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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

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