<<

. 14
( 16)



>>

2


1 1
’ ω•1 + ωψ (c1 ’ 4k)•ψ + ωp0 (c1 ’ 4k)ψ1 + ωp1 k•
+
4 2
1 1 ‚
+ ωp1 k(c1 + 12k)• + ωq 1 (’2kv + k(c1 + 12k)•q 1 ) .
2 2 ‚ψ
2
2




The action of U1 on the symmetries

‚ ‚ ‚ ‚
X1 = u 1 + v1 + •1 + ψ1 + ...,
‚u ‚v ‚• ‚ψ

X 1 = (c + 4k)(•1 q 1 + ψq 3 ) + 2k(c ’ 4k)¯1 v
p
‚u
2 2


+ (c + 4k)(ψ1 q 1 ’ •q 3 ) ’ 2k(c ’ 4k)¯1 u
p
‚v
2 2


+ ’ 4kuq 1 + 2ψ1 + 2k(c ’ 4k)¯1 ψ p
‚•
2


+ ’ 4kvq 1 ’ 2•1 ’ 2k(c ’ 4k)¯1 • p + ...,
‚ψ
2

‚ ‚ 1‚ 1‚
Y 1 = •1 + ψ1 +u +v + ...,
‚u ‚v 2 ‚• 2 ‚ψ
2

‚ ‚ 1 ‚ 1 ‚
’ q1 u + (q 1 ψ ’ u) + (’q 1 • ’ v)
Y1 = q1 v + ...,
2 ‚u ‚v 4k ‚• 4k ‚ψ
2 2 2 2


‚ ‚ ‚ ‚
’u ’•
X0 = v +ψ + ...,
‚u ‚v ‚• ‚ψ
‚ ‚ 1‚ 1 ‚
X 0 = ’q 1 ψ ’• ’ψ
+ q1 • + ...,
‚u ‚v 8k ‚• 8k ‚ψ
2 2

‚ ‚
’•
Y’ 1 = ψ + ...
‚u ‚v
2
320 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

creates hierarchies of symmetries in a similar way as in the preceding sub-
section. Note that X 1 is the nonlocal recursion symmetry constructed in
Section 8.2 of Chapter 6.

3. Supersymmetric Boussinesq equation
We discuss the construction of a supersymmetric extension of the Boussi-
nesq equation. Conservation laws, nonlocal variables, symmetries and re-
cursion operators for this supersymmetric system will be discussed too.

3.1. Construction of supersymmetric extensions. We start our
discussion from the classical system [14, 80]
1
ut = ’ uxx + uux + vx ,
2
1
vt = vxx + uvx + ux v. (7.30)
2
We construct a so-called fermionic extension [35] by setting
¦ = • + θu,
Ψ = ψ + θv, (7.31)
where • ψ, θ are odd variables.
Due to the classical grading of equation (7.30), i.e.,
deg(u) = 1, deg(v) = 2, deg(x) = ’1, deg(t) = ’2,
and the grading of the odd variables
1 1 3
deg(θ) = ’ , deg(•) = , deg(ψ) = ,
2 2 2
the variables ¦, Ψ are graded by
1 3
deg(¦) = , deg(Ψ) = .
2 2
Now we construct a formal extension of (7.30) by setting
ut = f1 [u, v, •, ψ]
vt = f2 [u, v, •, ψ]
•t = f3 [u, v, •, ψ]
ψt = f4 [u, v, •, ψ] (7.32)
where f1 , f2 , f3 , f4 are functions of degrees 3, 4, 5/2, 7/2 respectively
de¬ned on the jet bundle J ∞ (π), π : (x, t, u, v) ’ (x, t), extended by the
odd variables • and ψ. The construction of f1 and f2 should be done in
such a way that in the absence of odd variables f1 , f2 reduce to the right-
hand sides of (7.30). We now put on the following requirements on system
(7.32), see [88]:
3. SUPERSYMMETRIC BOUSSINESQ EQUATION 321

1. The existence of an odd symmetry of (7.32), i.e.,
‚ ‚ ‚ ‚
Y 1 = •1 + ψ1 +u +v + ...,
‚u ‚v ‚• ‚ψ
2

‚ ‚ ‚ ‚ ‚
.
) + · · · = ’2 .
[Y 1 , Y 1 ] = 2(u1 + v1 + •1 + ψ1
‚u ‚v ‚• ‚ψ ‚x
2 2

2. The existence of an even symmetry of (7.32) of appropriate degree
which reduces to the classical ¬rst higher order symmetry of (7.30)
in the absence of odd variables, i.e.,
1 ‚
clas
u3 ’ u2 + 2uv1 + 2vu1 ’ uu2 + u2 u1
X3 = 1
3 ‚u
1 ‚
v3 + u1 v1 + 2vv1 + uv2 + 2uu1 v + u2 v1
+ . (7.33)
3 ‚v
From the above requirements we obtained the following supersymmetric
extension of (7.30):
1
ut = ’ u2 + uu1 + v1 ,
2
1
vt = v2 + u1 v + uv1 + •1 ψ1 + •2 ψ,
2
1
•t = ’ •2 + ψ1 + u•1 ,
2
1
ψt = ψ2 + uψ1 + u1 ψ, (7.34)
2
while the symmetry X3 is given by
1 ‚
u3 ’ u2 + 2vu1 ’ uu2 + 2uv1 + u2 u1 + •1 ψ1 + •2 ψ
X3 = 1
3 ‚u
1
v3 + u1 v1 + 2vv1 + uv2 + 2uvu1 + u2 v1 + •2 ψ1 + •1 ψ2
+
3

+ 2u•1 ψ1 ’ ψψ2 + 2•2 ψu + 2u1 •1 ψ
‚v
1
•3 ’ u•2 + 2uψ1 + u2 •1 + v•1 ’ u1 •1 + u1 ψ ‚•
+
3
1 ‚
ψ3 + uψ2 + u2 ψ1 + vψ1 + u1 ψ1 + 2uu1 ψ + v1 ψ
+ . (7.35)
3 ‚ψ
The resulting supersymmetric extension of the Boussinesq equation is just
the same as mentioned in [67].
3.2. Construction of conserved quantities and nonlocal vari-
ables. For the supersymmetric extension (7.34) of the Boussinesq equation
we constructed the following set of conserved densities (X), associated con-

served quantities ( ’∞ X dx) and nonlocal variables D ’1 (X), i.e, the vari-
ables pi of degree i, qj of degree j:
p0 = D’1 (u),
322 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

p1 = D’1 (v),
p2 = D’1 (uv + •1 ψ),
p3 = D’1 (v 2 + uv1 + u2 v + 2u•1 ψ + •1 ψ1 ’ ψψ1 ),
p0 = D’1 (p1 ),
p1 = D’1 (ψq 1 + •ψ),
2
’1
(p1 •ψ ’ u•ψ + •1 ψ ’ uψq 1 + p1 •1 q 1 ),
p2 = D
2 2

p3 = 2D ’1 (p2 •1 ’ u2 ψ ’ 2vψ + u1 ψ ’ p1 v• ’ •v1 )q 1
2
2
p2
+ (uψ ’ p1 ψ)q 3 + (’u + p1 u ’ 2v + u1 ’ + p2 )•ψ
1
2

’ u2 v ’ uv1 ’ v 2
and
q 1 = D’1 (ψ),
2

q 3 = D’1 (uψ + v•),
2

q 1 1 = D’1 (q 1 v + p1 •1 ),
2 2
’1
(’•1 p2 + p2 (2ψ ’ 2•1 ) ’ 2(p1 v + v1 )q 1 ’ 2v•1 ),
q5 = D 1
2 2
1
q 5 = D’1 •1 p2 + p2 (’2ψ + •1 ) + (uv ’ 2uu1
1
2
2

+ u1 p1 + u2 )q 1 + v•1 .
2

Note that the variables p0 , p1 , . . . contain higher order nonlocalities.
In fact, introduction of the nonlocal variables p0 , p0 , . . . , q 1 , q 3 , q 3 , . . . is
2 2 2
essential for the construction of nonlocal symmetries, while the associated
Cartan forms ωp0 , ωp0 , . . . , ωq 1 , . . . play a signi¬cant role in the construction
2
of deformations or recursion operators.

3.3. Symmetries. We obtained the following symmetries for the su-
persymmetric extension of Boussinesq equation (7.34):
‚ ‚ ‚ ‚
Y 1 = •1 + ψ1 +u +v + ...,
‚u ‚v ‚• ‚ψ
2

‚ ‚ ‚
+ (u ’ p1 )
Y =ψ + ψ1 + ...,
1
‚u ‚v ‚•
2


‚ ‚ ‚ ‚
X1 = u 1 + v1 + •1 + ψ1 + ...,
‚u ‚v ‚• ‚ψ
‚ ‚
X 1 = (•ψ + •1 q 1 ) + (•ψ1 + •1 ψ + ψ1 q 1 )
2 ‚u 2 ‚v

‚ ‚
+ (’uq 1 ’ q 3 ’ •1 + u•) + (’vq 1 ’ ψ1 ’ uψ) + ...,
‚• ‚ψ
2 2 2
3. SUPERSYMMETRIC BOUSSINESQ EQUATION 323


Y 3 = (’2q 1 u1 ’ •2 + u•1 + p1 •1 ’ 3uψ + u1 •)
‚u
2 2


+ (’2q 1 v1 ’ ψ2 + 2uψ1 + p1 ψ1 ’ v•1 ’ vψ ’ 2u1 ψ + v1 •)
‚v
2


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


+ (’2q 1 ψ1 + 2•1 ψ + •ψ1 + uv + p1 v + v1 ) + ...,
‚ψ
2


= (’q 1 u1 ’ ψ1 ’ 2uψ + p1 ψ)
Y 3
‚u
2 2


+ (’q 1 v1 ’ ψ2 ’ 2uψ1 + p1 ψ1 ’ 2u1 ψ)
‚v
2

1 ‚ ‚
+ (’q 1 •1 ’ u2 + p1 u ’ v + u1 ’ p2 + p2 ) ’ q 1 ψ1 + ...,
21 ‚• ‚ψ
2 2


3.4. Deformation and recursion operator. In a way, analogously to
previous applications, we construct a deformation of the equation structure
U related to the supersymmetric Boussinesq equation, i.e.,

U1 = ωu1 ’ 2ωv ’ ωu u ’ ωp0 u1 ’ ω• ψ + ωq 1 (2ψ ’ •1 )
‚u
2

’ ωv1 ’ ωv u ’ 2ωu v ’ 2ω•1 ψ ’ ω• ψ1 + ωψ (•1 + ψ)
+

’ ω p 0 v1 + ω q 1 ψ 1
‚v
2

+ ω•1 ’ 2ωψ + ω• (2p1 ’ u) ’ ωp0 •1 + ωp1 (2q 1 + •)
2


’ ωq 3 ’ 2ωq 3 + ωq 1 u
‚•
2 2 2


’ ωψ1 ’ ωψ u ’ 2ωu ψ ’ ωp0 ψ1 + ωp1 ψ ’ ωq 1 v
+ .
‚ψ
2

From the deformation U , we obtain four hierarchies of (x, t)-independent
symmetries {Yn+ 1 }, {Y n+ 1 }, {Xn+1 }, {X n+1 }, n ∈ N, by
2 2

Yn+ 1 = (. . . (Y 1 U1 ) . . . U1 ),
2 2

Y n+ 1 = (. . . (Y U1 ) . . . U1 ),
1
2 2

Xn+1 = (. . . (X1 U1 ) . . . U1 ),
X n+1 = (. . . (X 1 U1 ) . . . U1 ),
and an (x, t)-dependent hierarchy de¬ned by
Sn = (. . . (S0 U1 ) . . . U1 ),
where S0 is de¬ned by
‚ ‚
S0 = (u + xu1 + 2tut ) + (2v + xv1 + 2tvt )
‚u ‚v
324 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

1 ‚ 3 ‚
+ ···
+ • + x•1 + 2t•t + ψ + xψ1 + 2tψt
2 ‚• 2 ‚ψ
In e¬ect, the hierarchies {Y n+ 1 } and {X n+1 } start at symmetries
2


Y ’1 =
‚•
2

and
‚ ‚
X 0 = (2q 1 ’ •) +ψ
‚• ‚ψ
2

respectively.

4. Supersymmetric extensions of the KdV equation, N = 2
In this chapter we shall discuss the supersymmetric extensions of the
classical KdV equation
ut = ’uxxx + 6uux (7.36)
with two odd variables, the situation N = 2. The construction of such
supersymmetric systems runs along similar lines as has been explained
for the supersymmetric extension of the classical nonlinear Schr¨dinger
o
equation, cf. Section 8 of Chapter 6. For additional references see also
[68, 87, 64, 65, 63, 82, 79].
The extension is obtained by considering two odd (pseudo) total deriv-
ative operators D1 and D2 given by
D1 = ‚ θ 1 + θ 1 Dx , D2 = ‚ θ 2 + θ 2 Dx , (7.37)
where θ1 , θ2 are two odd parameters. Obviously, these operators satisfy the
2 2
relations D1 = D2 = Dx and [D1 , D2 ] = 0.
The N = 2 supersymmetric extension of the KdV equation is obtained
by taking an even homogeneous ¬eld ¦
¦ = w + θ 1 ψ + θ 2 • + θ 2 θ1 u (7.38)
with degrees deg(¦) = 1, deg(u) = 2, deg(w) = 1, deg(•) = deg(ψ) = 3/2,
deg(θ1 ) = deg(θ2 ) = ’1/2, and considering the most general evolution
equation for ¦, which reduces to the KdV equation in the absence of the
odd variables •, ψ.
Proceeding in this way, we arrive at the system
1
¦t = Dx ’Dx ¦ + 3¦D1 D2 ¦ + (a ’ 1)D1 D2 ¦2 + a¦3 .
2
(7.39)
2
Rewriting this system in components, we arrive at a system of partial dif-
ferential equations for the two even variables u, w and the two odd variables
•, ψ, i.e.,
ut = Dx ’ u2 + 3u2 ’ 3••1 ’ 3ψψ1 ’ (a ’ 1)w1
2

’ (a + 2)ww2 + 3auw 2 + 6awψ• ,
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 325

•t = Dx ’ •2 + 3u• + 3aw 2 • ’ (a + 2)wψ1 ’ (a ’ 1)w1 ψ ,
ψt = Dx ’ ψ2 + 3uψ + 3aw 2 ψ + (a + 2)w•1 + (a ’ 1)w1 • ,
wt = Dx ’ w2 + aw3 + (a + 2)uw + (a ’ 1)ψ• , (7.40)
or equivalently,
ut = ’u3 + 6uu1 ’ 3••2 ’ 3ψψ2 ’ 3aw1 w2 ’ (a + 2)ww3 + 3au1 w2
+ 6auww1 + 6aw1 ψ• + 6awψ1 • + 6awψ•1 ,
•t = ’•3 + 3u1 • + 3u•1 + 6aww1 • + 3aw 2 •1 ’ (a + 2)w1 ψ1
’ (a + 2)wψ2 ’ (a ’ 1)w2 ψ ’ (a ’ 1)w1 ψ1 ,
ψt = ’ψ3 + 3u1 ψ + 3uψ1 + 6aww1 ψ + 3aw 2 ψ1 + (a + 2)w1 •1
+ (a + 2)w•2 + (a ’ 1)w2 • + (a ’ 1)w1 •1 ,
wt = ’w3 + 3aw 2 w1 + (a + 2)u1 w + (a + 2)uw1 + (a ’ 1)ψ1 •
+ (a ’ 1)ψ•1 . (7.41)
It has been demonstrated by several authors [87, 74] that the interesting
equations from the point of view of complete integrability are the special
cases a = ’2, 1, 4.
In Subsection 4.1 we discuss the case a = ’2. We shall present in
the respective subsections results for the construction of local and nonlocal
conservation laws, nonlocal symmetries and ¬nally present the recursion
operator for symmetries. A similar presentation is chosen for Subsections
4.2, where we deal with the case a = 4, and ¬nally in Subsections 4.3 we
present the results for the most intriguing case a = 1.
The structure is extremely complicated in this case, which can be illus-
trated from the fact that in order to ¬nd a good setting for the recursion
operator for symmetries, we had to introduce a total of 16 nonlocal variables
associated to the respective conservation laws, while the complete computa-
tion for the recursion operation required the introduction and ¬xing of more
than 20,000 constants.

4.1. Case a = ’2. In this subsection we discuss the case a = ’2,
which leads to the following system of partial di¬erential equations
ut = ’u3 + 6uu1 ’ 3••2 ’ 3ψψ2 + 6w1 w2 ’ 6u1 w2 ’ 12uww1
’ 12w1 ψ• ’ 12wψ1 • ’ 12wψ•1 ,
•t = ’•3 + 3u1 • + 3u•1 ’ 12ww1 • ’ 6w2 •1 + 3w2 ψ + 3w1 ψ1 ,
ψt = ’ψ3 + 3u1 ψ + 3uψ1 ’ 12ww1 ψ ’ 6w 2 ψ1 ’ 3w2 • ’ 3w1 •1 ,
wt = ’w3 ’ 6w2 w1 ’ 3ψ1 • ’ 3ψ•1 . (7.42)
The results obtained in this case for conservation laws, higher symmetries
and deformations or recursion operator will be presented in subsequent sub-
sections.
326 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

4.1.1. 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 de¬ned by
(p0,1 )x = w,
(p0,1 )t = 3•ψ ’ 2w 3 ’ w2 ;

(p0,2 )x = p1,1 ,
(p0,2 )t = 12p3,1 ’ u1 + 3ww1 (7.43)
(see the de¬nition of p1,1 and p3,1 below).
2. Nonlocal variables p1,1 , p1,2 , p1,3 , p1,4 of degree 1 de¬ned by the rela-
tions
(p1,1 )x = u,
(p1,1 )t = ’3ψψ1 ’ 3••1 + 12•ψw + 3u2 ’ 6uw2 ’ u2 + 3w1 ;
2


(p1,2 )x = ψq 1 ’ •q 1 ,
2 2

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

’ 6ψq 1 w ’ 3ψq 1 w1 ’ 2ψψ1 ’ 3•q 1 w1 ’ 3•q 1 u + 6•q 1 w2 + 2••1 ;
2
2 2 2 2 2


(p1,3 )x = ψq 1 ,
2

(p1,3 )t = ’ψ2 q 1 + 3ψq 1 u ’ 6ψq 1 w2 + •1 ψ ’ 3•q 1 w1 ’ •ψ1 ;
2 2 2 2


(p1,4 )x = •q 1 + w2 ,
2

(p1,4 )t = ’•2 q 1 + 3ψq 1 w1 + 3•q 1 u ’ 6•q 1 w2
2 2 2 2
4 2
’ 2••1 + 6•ψw ’ 3w ’ 2ww2 + w1 (7.44)
(the variables q 1 and q 1 are de¬ned below).
2 2
3. Nonlocal variable p2,1 of degree 2 de¬ned by
(p2,1 )x = q 1 q 1 u + ψ1 q 1 + ψq 1 w + •q 1 w,
2 2 2 2 2
2 2
2
(p2,1 )t = 3q 1 q 1 u ’ 6q 1 q 1 uw ’ q 1 q 1 u2 + 3q 1 q 1 w1 ’ ψ3 q 1 ’ ψ2 q 1 w
2 2 2 2 2 2 2 2 2 2
2
’ •2 q 1 w + ψ1 q 1 w1 + 4ψ1 q 1 u ’ 6ψ1 q 1 w ’ •1 q 1 u ’ 2•1 q 1 w1
2 2 2 2 2 2
3
+ •1 ψ1 + 3ψq 1 uw ’ 6ψq 1 w ’ ψq 1 w2 + 2ψq 1 u1 ’ 9ψq 1 ww1
2 2 2 2 2

’ 3ψψ1 q 1 q 1 ’ 2ψψ1 w + •q 1 u1 ’ 3•q 1 ww1 + 3•q 1 uw ’ 6•q 1 w3
2 2 2 2 2 2
’ 4•q 1 w2 ’ •ψ2 ’ 3••1 q 1 q 1 ’ 2••1 w + 12•ψq 1 q 1 w + •ψu.
2 22 22
(7.45)
4. Finally, the variable p3,1 of degree 3 de¬ned by
1
(p3,1 )x = (’ψψ1 ’ ••1 + 4•ψw + u2 ’ 2uw2 ’ ww2 ),
4
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 327

1
(p3,1 )t = (’2ψ1 ψ2 ’ 2•1 •2 ’ 2•1 ψ1 w + ψψ3 + 7ψ•2 w ’ 9ψψ1 u
4
+ 12ψψ1 w2 + 4•1 ψw1 + ••3 ’ 7•ψ2 w + 4•ψ1 w1 ’ 9••1 u
+ 12••1 w2 + 24•ψuw ’ 48•ψw 3 ’ 10•ψw2 + 4u3 ’ 12u2 w2
’ 2uu2 + 12uw 4 + 4uww2 + 4uw1 + u2 ’ 4u1 ww1 + 2u2 w2
2
1
+ 6w3 w2 + 6w2 w1 + ww4 ’ w1 w3 + w2 ).
2 2
(7.46)
Remark 7.1. It should be noted that the ¬rst lower index refers to the
degree of the object (in this case the nonlocal variable), while the second
lower index is referring to the numbering of the objects of that speci¬c
degree. The number of nonlocal variables of degree 3 is 4, since this num-
ber is the same as for nonlocal variables of degree 1, cf. (7.44). This total
number will arise after introduction of these nonlocal variables and com-
putation of the conservation laws and the associated nonlocal variables in
this augmented setting. These conservation laws and their associated non-
local variables are of a higher nonlocality. We shall not pursue this further
here, because the number of nonlocal variables found will turn out to be
su¬cient to compute the deformation of the system of equations (7.42), or
equivalently the construction of the recursion operator for symmetries. We
refer for a more comprehensive computation to Subsection 4.3, where all
nonlocal variables at the levels turn out to be essential in the computation
of the recursion operator for that case.
For the odd conservation laws and the associated odd nonlocal variables
we derived the following results.
1. At degree 1/2 we computed the variables q 1 and q 1 de¬ned by
2 2

(q 1 )x = •,
2

(q 1 )t = ’•2 + 3ψw1 + 3•u ’ 6•w 2 ;
2


(q 1 )x = ψ,
2

(q 1 )t = ’ψ2 + 3ψu ’ 6ψw 2 ’ 3•w1 . (7.47)
2

2. At degree 3/2 we have the variables q 3 and q 3 de¬ned by
2 2

(q 3 )x = q 1 u ’ •w,
2 2

(q 3 )t = 3q 1 u2 ’ 6q 1 uw2 ’ q 1 u2 + 3q 1 w1 + •2 w ’ ψ1 u ’ •1 w1 ’ 3ψψ1 q 1
2
2 2 2 2 2 2
3
+ ψu1 ’ 3ψww1 ’ 3••1 q 1 + 12•ψq 1 w ’ 3•uw + 6•w + •w2 ;
2 2


(q 3 )x = ’(q 1 u + ψw),
2 2

(q 3 )t = ’3q 1 u2 + 6q 1 uw2 + q 1 u2 ’ 3q 1 w1 + ψ2 w ’ ψ1 w1 + •1 u + 3ψψ1 q 1
2
2 2 2 2 2 2

’ 3ψuw + 6ψw 3 + ψw2 + 3••1 q 1 ’ 12•ψq 1 w ’ •u1 + 3•ww1 .
2 2
(7.48)
328 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

3. Finally, at degree 5/2 we obtained q 5 and q 5 de¬ned by the relations
2 2


(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 :
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
(7.74)
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 ,1 , q 1 ,2 , q 1 ,3 , and q 1 ,4 , de¬ned
2 2 2 2
by the relations
(q 1 ,1 )x = •,
2

(q 1 ,1 )t = ’•2 ’ 3ψ1 w + 3•u + 3•w 2 ;
2


(q 1 ,2 )x = ψ,
2

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


(q 1 ,3 )x = cos(2p0,1 )q 1 ,1 w + sin(2p0,1 )q 1 ,2 w,
2 2 2

(q 1 ,3 )t = cos(2p0,1 )(3q 1 ,1 uw + q 1 ,1 w ’ q 1 ,1 w2 ’ •1 w ’ ψw2 + •w1 )
3
2 2 2 2
4. SUPERSYMMETRIC EXTENSIONS OF THE KDV EQUATION, N = 2 339

+ sin(2p0,1 )(3q 1 ,2 uw + q 1 ,2 w3 ’ q 1 ,2 w2 ’ ψ1 w + ψw1 + •w2 );
2 2 2


(q 1 ,4 )x = cos(2p0,1 )q 1 ,2 w ’ sin(2p0,1 )q 1 ,1 w,
2 2 2

(q 1 ,4 )t = cos(2p0,1 )(3q 1 ,2 uw + q 1 ,2 w ’ q 1 ,2 w2 ’ ψ1 w + ψw1 + •w2 ) 3
2 2 2 2

+ sin(2p0,1 )(’3q 1 ,1 uw ’ q 1 ,1 w + q 1 ,1 w2 + •1 w + ψw2 ’ •w1 ) 3
2 2 2
(7.75)

2. At degree 3/2, we have q 3 ,1 and q 3 ,2 :
2 2


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

(q 3 ,1 )t = cos(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 2

+ q 1 ,2 u1 w + q 1 ,2 w2 w1 + 3q 1 ,1 u2 ’ q 1 ,1 uw2 ’ q 1 ,1 u2 ’ q 1 ,1 w4
2 2 2 2 2 2
2
’ q 1 ,1 ww2 ’ q 1 ,1 w1 ’ ψ2 q 1 ,1 q 1 ,2 ’ ψ2 w ’ ψ1 p1 w + ψ1 w1
2 2 2 2

+ •1 q 1 ,1 q 1 ,2 w ’ •1 u + 2•1 w2 + 3ψq 1 ,1 q 1 ,2 u + ψq 1 ,1 q 1 ,2 w2
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 )
+ sin(2p0,1 )(3q 1 ,2 u2 ’ q 1 ,2 uw2 ’ q 1 ,2 u2 ’ q 1 ,2 w4 ’ q 1 ,2 ww2
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 ,2 w1
2 2 2 2 2
2
’ q 1 ,1 u1 w ’ q 1 ,1 w w1 + •2 q 1 ,1 q 1 ,2 + •2 w + ψ1 q 1 ,1 q 1 ,2 w ’ ψ1 u
2 2 2 2 2 2
2
+ 2ψ1 w + •1 p1 w ’ •1 w1 + 2ψq 1 ,1 q 1 ,2 w1 ’ 3ψψ1 q 1 ,2
2 2 2

’ ψ•1 q 1 ,1 + ψp1 w + ψu1 + ψww1 ’ 3•q 1 ,1 q 1 ,2 u ’ •q 1 ,1 q 1 ,2 w2
2
2 2 2 2 2
’ •ψ1 q 1 ,1 ’ ••1 q 1 ,2 ’ 4•ψq 1 ,2 w ’ •p1 w1 ’ 3•uw
2 2 2
3

<<

. 14
( 16)



>>