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’ 2•w + •w2 );

(q 3 ,2 )x = cos(2p0,1 )(’q 1 ,2 u + q 1 ,2 w2 + q 1 ,1 p1 w + •q 1 ,1 q 1 ,2 + •w)
2 2 2 2 2 2
2
+ sin(2p0,1 )(q 1 ,2 p1 w + q 1 ,1 u ’ q 1 ,1 w + ψq 1 ,1 q 1 ,2 + ψw),
2 2 2 2 2

(q 3 ,2 )t = cos(2p0,1 )(’3q 1 ,2 u + q 1 ,2 uw + q 1 ,2 u2 + q 1 ,2 w4 + q 1 ,2 ww2
2 2
2 2 2 2 2 2
2 3
+ 3q 1 ,1 p1 uw + q 1 ,1 p1 w ’ q 1 ,1 p1 w2 ’ q 1 ,1 uw1 + q 1 ,1 u1 w
+ q 1 ,2 w1
2 2 2 2 2 2

+ q 1 ,1 w w1 ’ •2 q 1 ,1 q 1 ,2 ’ •2 w ’ ψ1 q 1 ,1 q 1 ,2 w + ψ1 u ’ 2ψ1 w2
2
2 2 2 2 2
’ •1 p1 w + •1 w1 ’ 2ψq 1 ,1 q 1 ,2 w1 + 3ψψ1 q 1 ,2 + ψ•1 q 1 ,1
2 2 2 2
2 2
’ ψp1 w ’ ψu1 ’ ψww1 + 3•q 1 ,1 q 1 ,2 u + •q 1 ,1 q 1 ,2 w + •ψ1 q 1 ,1
2 2 2 2 2
340 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

+ ••1 q 1 ,2 + 4•ψq 1 ,2 w + •p1 w1 + 3•uw + 2•w 3 ’ •w2 )
2 2

+ sin(2p0,1 )(3q 1 ,2 p1 uw + q 1 ,2 p1 w3 ’ q 1 ,2 p1 w2 ’ q 1 ,2 uw1
2 2 2 2

+ q 1 ,2 u1 w + q 1 ,2 w w1 + 3q 1 ,1 u ’ q 1 ,1 uw ’ q 1 ,1 u2 ’ q 1 ,1 w4
2 2 2
2 2 2 2 2 2
2
’ q 1 ,1 ww2 ’ ’ ψ2 q 1 ,1 q 1 ,2 ’ ψ2 w ’ ψ1 p1 w + ψ1 w1
q 1 ,1 w1
2 2 2 2

+ •1 q 1 ,1 q 1 ,2 w ’ •1 u + 2•1 w + 3ψq 1 ,1 q 1 ,2 u + ψq 1 ,1 q 1 ,2 w2 2
2 2 2 2 2 2
3
’ ψψ1 q 1 ,1 ’ ψ•1 q 1 ,2 + ψp1 w1 + 3ψuw + 2ψw ’ ψw2
2 2

+ 2•q 1 ,1 q 1 ,2 w1 ’ •ψ1 q 1 ,2 ’ 3••1 q 1 ,1 ’ 4•ψq 1 ,1 w + •p1 w2
2 2 2 2 2

+ •u1 + •ww1 ). (7.76)

3. At level 1 and 3/2 there exist three more higher nonlocal conservation
laws, of which we only shall present here the x-components:
(p1,5 )x = cos(2p0,1 )(wq 1 ,1 q 1 ,3 + wq 1 ,2 q 1 ,4 + p1,3 w)
2 2 2 2

+ sin(2p0,1 )(wq 1 ,2 q 1 ,3 ’ wq 1 ,1 q 1 ,4 ’ p1,4 w)
2 2 2 2
+ 2wq 1 ,1 q 1 ,2 + •q 1 ,1 ;
2 2 2


(q 3 ,3 )x = cos(2p0,1 )(q 1 ,4 (’2p1 w + w1 ) + q 1 ,3 (u + 2w 2 ) + q 1 ,2 (’2p1,1 w)
2 2 2 2

+ q 1 ,1 (2p1,2 w + 2p1,4 w) + ψp1,4 )
2

+ sin(2p0,1 )(q 1 ,4 (’u ’ 2w 2 ) + q 1 ,3 (’2p1 w + w1 ) + q 1 ,2 (2p1,2 w)
2 2 2

+ q 1 ,1 (2p1,1 w + 2p1,3 w + ψp1,3 )
2
’ q 1 ,1 w1 + q 1 ,2 u;
2 2


(q 3 ,4 )x = cos(2p0,1 )(q 1 ,4 (’u ’ 2w 2 ) + q 1 ,3 (’2p1 w + w1 ) + q 1 ,2 (’2p1,2 w)
2 2 2 2

+ q 1 ,1 (’2p1,1 w))
2

+ sin(2p0,1 )(q 1 ,4 (2p1 w ’ w1 ) + q 1 ,3 (’u ’ 2w 2 ) + q 1 ,2 (’2p1,1 w)
2 2 2

+ q 1 ,1 (2p1,2 w))
2

+ q 1 ,1 u + q 1 ,2 w1 + ψq 1 ,1 q 1 ,2 . (7.77)
2 2 2 2

Thus, we obtained the following 16 nonlocal variables:
p0,1 , p0,2 of degree 0,
p1 , p1,1 , p1,2 , p1,3 , p1,4 , p1,5 of degree 1,
p3,1 of degree 3,
1
q 1 ,1 , q 1 ,2 , q 1 ,3 , q 1 ,4 of degree ,
2
2 2 2 2

1
q 3 ,1 , q 3 ,2 , q 3 ,3 , q 3 ,4 of degree . (7.78)
2
2 2 2 2
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 341

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.
4.3.2. Higher and nonlocal symmetries. In this subsection, we present
results for higher and nonlocal symmetries for the N = 2 supersymmetric
extension of KdV equation (7.71) in the case a = ’1,
‚ ‚ ‚ ‚
Y =Yu +Yw +Y• +Yψ + ...
‚u ‚w ‚• ‚ψ
We obtained the following odd symmetries whose generating functions are:
u
Y 1 ,1 = ’ψ1 ,
2
w
Y 1 ,1 = •,
2

Y 1 ,1 = w1 ,
2
ψ
Y 1 ,1 = ’u;
2

u
Y 1 ,2 = cos(2p0,1 )(ψ1 ’ 2•w) + sin(2p0,1 )(’•1 ’ 2ψw),
2
w
Y 1 ,2 = cos(2p0,1 )• + sin(2p0,1 )ψ,
2

Y 1 ,2 = cos(2p0,1 )(2ψq 1 ,1 + w1 ) + sin(2p0,1 )(2ψq 1 ,2 ’ u) ’ 4ψq 1 ,4 ,
2 2 2
2
ψ
Y 1 ,2 = cos(2p0,1 )(’2•q 1 ,1 + u) + sin(2p0,1 )(’2•q 1 ,2 + w1 ) + 4•q 1 ,4 ;
2 2 2
2

u
Y 1 ,3 = cos(2p0,1 )(’•1 ’ 2ψw) + sin(2p0,1 )(’ψ1 + 2•w),
2
w
Y 1 ,3 = cos(2p0,1 )ψ ’ sin(2p0,1 )•),
2

Y 1 ,3 = cos(2p0,1 )(2ψq 1 ,2 ’ u) + sin(2p0,1 )(’2ψq 1 ,1 ’ w1 ) + 4ψq 1 ,3 ,
2 2 2
2
ψ
Y 1 ,3 = cos(2p0,1 )(’2•q 1 ,2 + w1 ) + sin(2p0,1 )(2•q 1 ,1 ’ u) ’ 4•q 1 ,3 ;
2 2 2
2

u
Y 1 ,4 = •1 ,
2
w
Y 1 ,4 = ψ,
2

Y 1 ,4 = u,
2
ψ
Y 1 ,4 = w1 ;
2

u
Y 3 ,1 = cos(2p0,1 )(’2q 1 ,2 u1 ’ 2q 1 ,2 ww1 + 2q 1 ,1 uw ’ q 1 ,1 w2
2 2 2 2
2
+ ψ2 + ψ1 p1 + •1 q 1 ,1 q 1 ,2 ’ 2•1 w + 2ψq 1 ,1 q 1 ,2 w ’ ψu
2 2 2 2
2
’ ψw ’ •p1 w ’ •w1 )
+ sin(2p0,1 )(2q 1 ,2 uw ’ q 1 ,2 w2 + 2q 1 ,1 u1 + 2q 1 ,1 ww1 ’ •2
2 2 2 2
+ 2ψ1 q 1 ,1 q 1 ,2 ’ 2ψ1 w ’ •1 p1 ’ ψp1 w ’ ψw1 ’ 2•q 1 ,1 q 1 ,2 w
2 2 2 2
342 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

+ •u + •w 2 ) ’ 2q 1 ,3 u1 ’ ψ1 p1,2 ’ ψ1 p1,4 + •1 p1,1 ,
2
w
= cos(2p0,1 )(’q 1 ,1 u + •1 ’ ψq 1 ,1 q 1 ,2 + ψw)
Y 3 ,1
2 2 2
2

+ sin(2p0,1 )(’q 1 ,2 u + ψ1 ’ •w) ’ 2q 1 ,3 w1 + ψp1,1 + •p1,2 + •p1,4 ,
2 2

= cos(2p0,1 )(q 1 ,1 q 1 ,2 u + ψ1 q 1 ,1 + ψq 1 ,1 p1 ’ •q 1 ,1 w
Y 3 ,1
2 2 2 2 2
2

’ •ψ + 2uw ’ w2 )
+ sin(2p0,1 )(’ψ1 q 1 ,2 ’ 2•1 q 1 ,1 + ψq 1 ,2 p1 ’ 2ψq 1 w + •q 1 ,2 w
2 2 2 2 2

’ p1 u + u1 + ww1 ) + 2•1 q 1 ,3 ’ 2ψq 3 ,1 + p1,1 u + p1,2 w1 + p1,4 w1 ,
2 2
ψ
Y 3 ,1 = cos(2p0,1 )(’q 1 ,1 q 1 ,2 w1 + 2ψ1 q 1 ,2 + •1 q 1 ,1 + ψq 1 ,1 w
2 2 2 2 2
2

’ 2•q 1 ,2 w ’ •q 1 ,1 p1 + p1 u ’ u1 ’ ww1 )
2 2

+ sin(2p0,1 )(2q 1 ,1 q 1 ,2 u ’ •1 q 1 ,2 ’ ψq 1 ,2 w ’ •q 1 ,2 p1 ’ •ψ + 2uw ’ w2 )
2 2 2 2 2

+ 2ψ1 q 1 ,3 + 2•q 3 ,1 + p1,1 w1 ’ p1,2 u ’ p1,4 u. (7.79)
2 2


We also have
u
Y 3 ,2 = cos(2p0,1 )(’2q 1 ,2 uw + q 1 ,2 w2 ’ 2q 1 ,1 u1 ’ 2q 1 ,1 ww1 + •2
2 2 2 2
2
’ 2ψ1 q 1 ,1 q 1 ,2 + 2ψ1 w + •1 p1 + ψp1 w + ψw1 + 2•q 1 ,1 q 1 ,2 w
2 2 2 2
2
’ •u ’ •w )
+ sin(2p0,1 )(’2q 1 ,2 u1 ’ 2q 1 ,2 ww1 + 2q 1 ,1 uw ’ q 1 ,1 w2 + ψ2 + ψ1 p1
2 2 2 2
2
+ •1 q 1 ,1 q 1 ,2 ’ 2•1 w + 2ψq 1 ,1 q 1 ,2 w ’ ψu ’ ψw ’ •p1 w ’ •w1 )
2 2 2 2
+ 2q 1 ,4 u1 ’ ψ1 p1,1 ’ ψ1 p1,3 ’ •1 p1,2 ,
2
w
= cos(2p0,1 )(q 1 ,2 u ’ ψ1 + •w)
Y 3 ,2
2
2

+ sin(2p0,1 )(’q 1 ,1 u + •1 ’ ψq 1 ,1 q 1 ,2 + ψw)
2 2 2
+ 2q 1 ,4 w1 ’ ψp1,2 + •p1,1 + •p1,3 ,
2

= cos(2p0,1 )(ψ1 q 1 ,2 + 2•1 q 1 ,1 ’ ψq 1 ,2 p1 + 2ψq 1 ,1 w ’ •q 1 ,2 w
Y 3 ,2
2 2 2 2 2
2

+ p1 u ’ u1 ’ ww1 )
+ sin(2p0,1 )(q 1 ,1 q 1 ,2 u + ψ1 q 1 ,1 + ψq 1 ,1 p1 ’ •q 1 ,1 w ’ •ψ + 2uw ’ w2 )
2 2 2 2 2
’ 2•1 q 1 ,4 ’ 2ψq 3 ,2 + p1,1 w1 ’ p1,2 u + p1,3 w1 ,
2 2
ψ
Y 3 ,2 = cos(2p0,1 )(’2q 1 ,1 q 1 ,2 u + •1 q 1 ,2 + ψq 1 ,2 w + •q 1 ,2 p1
2 2 2 2 2
2

+ •ψ ’ 2uw + w2 )
+ sin(2p0,1 )(’q 1 ,1 q 1 ,2 w1 + 2ψ1 q 1 ,2 + •1 q 1 ,1 + ψq 1 ,1 w ’ 2•q 1 ,2 w
2 2 2 2 2 2

’ •q 1 ,1 p1 + p1 u ’ u1 ’ ww1 )
2

’ 2ψ1 q 1 ,4 + 2•q 3 ,2 ’ p1,1 u ’ p1,2 w1 ’ p1,3 u (7.80)
2 2
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 343

and

u
Y 3 ,3 = cos(2p0,1 )(’4q 1 ,4 uw + 2q 1 ,4 w2 + 2q 1 ,3 u1 + 4q 1 ,3 ww1
2 2 2 2
2
’ 2ψ1 q 1 ,1 q 1 ,4 + ψ1 p1,2 + ψ1 p1,4 ’ 2•1 q 1 ,2 q 1 ,4 ’ 4•1 q 1 ,1 q 1 ,3 + •1 p1,1
2 2 2 2 2 2
+ •1 p1,3 ’ 4ψq 1 ,1 q 1 ,3 w + 2ψp1,1 w + 2ψp1,3 w + 4•q 1 ,1 q 1 ,4 w
2 2 2 2

’ 2•p1,2 w ’ 2•p1,4 w)
+ sin(2p0,1 )(’2q 1 ,4 u1 ’ 4q 1 ,4 ww1 ’ 4q 1 ,3 uw + 2q 1 ,3 w2
2 2 2 2
’ 2ψ1 q 1 ,1 q 1 ,3 + ψ1 p1,1 + ψ1 p1,3 ’ 2•1 q 1 ,2 q 1 ,3 + 4•1 q 1 ,1 q 1 ,4
2 2 2 2 2 2
’ •1 p1,2 ’ •1 p1,4 + 4ψq 1 ,1 q 1 ,4 w ’ 2ψp1,2 w ’ 2ψp1,4 w
2 2

+ 4•q 1 ,1 q 1 ,3 w ’ 2•p1,1 w ’ 2•p1,3 w)
2 2
+ 2q 1 ,2 u1 + 2q 1 ,2 ww1
2 2
+ 2q 1 ,1 uw ’ q 1 ,1 w2 ’ ψ2 ’ ψ1 p1 ’ 4•1 q 1 ,3 q 1 ,4
2 2 2 2
2
’ •1 q 1 ,1 q 1 ,2 + 2•1 w + ψu + ψw + •p1 w + •w1 ,
2 2
w
= cos(2p0,1 )(2q 1 ,4 u ’ 2q 1 ,3 w1 ’ 2ψq 1 ,2 q 1 ,4 ’ ψp1,1 ’ ψp1,3
Y 3 ,3
2 2 2 2
2

’ 2•q 1 ,1 q 1 ,4 + •p1,2 + •p1,4 )
2 2

+ sin(2p0,1 )(2q 1 ,4 w1 + 2q 1 ,3 u ’ 2ψq 1 ,2 q 1 ,3 + ψp1,2 + ψp1,4
2 2 2 2

’ 2•q 1 ,1 q 1 ,3 + •p1,1 + •p1,3 )
2 2
’ q 1 ,1 u + •1 ’ 4ψq 1 ,3 q 1 ,4 ’ ψq 1 ,1 q 1 ,2 + ψw,
2 2 2 2 2

= cos(2p0,1 )(’2q 1 ,2 q 1 ,4 u ’ 2q 1 ,1 q 1 ,4 w1 ’ 4q 1 ,1 q 1 ,3 u + 2ψ1 q 1 ,4
Y 3 ,3
2 2 2 2 2 2 2
2
+ 2•1 q 1 ,3 + 2ψq 1 ,4 p1 ’ 2ψq 1 ,2 p1,1 ’ 2ψq 1 ,2 p1,3 ’ 4ψq 1 ,1 q 1 ,2 q 1 ,3
2 2 2 2 2 2 2
+ 2ψq 1 ,1 p1,2 + 2ψq 1 ,1 p1,4 ’ 2•q 1 ,4 w + p1,1 u
2 2 2

+ p1,2 w1 + p1,3 u + p1,4 w1 )
+ sin(2p0,1 )(’2q 1 ,2 q 1 ,3 u + 4q 1 ,1 q 1 ,4 u ’ 2q 1 ,1 q 1 ,3 w1 + 2ψ1 q 1 ,3
2 2 2 2 2 2 2
’ 2•1 q 1 ,4 + 2ψq 1 ,3 p1 + 2ψq 1 ,2 p1,2 + 2ψq 1 ,2 p1,4
2 2 2 2
+ 4ψq 1 ,1 q 1 ,2 q 1 ,4 + 2ψq 1 ,1 p1,1 + 2ψq 1 ,1 p1,3 ’ 2•q 1 ,3 w + p1,1 w1
2 2 2 2 2 2

’ p1,2 u + p1,3 w1 ’ p1,4 u)
’ 4q 1 ,3 q 1 ,4 u ’ q 1 ,1 q 1 ,2 u ’ ψ1 q 1 ,1 ’ 4ψq 1 ,4 p1,2 ’ 4ψq 1 ,4 p1,4
2 2 2 2 2 2 2
’ 4ψq 1 ,3 p1,1 ’ 4ψq 1 ,3 p1,3 ’ ψq 1 ,1 p1 + •q 1 ,1 w ’ •ψ + 2uw ’ w2 ,
2 2 2 2
ψ
= cos(2p0,1 )(’2q 1 ,2 q 1 ,4 w1 ’ 2q 1 ,1 q 1 ,4 u ’ 2ψ1 q 1 ,3 + 2•1 q 1 ,4
Y 3 ,3
2 2 2 2 2 2
2
+ 2ψq 1 ,4 w ’ 2•q 1 ,4 p1 + 4•q 1 ,3 w + 2•q 1 ,2 p1,1 + 2•q 1 ,2 p1,3
2 2 2 2 2
+ 4•q 1 ,1 q 1 ,2 q 1 ,3 ’ 2•q 1 ,1 p1,2 ’ 2•q 1 ,1 p1,4 ’ p1,1 w1 + p1,2 u
2 2 2 2 2

’ p1,3 w1 + p1,4 u)
344 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

+ sin(2p0,1 )(’2q 1 ,2 q 1 ,3 w1 ’ 2q 1 ,1 q 1 ,3 u + 2ψ1 q 1 ,4 + 2•1 q 1 ,3
2 2 2 2 2 2
+ 2ψq 1 ,3 w ’ 4•q 1 ,4 w ’ 2•q 1 ,3 p1 ’ 2•q 1 ,2 p1,2 ’ 2•q 1 ,2 p1,4
2 2 2 2 2
’ 4•q 1 ,1 q 1 ,2 q 1 ,4 ’ 2•q 1 ,1 p1,1 ’ 2•q 1 ,1 p1,3 + p1,1 u
2 2 2 2 2

+ p1,2 w1 + p1,3 u + p1,4 w1 )
’ 4q 1 ,3 q 1 ,4 w1 ’ q 1 ,1 q 1 ,2 w1 ’ 2ψ1 q 1 ,2 ’ •1 q 1 ,1 ’ ψq 1 ,1 w
2 2 2 2 2 2 2
+ 4•q 1 ,4 p1,2 + 4•q 1 ,4 p1,4 + 4•q 1 ,3 p1,1 + 4•q 1 ,3 p1,3 + 2•q 1 ,2 w
2 2 2 2 2

+ •q 1 ,1 p1 ’ p1 u + u1 + ww1 , (7.81)
2


together with
u
Y 3 ,4 = cos(2p0,1 )(2q 1 ,4 u1 + 4q 1 ,4 ww1 + 4q 1 ,3 uw ’ 2q 1 ,3 w2 ’ 4ψ1 q 1 ,2 q 1 ,4
2 2 2 2 2 2
2
’ 2ψ1 q 1 ,1 q 1 ,3 + ψ1 p1,1 ’ 2•1 q 1 ,2 q 1 ,3 ’ •1 p1,2 ’ 4ψq 1 ,2 q 1 ,3 w
2 2 2 2 2 2

’ 2ψp1,2 w + 4•q 1 ,2 q 1 ,4 w ’ 2•p1,1 w)
2 2

+ sin(2p0,1 )(’4q 1 ,4 uw + 2q 1 ,4 w2 + 2q 1 ,3 u1 + 4q 1 ,3 ww1 ’ 4ψ1 q 1 ,2 q 1 ,3
2 2 2 2 2 2
+ 2ψ1 q 1 ,1 q 1 ,4 ’ ψ1 p1,2 + 2•1 q 1 ,2 q 1 ,4 ’ •1 p1,1 + 4ψq 1 ,2 q 1 ,4 w
2 2 2 2 2 2

’ 2ψp1,1 w + 4•q 1 ,2 q 1 ,3 w + 2•p1,2 w)
2 2
+ 2q 1 ,2 uw ’ q 1 ,2 w2 ’ 2q 1 ,1 u1 ’ 2q 1 ,1 ww1 + •2 ’ 4ψ1 q 1 ,3 q 1 ,4
2 2 2 2 2 2
’ 2ψ1 q 1 ,1 q 1 ,2 + 2ψ1 w + •1 p1 + ψp1 w + ψw1 + 2•q 1 ,1 q 1 ,2 w
2 2 2 2
2
’ •u ’ •w ,
w
Y 3 ,4 = cos(2p0,1 )(’2q 1 ,4 w1 ’ 2q 1 ,3 u + 2ψq 1 ,2 q 1 ,3 + ψp1,2
2 2 2 2
2

+ 2•q 1 ,1 q 1 ,3 + •p1,1 )
2 2

+ sin(2p0,1 )(2q 1 ,4 u ’ 2q 1 ,3 w1 ’ 2ψq 1 ,2 q 1 ,4 + ψp1,1
2 2 2 2

’ 2•q 1 ,1 q 1 ,4 ’ •p1,2 )
2 2
’ q 1 ,2 u + ψ1 + 4•q 1 ,3 q 1 ,4 ’ •w,
2 2 2

= cos(2p0,1 )(’2q 1 ,2 q 1 ,3 u + 2q 1 ,1 q 1 ,3 w1 + 2ψ1 q 1 ,3 ’ 2•1 q 1 ,4
Y 3 ,4
2 2 2 2 2 2
2
’ 4ψq 1 ,4 w ’ 2ψq 1 ,3 p1 + 2ψq 1 ,2 p1,2 + 2ψq 1 ,1 p1,1 ’ 2•q 1 ,3 w
2 2 2 2 2

+ p1,1 w1 ’ p1,2 u)
+ sin(2p0,1 )(2q 1 ,2 q 1 ,4 u ’ 2q 1 ,1 q 1 ,4 w1 ’ 2ψ1 q 1 ,4 ’ 2•1 q 1 ,3 + 2ψq 1 ,4 p1
2 2 2 2 2 2 2

’ 4ψq 1 ,3 w + 2ψq 1 ,2 p1,1 ’ 2ψq 1 ,1 p1,2 + 2•q 1 ,4 w ’ p1,1 u ’ p1,2 w1 )
2 2 2 2
+ 4q 1 ,3 q 1 ,4 w1 + ψ1 q 1 ,2 + 2•1 q 1 ,1 ’ 4ψq 1 ,4 p1,1 + 4ψq 1 ,3 p1,2 ’ ψq 1 ,2 p1
2 2 2 2 2 2 2
+ 2ψq 1 ,1 w ’ •q 1 ,2 w + p1 u ’ u1 ’ ww1 ,
2 2
ψ
= cos(2p0,1 )(’4q 1 ,2 q 1 ,4 u + 2q 1 ,2 q 1 ,3 w1 ’ 2q 1 ,1 q 1 ,3 u + 2ψ1 q 1 ,4
Y 3 ,4
2 2 2 2 2 2 2
2
+ 2•1 q 1 ,3 + 2ψq 1 ,3 w + 2•q 1 ,3 p1 ’ 2•q 1 ,2 p1,2 ’ 2•q 1 ,1 p1,1
2 2 2 2 2
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 345

+ p1,1 u + p1,2 w1 )
+ sin(2p0,1 )(’2q 1 ,2 q 1 ,4 w1 ’ 4q 1 ,2 q 1 ,3 u + 2q 1 ,1 q 1 ,4 u + 2ψ1 q 1 ,3
2 2 2 2 2 2 2
’ 2•1 q 1 ,4 ’ 2ψq 1 ,4 w ’ 2•q 1 ,4 p1 ’ 2•q 1 ,2 p1,1 + 2•q 1 ,1 p1,2
2 2 2 2 2

+ p1,1 w1 ’ p1,2 u)
’ 4q 1 ,3 q 1 ,4 u ’ 2q 1 ,1 q 1 ,2 u + •1 q 1 ,2 + ψq 1 ,2 w + 4•q 1 ,4 p1,1
2 2 2 2 2 2 2

’ 4•q 1 ,3 p1,2 + •q 1 ,2 p1 ’ •ψ + 2uw ’ w2 . (7.82)
2 2


Even symmetries are
u
Y1,1 = u1 ,
w
Y1,1 = w1 ,

Y1,1 = •1 ,
ψ
Y1,1 = ψ1 ;
u
Y1,2 = cos(2p0,1 )(’ψ1 q 1 ,1 ’ •1 q 1 ,2 + 2•q 1 ,1 w ’ 2uw + w2 )
2 2 2

+ sin(2p0,1 )(2•1 q 1 ,1 + 2ψq 1 ,1 w ’ u1 ’ 2ww1 ) ’ 2•1 q 1 ,3 ,
2 2 2
w
= cos(2p0,1 )(’ψq 1 ,2 ’ •q 1 ,1 + u)
Y1,2
2 2

+ sin(2p0,1 )w1 ’ 2ψq 1 ,3 ,
2

= cos(2p0,1 )(q 1 ,2 u + q 1 ,1 w1 ’ ψ1 ’ ψp1 + •w)
Y1,2
2 2

+ sin(2p0,1 )(’2q 1 ,1 u + •1 ’ 2ψq 1 ,1 q 1 ,2 ) + 2(q 1 ,3 u + ψp1,2 + ψp1,4 ),
2 2 2 2
ψ
= cos(2p0,1 )(q 1 ,2 w1 + q 1 ,1 u ’ •1 ’ ψw + •p1 )
Y1,2
2 2

+ sin(2p0,1 )(’ψ1 + 2•q 1 ,1 q 1 ,2 + 2•w) + 2(q 1 ,3 w1 ’ •p1,2 ’ •p1,4 );
2 2 2

u
Y1,3 = cos(2p0,1 )(2•1 q 1 ,1 + 2ψq 1 w ’ u1 ’ 2ww1 )
2 2

+ sin(2p0,1 )(ψ1 q 1 ,1 + •1 q 1 ,2 ’ 2•q 1 ,1 w + 2uw ’ w2 ) ’ 2•1 q 1 ,4 ,
2 2 2 2
w
Y1,3 = cos(2p0,1 )w1
+ sin(2p0,1 )(ψq 1 + •q 1 ,1 ’ u) ’ 2ψq 1 ,4 ,
2 2 2

= cos(2p0,1 )(’2q 1 ,1 u + •1 ’ 2ψq 1 ,1 q 1 ,2 )
Y1,3
2 2 2

+ sin(2p0,1 )(’q 1 ,2 u ’ q 1 ,1 w1 + ψ1 + ψp1 ’ •w)
2 2

+ 2(q 1 ,4 u ’ ψp1,1 ’ ψp1,3 ),
2
ψ
Y1,3 = cos(2p0,1 )(’ψ1 + 2•q 1 ,1 q 1 ,2 + 2•w)
2 2

+ sin(2p0,1 )(’q 1 ,2 w1 ’ q 1 ,1 u + •1 + ψw ’ •p1 )
2 2

+ 2(q 1 ,4 w1 + •p1,1 + •p1,3 ); (7.83)
2

u
Y1,4 = cos(2p0,1 )(’2ψ1 q 1 ,2 + 2•q 1 ,2 w + u1 + 2ww1 )
2 2
346 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

+ sin(2p0,1 )(ψ1 q 1 ,1 + •1 q 1 ,2 + 2ψq 1 ,2 w ’ 2uw + w2 ) ’ 2ψ1 q 1 ,3 ,
2 2 2 2
w
= ’ cos(2p0,1 )w1
Y1,4
+ sin(2p0,1 )(’ψq 1 ,2 ’ •q 1 ,1 + u) + 2•q 1 ,3 ,
2 2 2

Y1,4 = cos(2p0,1 )(•1 + 2ψw)
+ sin(2p0,1 )(’q 1 ,2 u + q 1 ,1 w1 + ψ1 ’ ψp1 ’ •w)
2 2

+ 2(’q 1 ,3 w1 + ψp1,1 ),
2
ψ
Y1,4 = cos(2p0,1 )(2q 1 ,2 u ’ ψ1 )
2

+ sin(2p0,1 )(q 1 ,2 w1 ’ q 1 ,1 u + •1 + ψw + •p1 ) + 2(q 1 ,3 u ’ •p1,1 ).
2 2 2
(7.84)

Finally, we got

u
Y1,5 = cos(2p0,1 )(ψ1 q 1 ,1 + •1 q 1 ,2 + 2ψq 1 ,2 w ’ 2uw + w2 )
2 2 2

+ sin(2p0,1 )(2ψ1 q 1 ,2 ’ 2•q 1 ,2 w ’ u1 ’ 2ww1 ) ’ 2ψ1 q 1 ,4 ,
2 2 2
w
= cos(2p0,1 )(’ψq 1 ,2 ’ •q 1 ,1 + u)
Y1,5
2 2

+ sin(2p0,1 )w1 + 2•q 1 ,4 ,
2

= cos(2p0,1 )(’q 1 ,2 u + q 1 ,1 w1 + ψ1 ’ ψp1 ’ •w)
Y1,5
2 2

+ sin(2p0,1 )(’•1 ’ 2ψw) + 2(’q 1 ,4 w1 + ψp1,2 ),
2
ψ
= cos(2p0,1 )(q 1 ,2 w1 ’ q 1 ,1 u + •1 + ψw + •p1 )
Y1,5
2 2

+ sin(2p0,1 )(’2q 1 ,2 u + ψ1 ) + 2(q 1 ,4 u ’ •)p1,2 );
2 2

u
Y1,6 = cos(2p0,1 )(’ψ1 q 1 ,3 + •1 q 1 ,4 + 2ψq 1 ,4 w + 2•q 1 ,3 w)
2 2 2 2

+ sin(2p0,1 )(ψ1 q 1 ,4 + •1 q 1 ,3 + 2ψq 1 ,3 w ’ 2•q 1 ,4 w)
2 2 2 2
’ ψ1 q 1 ,2 ’ •1 q 1 ,1 ’ ψq 1 ,1 w + •q 1 ,2 w,
2 2 2 2
w
= ’ cos(2p0,1 )(ψq 1 ,4 + •q 1 ,3 )
Y1,6
2 2

+ sin(2p0,1 )(’ψq 1 ,3 + •q 1 ,4 ),
2 2

Y1,6 = cos(2p0,1 )(’q 1 ,4 u + q 1 ,3 w1 + 2ψq 1 ,2 q 1 ,4 + 2ψq 1 ,1 q 1 ,3 )
2 2 2 2 2 2

+ sin(2p0,1 )(’q 1 ,4 w1 ’ q 1 ,3 u + 2ψq 1 ,2 q 1 ,3 ’ 2ψq 1 ,1 q 1 ,4 )
2 2 2 2 2 2
+ q 1 ,1 u ’ •1 + 4ψq 1 ,3 q 1 ,4 + ψq 1 ,1 q 1 ,2 ’ ψw,
2 2 2 2 2
ψ
Y1,6 = cos(2p0,1 )(q 1 ,4 w1 + q 1 ,3 u ’ 2•q 1 ,2 q 1 ,4 ’ 2•q 1 ,1 q 1 ,3 )
2 2 2 2 2 2

+ sin(2p0,1 )(’q 1 ,4 u + q 1 ,3 w1 ’ 2•q 1 ,2 q 1 ,3 + 2•q 1 ,1 q 1 ,4 )
2 2 2 2 2 2

+ q 1 ,2 u ’ ψ1 ’ 4•q 1 ,3 q 1 ,4 ’ •q 1 ,1 q 1 ,2 + •w. (7.85)
2 2 2 2 2
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 347

4.3.3. Recursion operator. Here we shall discuss brie¬‚y the recursion
properties of the nonlocal symmetries Y1,2 , Y1,3 , Y1,4 , Y1,5 , Y1,6 given in
(7.83) and (7.85).
We shall discuss their action on the supersymmetry Y 1 ,1 of degree 1/2.
2
In order to compute the Lie bracket of these symmetries, we have to
derive the nonlocal components, just for the vector ¬eld Y 1 ,1 .
2
Due to the invariance of the equations, de¬ning the nonlocal variables
p0,1 , p1 , q 1 ,1 , q 1 ,2 , q 1 ,3 , q 1 ,4 and p1,1 , p1,2 , p1,3 , p1,4 , the nonlocal components
2 2 2 2
can be obtained.
The prolongation of the vector ¬eld Y 1 ,1 is then given as
2


‚ ‚ ‚ ‚
Y 1 ,1 = ’ψ1 ’u
+• + w1
‚u ‚w ‚• ‚ψ
2

‚ ‚ ‚ ‚
’ p1 ’ (p1,1 + p1,3 )
+w + (p1,2 + p1,4 )
‚q 1 ,1 ‚q 1 ,2 ‚q 1 ,3 ‚q 1 ,4
2 2 2 2
‚ ‚
’ψ
+ q 1 ,1
‚p0,1 ‚p1
2


+ (cos(2p0,1 )(2q 1 ,1 p1 + q 1 ,2 w) + sin(2p0,1 )(2q 1 ,2 p1 ) ’ 2q 3 ,1 )
‚p1,1
2 2 2 2


+ (cos(2p0,1 )(2q 1 ,2 p1 ) ’ sin(2p0,1 )(2q 1 ,1 p1 + q 1 ,2 w) + 2q 3 ,2 )
‚p1,2
2 2 2 2

+ (’ cos(2p0,1 )(2q 1 ,1 p1 + 2q 1 ,2 w) + sin(2p0,1 (q 1 ,1 w ’ q 1 ,2 p1 )
2 2 2 2

+ 2q 3 ,1 )
‚p1,3
2

+ (cos(2p0,1 )(q 1 ,1 w ’ q 1 ,2 p1 ) + sin(2p0,1 )(2q 1 ,1 p1 + 2q 1 ,2 w)
2 2 2 2

’ 2q 3 ,1 ) . (7.86)
‚p1,4
2


For the vector ¬elds Y1,i , i = 2, . . . , 6, prolongation is not required due to
the locality of Y 1 ,1 .
2
We obtain the following commutators:

[Y1,2 , Y 1 ,1 ] = 0,
2

[Y1,3 , Y 1 ,1 ] = 0,
2

[Y1,4 , Y 1 ,1 ] = 2Y 3 ,1 ,
2 2

[Y1,5 , Y 1 ,1 ] = ’2Y 3 ,2 ,
2 2

[Y1,6 , Y 1 ,1 ] = ’2Y 3 ,3 , (7.87)
2 2


meaning that Y1,i , i = 2, . . . , 6, take symmetry Y 1 ,1 higher into the hierarchy.
2
Similar results are obtained for the local symmetry Y 1 ,4 .
2
348 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

In order to compute the Lie brackets for Y 1 ,2 and Y 1 ,3 , leading to similar
2 2
results too, prolongations of these vector ¬elds are required.
The results are related to a similar action of the recursion symmetry for
the n = 1 supersymmetric KdV equation, discussed in Section 1.
The work on the contruction of the recursion operator as obtained for
the cases a = ’2 by (7.56) and a = 4 by (7.70), is still in progress and will
be published elsewhere.
CHAPTER 8


Symbolic computations in di¬erential geometry

To introduce this subject, it is nice to tell the story how NN computed
the tenth conservation law of the classical KdV equation at the end of the
sixties.
From previous results one had obtained nine conservation laws for the
KdV equation and the idea was that if one would be able to compute the
tenth then people would be convinced that there existed an in¬nite hierarchy
of conservation laws for the KdV equation. At that time, the notion of
recursion operators (the ¬rst one obtained by Lenard) was not yet known.
Then NN took the decision to retire for two weeks to a nice cabin some-
where high up in the mountains and to try to ¬gure out whether he would be
able to ¬nd number ten. After two weeks he returned from his exile position
having found the next one in the hierarchy, thus “proving” the existence of
an in¬nite hierarchy.
With nowadays modern facilities it is possible to construct the ¬rst ten
or twenty in few seconds. This is just one of the examples demonstrating
the need for computer programs to do in principle simple, but in e¬ect huge
algebraic computations to get to ¬nal results.
Towards the end of the seventies the ¬rst computer programs were con-
structed. Among them Gragert [22], Schr¨fer [90], Schwarz [91], Kersten
u
[34], . . . , just doing part of the work on computations on di¬erential forms,
vector ¬elds, solutions of overdetermined systems of partial di¬erential equa-
tions, covering conditions, etc.
Since then, quite a number of programs has been constructed and it
seems that nowadays each individual researcher in this ¬eld of mathematical
physics uses his or her own developed software to do the required computa-
tions in more or less the most or almost most e¬cient way. An overview of
existing programs in all distinct related areas was recently given by Hereman
in his extensive paper [30].
In the following sections, we shall discuss in some detail a number of
types of computations which can be carried through on a computer sys-
tem. The basis of these programs has been constructed by Gragert [22],
Kersten [37], Gragert, Kersten and Martini [24, 25], Roelofs [85, 86], van
Bemmelen [9, 8] at the University of Twente, starting in 1979 with exterior
di¬erential forms, construction and solution of overdetermined systems of
partial di¬erential equations arising from symmetry computations, exten-
sion of the software to work in a graded setting, meaning supercalculus,

349
350 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

required for the interesting ¬eld of super and supersymmetric extensions of
classical di¬erential equations and at the end a completely new package,
being extremely suitable for classical as well as supersymmetrical systems,
together with packages for computation of covering structures of completely
integrable systems, and a package to handling the computations with to-
tal derivative operators. We should mention here too (super) Lie algebra
computations for covering structures by Gragert and Roelofs [23, 26].
We prefer to start in Section 1 with setting down the basic notions of the
graded or supercalculus, since classical di¬erential geometric computations
can be embedded in a very e¬ective way in this more general setting, which
will be done in Section 2.
In Section 3 we shall give an idea how the software concerning construc-
tion of solutions of overdetermined systems of partial di¬erential equations
works, and what the facilities are.
Finally we shall present in Subsection 3.2 a computer session concerning
the construction of higher symmetries of third order of the Burgers equation,
i.e., de¬ning functions involving derivatives (with respect to x up to order 3),
cf. Chapter 2.

1. Super (graded) calculus
We give here a concise exposition of super (or graded) calculus needed
for symbolic computations.
At ¬rst sight the introduction of graded calculus requires a completely
new set of de¬nitions and objects. It has been shown that locally a graded
manifold, or equivalently the algebra of functions de¬ned on it, is given as
C ∞ (U ) — Λ(n), where Λ(n) is the exterior algebra of n (odd) variables, [50].
Below we shall set down the notions involved in the graded calculus and
graded di¬erential geometry.
Thus we give a short review of the notions of graded di¬erential ge-
ometry as far as they are needed for implementation by means of software
procedures, i.e., graded commutative algebra, graded Lie algebra, graded man-
ifold, graded derivation, graded vector ¬eld, graded di¬erential form, exterior
di¬erentiation, inner di¬erentiation or contraction by a vector ¬eld, Lie de-
rivative along a vector ¬eld, etc.
The notions and notations have been taken from Kostant [50] and the
reader is referred to this paper for more details, compare with Chapter 6.
Throughout this section, the basic ¬eld is R or C and the grading will be
with respect to Z2 = {0, 1}.
1. A vector space V over R is a graded vector space if one has V0 and V1
subspaces of V , such that
V = V0 • V1 (8.1)
is a direct sum. Elements of V0 are called even, elements of V1 are
called odd. Elements of V0 or V1 are called homogeneous elements.
1. SUPER (GRADED) CALCULUS 351

If v ∈ Vi , i = 0, 1, then i is called the degree of v, i.e.,
|v| = i, i = 0, 1, or i ∈ Z2 . (8.2)
The notation |v| is used for homogeneous elements only.
2. A graded algebra B is a graded vector space B = B0 • B1 with a
multiplication such that
Bi · Bj ‚ Bi+j , i, j ∈ Z2 . (8.3)
3. A graded algebra B is called graded commutative if for any two ho-
mogeneous elements x, y ∈ B we have
xy = (’1)|x||y| yx. (8.4)
4. A graded space V is a left module over the graded algebra B, if V is
a left module in the usual sense and
Bi · Vj ‚ Vi+j , i, j ∈ Z2 ; (8.5)
right modules are de¬ned similarly.
5. If V is a left module over the graded commutative algebra B, then V
inherits a right module structure, where we de¬ne
def
v · b = (’1)|v||b| b · v, v ∈ V, b ∈ B. (8.6)
Similarly, a left module structure is determined by a right module
structure.
6. A graded vector space g = g0 • g1 , together with a bilinear operation
[·, ·] on g such that [x, y] ∈ g|x|+|y| is called a graded Lie algebra if
[x, y] = ’(’1)|x||y| [y, x],
(’1)|x||z| [x, [y, z]] + (’1)|z||y| [z, [x, y]] + (’1)|y||x| [y, [z, x]] = 0, (8.7)
where the last equality is called the graded Jacobi identity.
If V is a graded vector space, then End(V ) has the structure of a
graded Lie algebra de¬ned by
[±, β] = ±β ’ (’1)|±||β| β±, ±, β ∈ End(V ). (8.8)
7. If B is a graded algebra, an operator h ∈ Endi (B) is called a graded
derivation of B if
h(xy) = h(x)y + (’1)|i||x| xh(y). (8.9)
An operator h ∈ End(B) is a derivation if its homogeneous compo-
nents are so.
The graded vector space of derivations of B, denoted by Der(B),
is a graded Lie subalgebra of End(B). Equality (8.9) is called graded
Leibniz rule. If B is a graded commutative algebra then Der(B) is a
left B-module: if ζ ∈ Der(B), f, g ∈ B, then f ζ ∈ Der(B), where
(f ζ)g = f (ζg). (8.10)
352 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

8. The local picture of a graded manifold is an open neighborhood U ‚
Rm together with the graded commutative algebra
C ∞ (U ) — Λ(n), (8.11)
where Λ(n) is the antisymmetric (exterior) algebra on n elements
|si | = 1, si sj = ’sj si ,
s1 , . . . , sn , i, j = 1, . . . , n. (8.12)
The pair (m|n) is called the dimension of the graded manifold at
hand. A particular element f ∈ C ∞ (U ) — Λ(n) is represented as
f= fµ s µ , (8.13)
µ

where µ is a multi-index: µ ∈ Mn = {µ = (µ1 , . . . , µn ) | µi ∈ N, 1 ¤
µ1 ¤ µ2 · · · ¤ µk ¤ n},
fµ ∈ C ∞ (U ).
s µ = s µ1 · s µ2 · . . . · s µk , (8.14)
9. Graded vector ¬elds on a graded manifold (U, C ∞ (U ) — Λ(n)) are
introduced as graded derivations of the algebra C ∞ (U ) — Λ(n). They
constitute a left C ∞ (U ) — Λ(n)-module. Locally, a graded vector ¬eld
V is represented as
m n
‚ ‚
V= fi + gj , (8.15)
‚ri ‚sj
i=1 j=1

where fi , gj ∈ C ∞ (U ) — Λ(n), and ri , i = 1, . . . , m, are local coordi-
nates in U ‚ Rm .
The derivations ‚/‚ri , i = 1, . . . , m, are even, while the deriva-
tions ‚/‚sj , j = 1, . . . , n, are odd. They satisfy the relations
‚sj
‚rk ‚rk ‚sl
= δik , = 0, = 0, = δjl (8.16)
‚ri ‚ri ‚sj ‚sj
for all i, k = 1, . . . , m, j, l = 1, . . . , n.
10. A graded di¬erential k-form is introduced as k-linear mapping β on
Der(C ∞ (U ) — Λ(n)) which has to satisfy the identities
Pl’1
ζ1 , . . . , f ζl , . . . , ζk | β = (’1)|f | |ζi |
f ζ 1 , . . . , ζk | β (8.17)
i=1


and

ζ1 , . . . , ζj , ζj+1 . . . , ζk | β
= (’1)1+|ζj ||ζj+1 | ζ1 , . . . , ζj , ζj+1 , . . . , ζk | β , (8.18)
for all ζi ∈ Der(C ∞ (U ) — Λ(n)) and f ∈ C ∞ (U ) — Λ(n). The set of
k-forms is denoted by „¦k (U ).
Remark 8.1. Actually we have to write „¦k (U, C ∞ (U ) — Λ(n)),
but we made our choice for the abbreviated „¦k (U ).
1. SUPER (GRADED) CALCULUS 353

The set „¦k (U ) has the structure of a right C ∞ (U ) — Λ(n)-module
by
ζ1 , . . . , ζk | βf = ζ1 , . . . , ζk | β f. (8.19)
We also set „¦0 (U ) = C ∞ (U ) — Λ(n) and „¦(U ) = •∞ „¦k (U ).
k=0
Moreover „¦(U ) can be given a structure of a bigraded (Z+ , Z2 )-com-
mutative algebra, that is, if βi ∈ „¦ki (U )ji , i = 1, 2, then
β1 β2 ∈ „¦k1 +k2 (U )j1 +j2 (8.20)
and
β1 β2 = (’1)k1 k2 +j1 j2 β2 β1 . (8.21)
For the general de¬nition of β1 β2 see [50].
11. One de¬nes the exterior derivative (or de Rham di¬erential )
d : „¦0 (U ) ’ „¦1 (U ), f ’ df, (8.22)
by the condition
ζ | df = ζf (8.23)
for ζ ∈ Der(C ∞ (U ) — Λ(n)) and f ∈ „¦0 (U ) = C ∞ (U ) — Λ(n). By
[50] and the de¬nition of β1 β2 ,
dri , i = 1, . . . , m, dsj , j = 1, . . . , n, (8.24)
de¬ned by
‚ ‚
| dri = δik , | dri = 0,
‚rk ‚sj
‚ ‚
| dsl = 0, | dsl = δjl , (8.25)
‚rk ‚sj
generate „¦(U ) and any β ∈ „¦(U ) can be uniquely written as
drµ dsν fµ,ν ,
β= (8.26)
µ,ν

where
1 ¤ µ1 ¤ . . . ¤ µk ¤ n,
µ = (µ1 , . . . , µk ), l(µ) = k,
νi ∈ N = Z+ \ {0},
ν = (ν1 , . . . , νn ),
n
fµ,ν ∈ C ∞ (U ) — Λ(n).
|ν| = νi , (8.27)
i=1
Note in particular that by (8.21),
dri drj = ’drd ri , dri dsj = ’dsj dri , dsj dsk = dsk dsj , (8.28)
and by consequence
dsj . . . dsj = 0. (8.29)
k times
354 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

By means of (8.22) and (8.23), the operator d : „¦0 (U ) ’ „¦1 (U ) has
the following explicit representation
m n
‚f ‚f
df = dri + dsj . (8.30)
‚ri ‚sj
i=1 j=1

12. Since „¦(U ) is a (Z+ , Z2 )-bigraded commutative algebra, the algebra
End(„¦(U )) is bigraded too and if u ∈ End(„¦(U )) is of bidegree (b, j) ∈
(Z+ , Z2 ), then
u(„¦a (U )i ) ∈ „¦a+b (U )i+j . (8.31)
Now, an element u ∈ End(„¦(U )) of bidegree (b, j) is a bigraded deriva-
tion of „¦(U ), if for any ± ∈ „¦a (U )i , β ∈ „¦(U ) one has the Leibniz
rule
u(±β) = u(±)β + (’1)ab+ij ±u(β). (8.32)
There exists a unique derivation, the exterior di¬erentiation,
d : „¦(U ) ’ „¦(U ) (8.33)

of bidegree (1, 0), such that d is de¬ned by (8.22), (8.30), and
„¦0 (U )

d2 = 0. (8.34)
If β ∈ „¦(U ),

drµ dsν fµ,ν ,
β= (8.35)
µ,ν

then
(’1)l(µ)+|ν| drµ dsν dfµ,ν .
dβ = (8.36)
µ,ν

Other familiar operations on ordinary manifolds have their counter-
parts in the graded case too.
13. If ζ ∈ Der(C ∞ (U ) — Λ(n)), inner di¬erentiation by ζ, or contraction
by ζ, iζ is de¬ned by
Pb
|ζ| |ζi |
ζ1 , . . . , ζb | iζ β = (’1) ζ, ζ1 , . . . , ζb | β (8.37)
i=1



for ζ, ζ1 , . . . , ζb ∈ Der(C ∞ (U ) — Λ(n)) and β ∈ „¦b+1 (U ). Moreover
iζ : „¦(U ) ’ „¦(U ), β ∈ „¦b+1 (U ), iζ β ∈ „¦b (U ), is a derivation of
bidegree (’1, |ζ|).
Bigraded derivations on „¦(U ) can be shown to constitute a bi-
graded Lie algebra Der „¦(U ) by the following Lie bracket. If u1 , u2 ∈
Der „¦(U ) of bidegree (bi , bj ), i = 1, 2, then
[u1 , u2 ] = u1 u2 ’ (’1)b1 b2 +j1 j2 u2 u1 ∈ Der „¦(U ). (8.38)
2. CLASSICAL DIFFERENTIAL GEOMETRY 355

14. From (8.38) we have that Lie derivative by the vector ¬eld ζ de¬ned
by
Lζ = diζ + iζ d (8.39)

is a derivation of „¦(U ) of bidegree (0, |ζ|).
The fact that exterior di¬erentiation d, inner di¬erentiation by ζ,
iζ , and Lie derivative by ζ, Lζ are derivations, has been used to imple-
ment them on the computer system starting from the representation
of vector ¬elds and di¬erential forms (8.15) and (8.35).
15. If one has a graded manifold (U, C ∞ (U ) — Λ(n)), the exterior deriv-
ative is easy to be represented as an odd vector ¬eld in the following
way
m n
‚ ‚
dri § dsj §
d= + , (8.40)
‚ri ‚sj
i=1 j=1

where now the initial system has been augmented by n even variables
ds1 , . . . , dsn and m odd variables dr1 , . . . , drm . The implementation
of the supercalculus package is based on the theorem proved in [50]
that locally a supermanifold, or a graded manifold, is represented as
U, C ∞ (U ) — Λ(n), U ‚ Rn , from which it is now easy to construct
the di¬erential geometric operations.
Suppose we have a supermanifold of dimension (m|n). Local
variables are given by (r, s) = (ri , sj ), i = 1, . . . , m, j = 1, . . . , n.
Associated to these coordinates, we have (dri , dsj ), i = 1, . . . , m,
j = 1, . . . , n. We have to note that dri , i = 1, . . . , m, are odd while
dsj , j = 1, . . . , n, are even.
So the exterior algebra is
C ∞ (Rm ) — R[ds] — Λ(n) — Λ(m), (8.41)

where in (8.41) a speci¬c element is given by

dsk1 . . . dskm drµ1 . . . drµr fk,µ
f= (8.42)
m
1

while in (8.42) ki ≥ 0, i = 1, . . . , m, 1 ¤ µ1 < · · · < µr ¤ n, while
fk,µ ∈ C ∞ (Rm ) — Λ(n).


2. Classical di¬erential geometry
We shall describe here how classical di¬erential geometric objects are
realised in the graded setting of the previous section. We start at a super-
algebra A on n even elements, r1 , . . . , rn , and n odd elements s1 , . . . , sn ,
i.e.,
A = C ∞ (Rn ) — Λ(n), (8.43)
356 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

where Λ(n) is the exterior algebra on n elements, s1 , . . . , sn . A particular
element f ∈ A = C ∞ (Rn ) — Λ(n) is represented as
f= fµ s µ , (8.44)
µ

where µ is a multi-index µ ∈ Mn = {µ = (µ1 , . . . , µk ) | µi ∈ N, 1 ¤ µ1 ¤
· · · ¤ µk ¤ n} and
fµ ∈ C ∞ (Rn ),
s µ = s µ1 s µ2 . . . s µk , (8.45)
where we in e¬ect formally assume:
si = dri , i = 1, . . . , n. (8.46)
1. Functions are represented as elements of the algebra A0 = C ∞ (Rn ).
2. Derivations of A0 can be identi¬ed with vector ¬elds
‚ ‚
+ · · · + Vn
V = V1 , (8.47)
‚r1 ‚rn
where Vi ∈ C ∞ (Rn ), i = 1, . . . , n.
3. Di¬erential forms are just speci¬c elements of A.
4. Exterior derivative is a derivation of A which is odd and can be
represented as the vector ¬eld
‚ ‚
+ · · · + drn
d = dr1 , dri = si . (8.48)
‚r1 ‚rn
5. Contraction by a V , where V is given by (8.47), can be represented
as an odd derivation of A by
‚ ‚
+ · · · + Vn
V ±= V1 (±). (8.49)
‚s1 ‚sn
6. The Lie derivative by V can be easily implemented by the formula
LV (±) = V d(±) + d(V ±). (8.50)

3. Overdetermined systems of PDE
In construction of classical and higher symmetries, nonlocal symmetries
and deformations or recursion operators, one is always left with an overde-
termined system of partial di¬erential equations for a number of so-called
generating functions (or sections). The ¬nal result is obtained as the general
solution to this resulting system.
In Section 3.1 we shall describe how by the procedure which is called
here solve equation, written in the symbolic language LISP, one is able
to solve the major part of the construction of the general solution of the
overdetermined system of partial di¬erential equations resulting from the
symmetry condition (2.29) on p. 72 or the deformation condition (6.42) on
p. 266.
It should be noted that each speci¬c equation or system of equations
arising from mathematical physics has its own speci¬cs, e.g., the sine-Gordon
3. OVERDETERMINED SYSTEMS OF PDE 357

equation is not polynomial but involves the sine function, similar to the
Harry Dym equations, where radicals are involved.
In Subsection 3.2 we discuss, as an application, symmmetries of the
Burgers equation, while ¬nally in Subection 3.3 we shall devote some words
to the polynomial and graded cases.

3.1. General case. Starting at the symmetry condition (2.29), one
arrives at an overdetermined system of homogeneous linear partial di¬eren-
tial equations for the generating functions Fi , i = 1, . . . , m. First of all,
one notes that in case one deals with a di¬erential equation1 E k ‚ J k (x, u),
x = (x1 , . . . , xn ), u = (u1 , . . . , um ), then the r-th prolongation E k+r is always
polynomial with respect to the higher jet variables in the ¬bre E k+r ’ E k .
The symmetry condition (2.29) is also polynomial with respect to these
variables, cf. Subsection 3.2. So the overdetermined system of partial di¬er-
ential equations can always be splitted with respect to the highest variables
leading to a new system of equations.
These equations are stored in the computer system memory as right-
hand sides of operators equ(1), . . . , equ(te), where the variable te stands
for the Total Number of Equations involved.
If at a certain stage, the computer system constructs new expressions
which have to vanish in order to generate the general solution to the system
of equations (for instance, the derivative of an equation is a consequence,
which might be easier to solve). These new equations are added to the
system as equ(te + 1), . . . and the value of te is adjusted automatically to
the new situation.
In the construction of solutions to the system of equations we distinguish
between a number of di¬erent cases:
1. CASE A: A partial di¬erential equation is of a polynomial type in
one (or more) of the variables, the functions F— appearing in this
equation are independent of this (or these) variable(s). By conse-
quence, each of the coe¬cients of the polynomial has to be zero, and
the partial di¬erential equation decomposes into some new additional
and smaller equations.
Example 8.1. The partal di¬erential equation is
equ(.) :=x2 (F1 )x2 + x1 F2 , (8.51)
1

where in (8.51) the functions F1 , F2 are independent of x1 .
By consequence, the coe¬cients of the polynomial in x1 have to
be zero, i.e., (F1 )x2 and F2 . So equation (8.51) is equivalent to the
system
equ(.):=(F1 )x2 ,
(8.52)
equ(.):=F2

1
We use the notation J k (x, u) as a synonim for J k (π), where π : (x, u) ’ (x).
358 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

2. CASE B: The partial di¬erential equations equ(.) represents a de-
rivative of a function F— . In general
equ(.) :=(F— )xk1 ,...,xkr , (8.53)
i1 ir


is a mixed (k1 + · · · + kr )-th order derivative.
The general solution of (8.53) is
r ks ’1
Fis,t xts ,
F— := (8.54)
i
s=1 t=0

whereas in (8.54) Fis,t depends on the same variables as F— , except
for xis , t = 0, . . . , ks ’ 1, s = 1, . . . , r.
Example 8.2.
equ(.) :=(F1 )x1 ,x2 . (8.55)
The general solution to this equation is given by
F1 := F2 + F3 , (8.56)
where F2 depends on the same variables as F1 , except for x1 , while
F3 depends on the same variables as F1 , except for x2 .
3. CASE C: The partial di¬erential equation equ(.) contains a func-
tion F— , depending on all variables present as arguments of some
other function(s) F—— , occuring in this equation, whereas there is no
derivative of a function F— present in the equation.
The partial di¬erential equation can then be solved for the func-
tion F— .
Example 8.3.
equ(.) :=x1 F1 + x2 (F2 )x1 , (8.57)
where in (8.57) F1 , F2 are dependent on x1 , x2 , x3 . The solution is
F1 := (’x2 (F2 )x1 )/x1 (8.58)
We have to make a remark here. There is a switch in the system
that checks for the coe¬cient for the function F— to be a number. In
case the switch coefficient check is on, equ(.) will not be solved.
In case the switch coefficient check is o¬, the result is given as in
(8.58).
4. CASE D: In the partial di¬erential equation there is a derivative
of a function F— with respect to variables which are not present as
argument of any other function F—— , while the coe¬cient of F— is a
number. By the assumption that x1 , . . . , xn appear as polynomials,
the partial di¬erential equation can be integrated.
3. OVERDETERMINED SYSTEMS OF PDE 359

Example 8.4. Let the partial di¬erential equation be given by
equ(.) :=(F1 )x3 + x2 F2 , (8.59)
where F1 depends on x1 , x2 , x3 and F2 depends on x1 , x2 .
The solution to (8.59) is
F1 := ’x2 x3 F2 + F3 , (8.60)
whereas F3 depends on x1 , x2 and is independent of x3 .
5. CASE E: In the partial di¬erential equation a speci¬c variable xi is
present just once as argument of some function F— . By appropriate
di¬erentiation, one may arrive at a simple equation, which can be
solved.
Evaluation of the original equation can result in an equation which
can be solved too.
Example 8.5.
equ(.) :=x2 (F1 )x2 ,x3 + x3 F2 , (8.61)
where F1 depends on x1 , x2 , x3 and F2 depends on x1 , x2 .
Di¬erentiation with respect to x3 twice results in
equ(.) :=x2 (F1 )x2 ,x3 . (8.62)
3

The solution to (8.62) is CASE B:
F1 := F3 x2 + F4 x3 + F5 + F6 , (8.63)
3
where F1 , F4 , F5 are dependent on x1 , x2 , F6 depends on x1 , x3 .
Substitution of the result (8.63) into the original equation (8.61)
leads to
equ(.) :=2x2 x3 (F3 )x2 + x2 (F4 )x2 + x3 F2 . (8.64)
Due to CASE A, the procedure solve equation constructs two new
equations
equ(.):=2x2 (F3 )x2 + F2 ,
equ(.):=x2 (F4 )x2 (8.65)
The complete result of the procedure solve equation will in this case
be (8.63) and (8.65).
Now the procedure solve equation is then useful to solve the last
two equations (8.65) constructed before; this last step is not carried
through automatically.
For this case there is a switch “differentiation” too, similar to
the previous case.
In practical situations, one is able to solve the overdetermined system
of partial di¬erential equations, using the methods described in the CASES
A, B, C, D, E and some additional considerations, which are speci¬c for the
problem at hand.
360 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

3.2. The Burgers equation. We shall discuss the construction of
higher symmetries of order three of the Burgers equation in order to demon-
strate the facilities of the INTEGRATION package, in e¬ect the procedure
solve equation described in the previous subsection.
The Burgers equation is given by the following partial di¬erential equa-
tion
ut = uu1 + u2 , (8.66)
where partial derivatives with respect to x are given by indices 1, 2, . . . We
start this example by introduction of the vector ¬elds Dx , Dt in the jet
bundle where local coordinates are given by x, t, u, u1 , u2 , u3 , u4 , u5 , u6 ,
u7 , u8 and a generating function F1 , which is dependent on the jet variables
x, t, u, u1 , u2 , u3 .
So the representation of the vector ¬elds Dx , Dt is given by
‚ ‚ ‚ ‚ ‚ ‚ ‚
Dx = + u1 + u2 + u3 + u4 + u5 + u6
‚x ‚u ‚u1 ‚u2 ‚u3 ‚u4 ‚u5
‚ ‚
+ u7 + u8 ,
‚u6 ‚u7
‚ ‚ ‚ ‚ ‚ ‚
Dt = + (ut ) + (ut )1 + (ut )2 + (ut )3 + (ut )4
‚t ‚u ‚u1 ‚u2 ‚u3 ‚u4
‚ ‚
+ (ut )5 + (ut )6 , (8.67)
‚u5 ‚u6
where (ut ), . . . , (ut )6 are given by
(ut ) = uu1 + u2 ,
(ut )1 = uu2 + u2 + u3 ,
1
(ut )2 = uu3 + 3u1 u2 + u4 ,
(ut )3 = uu4 + 4u1 u3 + 3u2 + u5 ,
2
(ut )4 = uu5 + 5u1 u4 + 10u2 u3 + u6 ,
(ut )5 = uu6 + 6u1 u5 + 15u2 u4 + 10u2 + u7 ,
3
(ut )6 = uu7 + 7u1 u6 + 21u2 u5 + 35u3 u4 + u8 . (8.68)
In the remaining part of this section we shall present in e¬ect a computer
session and give some comments on the construction and use of the procedure
solve equation. We shall stick as close as possible to the real output of
the computer system. Boldtext will refer to real input to the system, while
the rest is just the output on screen.
Now from the symmetry condition (2.29) we obtain
2 : equ(1) = Dt (F1 ) ’ Dx (Dx F1 ) ’ uDx (F1 ) ’ u1 F1 ; (8.69)
where the solution is to be determined in such a way that the right-hand
side in (8.69) has to vanish.
3. OVERDETERMINED SYSTEMS OF PDE 361

The resulting equation is now given by
equ(1) = (F1 )t ’ 2(F1 )u,u1 u1 u2 ’ 2(F1 )u,u2 u1 u3 ’ 2(F1 )u,u3 u1 u4
’ 2(F1 )u,x u1 ’ (F1 )u2 u2 ’ 2(F1 )u1 ,u2 u2 u3 ’ 2(F1 )u1 ,u3 u2 u4
1
’ 2(F1 )u1 ,x u2 ’ (F1 )u2 u2 + (F1 )u1 u2 ’ 2(F1 )u2 ,u3 u3 u4
12 1

’ 2(F1 )u2 ,x u3 ’ (F1 )u2 u2 + 3(F1 )u2 u1 u2 ’ 2(F1 )u3 ,x u4 ’ (F1 )u2 u2
23 34

+ 4(F1 )u3 u1 u3 + 3(F1 )u3 u2 ’ (F1 )x2 ’ (F1 )x u ’ F1 u1 $ (8.70)
2

The dependency of the function F1 is stored on a depl!* (dependency list):
3 : lisp depl!—;
(((f 1) u3 u2 u1 u t x)) (8.71)

Equ(1) is an equation, which is a polynomial with respect to the variable u4 ,
so in order to be 0, its coe¬cients should be zero.
These coe¬cients will be detected by the procedure solve equation(*),
i.e., CASE A:
4 : solve equation(1);
equ(1) breaks into equ(2), . . . , equ(4) by u4 , u5 , u6 , u7 , u8

5 : print equations(2, 4);
equ(2) := ’ (F1 )u2 $
3
Functions occurring :
F1 (u3 , u2 , u1 , u, t, x)

equ(3) := ’ 2((F1 )u,u3 u1 + (F1 )u1 ,u3 u2 + (F1 )u2 ,u3 u + (F1 )u3 ,x )$
Functions occurring :
F1 (u3 , u2 , u1 , u, t, x)

equ(4) :=(F1 )t ’ 2(F1 )u,u1 u1 u2 ’ 2(F1 )u,u2 u1 u3 ’ 2(F1 )u,x u1
’ (F1 )u2 u2 ’ 2(F1 )u1 ,u2 u2 u3 ’ 2(F1 )u1 ,x u2 ’ (F1 )u2 u2 + (F1 )u1 u2
1 12 1

’ 2(F1 )u2 ,x u3 ’ (F1 )u2 u2 + 3(F1 )u2 u1 u2 + 4(F1 )u3 u1 u3 + 3(F1 )u3 u2
23 2
’ (F1 )x2 ’ (F1 )x u ’ F1 u1 $

Functions occurring :
F1 (u3 , u2 , u1 , u, t, x) (8.72)
We now are left with a system of three partial di¬erential equations for the
function F1 .
Equ(2) can now be solved, the result being a polynomial of degree 1 with
respect to the variable u3 , while coe¬cients are functions still dependent on
x, t, u, u1 , u2 .
362 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

The construction of this solution, as in CASE B, is performed by the
procedure solve equation(*) too, i.e.:
6 : solve equation(2);
equ(2) : Homogeneous integration of (F1 )u2
3

7 : F1 := F1 ;
F1 := F3 u3 + F2 $ (8.73)
Substitution of this result into the third equation leads to:
8 : equ(3) := equ(3);
equ(3) := ’2((F3 )u u1 + (F3 )u1 u2 + (F3 )u2 u3 + (F3 )x )$ (8.74)
and this equation splits up, since it is a polynomial of degree 1 with respect
to u3 , CASE A:
9 : solve equation(3);
equ(3) breaks into equ(5), ..., equ(6) by u3 , u4 , u5 , u6 , u7 , · · ·

10 : print equations(5, 6);
equ(5) := ’2(F3 )u2 $
Functions occurring :
F3 (u2 , u1 , u, t, x)

equ(6) := ’2((F3 )u u1 + (F3 )u1 u2 + (F3 )x )$
Functions occurring :
F3 (u2 , u1 , u, t, x) (8.75)
Now the procedure can be repeated, since equ(5) indicates that F3 is inde-
pendent of u2 , in e¬ect a polynomial of degree 0, and equ(6) can be splitted
with respect to u2 :
11 : solve equation(5);
equ(5) : Homogeneous integration of (F3 )u2

12 : solve equation(6);
equ(6) breaks into equ(7), . . . , equ(8) by u2 , u3 , u4 , u5 , u6 , . . .

13 : print equations(7, 8);
equ(7) := ’2(F4 )u1 $
Functions occurring :
F4 (u1 , u, t, x)

equ(8) := ’2((F4 )u u1 + (F4 )x )$
Functions occurring :
3. OVERDETERMINED SYSTEMS OF PDE 363

F4 (u1 , u, t, x) (8.76)
From equ(7) we have that F4 is independent of u1 and combination with
equ(8) then results in the fact that F4 is independent of u and x too:
14 : solve equation(7);
equ(7) : Homogeneous integration of (F4 )u1

15 : solve equation(8);
equ(8) breaks into equ(9), ..., equ(10) by u1 , u2 , u3 , u4 , u5 , . . .

16 : print equations(9, 10);
equ(9) := ’2(F5 )u $
Functions occurring :
F5 (u, t, x)

equ(10) := ’2(F5 )x $
Functions occurring :
F5 (u, t, x)

17 : solve equation(9);
equ(9) : Homogeneous integration of (F5 )u

18 : solve equation(10);
equ(10) : Homogeneous integration of (F6 )x (8.77)
Summarising the results obtained thusfar, we are left with an expression for
the function F1 in terms of F2 and F7 and one equation, equ(4), which is
polynomial with respect to u3 :
19 : f(1) := f(1);
F1 := F7 u3 + F2 $

20 : print equations(1, te);
equ(4) := (F7 )t u3 + (F2 )t ’ 2(F2 )u,u1 u1 u2 ’ 2(F2 )u,u2 u1 u3
’ 2(F2 )u,x u1 ’ (F2 )u2 u2 ’ 2(F2 )u1 ,u2 u2 u3 ’ 2(F2 )u1 ,x u2 ’ (F2 )u2 u2
1 12

+ (F2 )u1 u2 ’ 2(F2 )u2 ,x u3 ’ (F2 )u2 u2 + 3(F2 )u2 u1 u2 ’ (F2 )x2
1 23

’ (F2 )x u + 3F7 u1 u3 + 3F7 u2 ’ F2 u1 $
2
Functions occurring :
F2 (u2 , u1 , u, t, x)
F7 (t)

21 : solve equation(4);
equ(4) breaks into equ(11), . . . , equ(13) by u3 , u4 , u5 , u6 , u7 , . . .
364 8. SYMBOLIC COMPUTATIONS IN DIFFERENTIAL GEOMETRY

22 : print equations(11, 13);
equ(11) := ’(F2 )u2 $
2
Functions occurring :
F2 (u2 , u1 , u, t, x)

equ(12) := (F7 )t ’ 2(F2 )u,u2 u1 ’ 2(F2 )u1 ,u2 u2 ’ 2(F2 )u2 ,x + 3F7 u1 $
Functions occurring :
F2 (u2 , u1 , u, t, x)
F7 (t)

equ(13) := (F2 )t ’ 2(F2 )u,u1 u1 u2 ’ 2(F2 )u,x u1 ’ (F2 )u2 u2
1
’ 2(F2 )u1 ,x u2 ’ (F2 )u2 u2 + (F2 )u1 u2 + 3(F2 )u2 u1 u2 ’ (F2 )x2
12 1

’ (F2 )x u + 3F7 u2 ’ F2 u1 $
2
Functions occurring :
F7 (t)
F2 (u2 , u1 , u, t, x) (8.78)

The remaining system, equ(11), equ(12), equ(13), can be handled in a sim-
ilar way as before, leading to an expression for the function F2 :

23 : solve equation(11);
equ(11) : Homogeneous integration of (F2 )u2
2


24 : equ(12) := equ(12);
equ(12) := ’2(F9 )u u1 ’ 2(F9 )u1 u2 ’ 2(F9 )x + (F7 )t + 3F7 u1 $

25 : solve equation(12);
equ(12) breaks into equ(14), . . . , equ(15) by u2 , u3 , u4 , u5 , u6 , . . .

26 : equ(14);
’ 2(F9 )u1 $

27 : solve equation(14);
equ(14) : Homogeneous integration of (F9 )u1

28 : equ(15);
’ 2(F10 )u u1 ’ 2(F10 )x + (F7 )t + 3F7 u1 $

29 : solve equation(15);
equ(15) breaks into equ(16), . . . , equ(17) by u1 , u2 , u3 , u4 , u5 , . . .

30 : print equations(16, 17);
3. OVERDETERMINED SYSTEMS OF PDE 365

equ(16) := ’2(F10 )u + 3F7 $
Functions occurring :
F7 (t)
F10 (u, t, x) (8.79)
and
equ(17) := ’2(F10 )x + (F7 )t $
Functions occurring :
F7 (t)
F10 (u, t, x)

31 : solve equation(16);
CASE C :
equ(16) : Inhomogeneous integration of (F10 )u

32 : solve equation(17);
equ(17) : Inhomogeneous integration of (F11 )x

33 : f(2) := f(2);
F2 := ((F7 )t u2 x + 2F12 u2 + 2F8 + 3F7 uu2 )/2$ (8.80)
while the original de¬ning function F1 , and the remaining equation, equ(13),
are given by:
34 : f(1) := f(1);
F1 := ((F7 )t u2 x + 2F12 u2 + 2F8 + 3F7 uu2 + 2F7 u3 )/2$

35 : print equations(1, te);
equ(13) := (2(F12 )t u2 + 2(F8 )t ’ 4(F8 )u,u1 u1 u2 ’ 4(F8 )u,x u1
’ 2(F8 )u2 u2 ’ 4(F8 )u1 ,x u2 ’ 2(F8 )u2 u2 + 2(F8 )u1 u2 ’ 2(F8 )x2
1 12 1
’ 2(F8 )x u + (F7 )t2 u2 x + 2(F7 )t uu2 + 2(F7 )t u1 u2 x + 4F12 u1 u2
’ 2F8 u1 + 6F7 uu1 u2 + 6F7 u2 )/2$
2
Functions occurring :
F7 (t)
F8 (u1 , u, t, x)
F12 (t) (8.81)
Equ(13) is a polynomial with respect to the variable u2 , and the result is
again a system of three equations, the ¬rst two of them can be solved in

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