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o
[88]
ut = ’v2 + kv(u2 + v 2 ) ’ u1 (c1 ’ c2 )•ψ ’ 4kvψψ1
316 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

’ u(c1 + c2 + 4k)ψ•1 + c2 u•ψ1 + c1 v••1 ,
vt = u2 ’ ku(u2 + v 2 ) ’ v1 (c1 ’ c2 )•ψ + 4ku••1
+ v(c1 + c2 + 4k)•ψ1 ’ c1 uψψ1 ’ c2 vψ•1 ,
1 1
•t = ’ψ2 + ( c2 u2 + ku2 + kv 2 )ψ ’ c2 uv• ’ (c1 ’ c2 )•ψ•1 ,
2 2
1 1
ψt = •2 ’ ( c2 v 2 + ku2 + kv 2 )• + c2 uvψ ’ (c1 ’ c2 )•ψψ1 ,
2 2
where in
c1 = ’4k,
Case A: c2 = 0,
Case B: c1 = c, c2 = 4k.
The construction of deformations will follow exactly the same lines as for the
supersymmetric KdV equation presented in Section 1, so for the nonlinear
Schr¨dinger equation we shall only present the results.
o

2.1. Case A. In order to work in the appropriate covering for the su-
persymmetric extension of the Nonlinear Schr¨dinger Equation we did con-
o
struct the following set of nonlocal variables, associated to conserved quan-
tities
p0 , p1 , p2 , p0 , p1 , p2 ,
q1 , q 1 , q3 , q 3 , q5 , q 5 ,
2 2 2 2 2 2

which are de¬ned by
p0 = D’1 (•ψ),
p0 = D’1 (p1 ),
p1 = D’1 (u2 + v 2 ’ 2••1 ’ 2ψψ1 ),
p1 = D’1 k(ψv + •u)q 1 + k(ψu ’ •v)q 1 ’ 2ψψ1 ’ 2••1 ,
2 2
’1
p2 = D (uv1 + 2•1 ψ1 ),
p2 = D’1 k(2ψ1 v + 2•1 u + kψvp1 + k•up1 )q 1
2

+ k(’2ψ1 u + 2•1 v ’ kψup1 + k•vp1 )q 1 + 2uv1 ,
2
’1
(ψu ’ •v),
q1 = D
2

q 1 = D’1 (ψv + •u),
2

q 3 = D’1 (kψup1 ’ k•vp1 + 2ψ1 u ’ 2•1 v),
2

q 3 = D’1 (kψvp1 + k•up1 + 2ψ1 v + 2•1 u).
2

After introduction of the associated Cartan forms, we found the deformation,
or Nijenhuis operator, for this case to be
U1 = ωv1 + ωp1 (’kv) ’ 2ωp0 ku1 + ωu (’2k•ψ)
2. SUPERSYMMETRIC EXTENSIONS OF NLS 317

+ ω• (’kuψ ’ kv•) + ωψ (’kvψ + ku•)

’ ωq 1 (k•1 ) + ωq 1 (kψ1 )
‚u
2 2

’ ωu1 + ωp1 (ku) ’ 2ωp0 kv1 + ωv (’2k•ψ)
+
+ ω• (’kvψ + ku•) + ωψ (kuψ + kv•)

+ ωq 1 (’kψ) + ωq 1 (’k•1 )
‚v
2 2
k
+ ωψ1 + ω• (k•ψ) + ωp1 (’ ψ) + ωp0 (’2k•1 )
2
k k ‚
+ ωq 1 (’ u) + ωq 1 (’ v)
2 2 ‚•
2 2

k
+ ’ ω•1 + ωψ (’k•ψ) + ωp1 (+ •) + ωp0 (’2kψ1 )
2
k k ‚
+ ωq 1 (’ v) + ωq 1 ( u) .
2 22 ‚ψ
2

By starting at the symmetries (see [88])
‚ ‚
’u
X0 = v + ...,
‚u ‚v
‚ ‚
’•
X0 = ψ + ...,
‚• ‚ψ
‚ ‚ 1‚ 1‚
Y 1 = ’ψ1 ’u
+ •1 +v + ...,
‚u ‚v 2 ‚• 2 ‚ψ
2

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

and
‚ ‚
S0 = (u + xu1 + 2tut ) + (v + xv1 + 2tvt )
‚u ‚v
1 ‚ 1 ‚
+ ( • + x•1 + 2t•t ) + ( ψ + xψ1 + 2tψt ) + ...,
2 ‚• 2 ‚ψ
the recursion operator U1 = R generates ¬ve hierarchies of symmetries
Xn = X 0 Rn ,
Yn+ 1 = Y 1 Rn ,
2 2

X n = X 0 Rn ,
Y n+ 1 = Y 1 Rn ,
2 2

S n = S 0 Rn ,
where X0 Rn , . . . should be understood as
Xn = X0 Rn = (. . . ((X0 U1 ) U1 ) ...) U1 .
n times
318 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

2.2. Case B. In this case the supersymmetric nonlinear Schr¨dinger
o
equation is
ut = ’v2 + kv(u2 + v 2 ) ’ (c1 ’ 4k)u1 •ψ ’ 4kvψψ1 ,
’ (c1 + 8k)uψ•1 + 4ku•ψ1 + c1 v••1 ,
vt = u2 ’ ku(u2 + v 2 ) ’ (c1 ’ 4k)v1 •ψ + 4ku••1 ,
(c1 + 8k)v•ψ1 ’ c1 uψψ1 ’ 4kvψ•1 ,
•1 = ’ψ2 + (3ku2 + kv 2 )ψ ’ 2kuv• ’ (c1 ’ 4k)•ψ•1 ,
ψ1 = •2 ’ (ku2 + 3kv 2 )• + 2kuvψ ’ (c1 ’ 4k)•ψψ1 .
We introduce the following nonlocal variables, resulting from computed con-
servation laws,
p0 = D’1 (•ψ),
p0 = D’1 p1 + (c1 + 4k)p1 ,
1
p1 = D’1 u2 + v 2 + (c1 + 4k)(••1 + ψψ1 ) ,
2k
1
p1 = D’1 (uψ ’ v•)q 1 ’ (••1 + ψψ1 ) ,
2k
2


q 1 = D’1 (uψ ’ v•),
2
1
q 1 = D’1 q 3 + (c1 + 4k)•ψq 1 ,
2
2 2 2

q 3 = D’1 (vψ1 + u•1 )
2

and additionally
q’ 1 = D’1 (q 1 ),
2 2
1 1
p2 = D’1 ’ uv1 + (c1 + 12k)(u2 + v 2 )•ψ ,
(c1 + 4k)•1 ψ1 +
4k 4
1
p2 = D’1 ’ (vψ1 + u•1 )q 1 + •1 ψ1 .
2k
2


Within this covering, we constructed a deformation of the form
1
(c1 ’ 4k)•ψ
U 1 = ωv 1 + ω u
2
1 1
’ 4kuψ + (c1 ’ 4k)v• + ωψ (c1 ’ 4k)vψ + 4ku•
+ ω•
4 4
1 1
(c1 ’ 4k)u1 + ωp1 ’ kv + ωp1 ’ k(c1 + 12k)v
+ ω p0
2 2
1 1
+ ωq 1 k(c1 + 12k)vq 1 + (c1 + 4k)•1
2 2
2
2
1 ‚
’ (c1 + 4k)ψ
+ ωq 3
2 ‚u
2
2. SUPERSYMMETRIC EXTENSIONS OF NLS 319

1 1
’ ωu1 + ωv (c1 ’ 4k)•ψ + ω• (’4kvψ ’ (c1 ’ 4k)u•)
+
2 4
1
+ ωψ (’ (c1 ’ 4k)uψ + 4kv•)
4
1 1
+ ωp0 (c1 ’ 4k)v1 + ωp1 (ku) + ωp1 k(c1 + 12k)u
2 2
1 1
+ ωq 1 (’ k(c1 + 12k)uq 1 + (c1 + 4k)ψ1 )
2 2
2
2
1 ‚
+ ωq 3 (c1 + 4k)•
22 ‚v
1 1
+ ωψ1 + ω• (c1 ’ 4k)•ψ + ωp0 (c1 ’ 4k)•1 ’ ωp1 kψ
4 2
1 1 ‚
’ ωp1 k(c1 + 12k)ψ + ωq 1 (’2ku ’ k(c1 + 12k)ψq 1 )
2 2 ‚•
2
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-

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