , p1,2 , p1,3 , and p1,4 of degree 1 are de¬ned by
(p1 )x = u,
(p1 )t = ’3ψψ1 ’ 3••1 ’ 6•ψw + 3u2 + 3uw2 ’ u2 ’ 3ww2 ;

(p1,1 )x = cos(2p0,1 )(•q 1 ,2 + p1 w) + sin(2p0,1 )(ψq 1 ,2 + w2 ),
2 2

(p1,1 )t = cos(2p0,1 )(’•2 q 1 ,2 ’ ψ1 q 1 ,2 w ’ 2ψq 1 ,2 w1 ’ ψ•1 + 3•q 1 ,2 u
2 2 2 2

+ •q 1 ,2 w ’ •ψ1 + 3p1 uw + p1 w ’ p1 w2 + uw1 ’ u1 w ’ w2 w1 )
2 3
2

+ sin(2p0,1 )(’ψ2 q 1 ,2 + •1 q 1 ,2 w + 3ψq 1 ,2 u + ψq 1 ,2 w2 ’ 2ψψ1
2 2 2 2
2 4 2
+ 2•q 1 ,2 w1 ’ 2•ψw + 4uw + w ’ 2ww2 + w1 );
2


(p1,2 )x = cos(2p0,1 )(ψq 1 ,2 + w2 ) ’ sin(2p0,1 )(•q 1 ,2 + p1 w),
2 2

(p1,2 )t = cos(2p0,1 )(’ψ2 q 1 ,2 + •1 q 1 ,2 w + 3ψq 1 ,2 u
2 2 2

+ ψq 1 ,2 w ’ 2ψψ1 + 2•q 1 ,2 w1 ’ 2•ψw + 4uw 2 + w4 ’ 2ww2 + w1 )
2 2
2 2

+ sin(2p0,1 )(•2 q 1 ,2 + ψ1 q 1 ,2 w + 2ψq 1 ,2 w1 + ψ•1 ’ 3•q 1 ,2 u
2 2 2 2

’ •q 1 ,2 w + •ψ1 ’ 3p1 uw ’ p1 w + p1 w2 ’ uw1 + u1 w + w2 w1 );
2 3
2
338 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

(p1,3 )x = ’2 cos(2p0,1 )•q 1 ,2 + sin(2p0,1 )(2q 1 ,1 q 1 ,2 w ’ ψq 1 ,2 + •q 1 ,1 ),
2 2 2 2 2

(p1,3 )t = 2 cos(2p0,1 )(•2 q 1 ,2 + 2ψ1 q 1 ,2 w + •1 q 1 ,1 w + ψq 1 ,2 w1
2 2 2 2
2 2
+ ψq 1 ,1 w + ψ•1 ’ 3•q 1 ,2 u ’ 2•q 1 ,2 w ’ •q 1 ,1 w1 + •ψ1 )
2 2 2 2
3
+ sin(2p0,1 )(6q 1 ,1 q 1 ,2 uw + 2q 1 ,1 q 1 ,2 w ’ 2q 1 ,1 q 1 ,2 w2 + ψ2 q 1 ,2
2 2 2 2 2 2 2
2
’ •2 q 1 ,1 ’ ψ1 q 1 ,1 w ’ •1 q 1 ,2 w ’ 3ψq 1 ,2 u ’ ψq 1 ,1 w
2 2 2 2 2

’ 2ψq 1 ,1 w1 + 2ψψ1 ’ 2•q 1 ,2 w1 + 3•q 1 ,1 u + •q 1 ,1 w2 ’ 2••1 );
2 2 2 2


(p1,4 )x = cos(2p0,1 )(2q 1 ,1 q 1 ,2 w ’ ψq 1 ,2 + •q 1 ,1 ) + 2 sin(2p0,1 )•q 1 ,2 ,
2 2 2 2 2
3
(p1,4 )t = cos(2p0,1 )(6q 1 ,1 q 1 ,2 uw + 2q 1 ,1 q 1 ,2 w ’ 2q 1 ,1 q 1 ,2 w2 + ψ2 q 1 ,2
2 2 2 2 2 2 2
2
’ •2 q 1 ,1 ’ ψ1 q 1 w ’ •1 q 1 ,2 w ’ 3ψq 1 ,2 u ’ ψq 1 ,2 w ’ 2ψq 1 ,1 w1
2 2 2 2 2 2
2
+ 2ψψ1 ’ 2•q 1 ,2 w1 + 3•q 1 ,1 u + •q 1 ,1 w ’ 2••1 )
2 2 2

+ 2 sin(2p0,1 )(’•2 q 1 ,2 ’ 2ψ1 q 1 ,2 w ’ •1 q 1 ,1 w ’ ψq 1 ,2 w1 ’ ψq 1 ,1 w2
2 2 2 2 2
2
’ ψ•1 + 3•q 1 ,2 u + 2•q 1 ,2 w + •q 1 ,1 w1 ’ •ψ1 ). (7.73)
2 2 2

3. The variable p3,1 of degree 3 is
1
(p3,1 )x = (ψψ1 + ••1 + 2•ψw ’ u2 ’ uw2 + ww2 ),
2
1
(p3,1 )t = (2ψ1 ψ2 + 2•1 •2 + 8•1 ψ1 w ’ ψψ3 + 5ψ•2 w + 9ψψ1 u
2
+ 12ψψ1 w2 + ψ•1 w1 ’ ••3 ’ 5•ψ2 w ’ •ψ1 w1 + 9••1 u
+ 12••1 w2 + 18•ψuw + 12•ψw 3 ’ 2•ψw2 ’ 4u3 ’ 9u2 w2 + 2uu2
’ 3uw4 + 11uww2 ’ uw1 ’ u2 + u1 ww1 + 4u2 w2 + 6w3 w2
2
1
+ 3w2 w1 ’ ww4 + w1 w3 ’ w2 ).
2 2

<< . .

. 51
( : 58)



. . >>