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P0 = •ψ dx,
’∞

(uψ ’ v•) dx,
Q1 =
2
’∞

1 ’1
(c + 4k)(••1 + ψψ1 ) + 2k(u2 + v 2 ) dx,
P1 = k
2
’∞

Q3 = (u•1 + vψ1 ) dx,
2
’∞

1 ’1
(c + 4k)•1 ψ1 + (c + 12k)k(u2 + v 2 )•ψ ’ 4kuv1 dx,
P2 = k
4
’∞

’k ’1 uψ2 ’ v•2
Q5 =
2
’∞

’ k(u2 + v 2 )(uψ ’ v•) ’ 4k•ψ(u•1 + vψ1 ) dx. (6.197)
Moreover, in the case where c = ’4k we have an additional local conserved
quantity of degree 3 given by

16uv(•ψ1 ’ ψ•1 )
P3 =
’∞

+ 32uv•ψ + (u2 + v 2 )2 ’ 2k ’1 (uu2 + vv2 ) dx. (6.198)

Motivated by the nonlocal results in case A, we introduce the nonlocal
variables p0 , q 1 , p1 , q 3 , p2 and q 5 as formal integrals associated to the
2 2 2
conserved quantities given in (6.197).
Including these new nonlocal variables in our computations, we get an
additional set of nonlocal conserved quantities

1
¯
Q1 = 2q 3 + (c + 4k)•ψq 1 dx,
2
2 2 2
’∞

1 ’1
¯ 2k(uψ ’ v•)q 1 ’ (••1 + ψψ1 ) dx,
P1 = k
2 2
’∞
304 6. SUPER AND GRADED THEORIES


1
¯ ’ k ’1 2k(u•1 + vψ1 )q 1 ’ •1 ψ1 dx,
P2 = (6.199)
2 2
’∞
as well as an additional conserved quantity in the case c = ’4k, namely

¯
P0 = p1 dx.
’∞
This situation can be described as higher nonlocalities, or covering of a
covering, and as it will be shown lead to new interesting results.
Remark 6.15. The results (6.197), (6.199) indicate the existence of a
double hierarchy of odd conserved quantities {Qn+ 1 }n∈N as well as a double
2
hierarchy of even conserved quantities {Pn }n∈N .
In order to obtain any further results, we also need the conserved quan-
tity Q 7 of degree 7/2 which is given by
2


1 ’1
2u•3 + 2vψ3 ’ 2kv 3 ψ1 ’ 2ku3 •1 ’ 6kuv 2 •1 + 6ku2 v1 ψ
Q7 = k
6
2
’∞

’ 12kuvv1 • + 2c(uψ ’ v•)•1 ψ1 ’ (c ’ 12k)•ψ(uψ2 ’ v•2 ) dx. (6.200)
Let us stress that now, by the introduction of the nonlocal variables q 1 , p1 ,
¯¯
2
p2 and q 7 , associated to the appropriate conserved quantities, we are able to
¯
2
remove the condition c = ’4k on the existence of the conserved quantities
¯
P3 and P0 . By also including q 1 , p1 , p2 and q 7 in our computations, we
¯¯¯
2 2
¬nd four additional conserved quantities given by

¯
P0 = p1 + (c + 4k)¯1 dx,
p
’∞

1
¯ ’ k ’1 2kq 5 + 2k(u2 + v 2 )q 1 + (••1 + ψψ1 )q 1 dx,
Q3 =
2
2 2 2 2
’∞

k ’1 2k(c + 4k)(uψ ’ v•)q 5 + 2(c + 4k)(u2 ψψ1 + v 2 ••1 )
P3 =
2
’∞
’ 2(c + 12k)uvψ•1 ’ 2(c ’ 4k)uv•ψ1
+ 32ku1 v•ψ + k(u2 + v 2 )2 ’ 2(uu2 + vv2 ) dx,

1 ’1
¯ 4k 2 (uψ ’ v•)q 5 + 2k(u•1 + vψ1 )q 3
P3 = k
2 2 2
’∞

’ (•1 •2 + ψ1 ψ2 ) + (c ’ 4k)•ψ•1 ψ1 dx. (6.201)

Note that the ¬rst equation in (6.201) and the third one in (6.201) reduce
to the second equations in (6.199) and (6.197) respectively under the con-
dition c = ’4k. Furthermore, from the computation of the t-component
¯
of the conservation law q 3 associated to Q 3 , it becomes apparent why the
¯
2 2
introduction of the nonlocal variable q 7 is required in its construction.
2
8. SUPERSYMMETRIC EXTENSIONS OF THE NLS 305

So P3 is just an ordinary conserved quantity of this supersymmetric ex-
tension; for c = ’4k it is just a local conserved quantity, while for other
values of c it is a nonlocal one.
We now turn to the construction of the Lie algebra of even and odd
symmetries for the supersymmetric NLS equation (6.172). According to the
introduction of the nonlocal variables associated to the conserved quantities
obtained earlier in this section we ¬nd the following result.
Theorem 6.43. The supersymmetric NLS equation (6.172) admits the
following set of even and odd symmetries of degree ¤ 2. The symmetries of
degree 0 are given by
‚ ‚ ‚ ‚
’u ’• ,
X0 = v +ψ
‚u ‚v ‚• ‚ψ
‚ ‚ 1 ‚ 1 ‚
¯ ’ k ’1 • ’ k ’1 ψ ;
’ •q 1
X0 = ψq 1 (6.202)
2 ‚u 2 ‚v 8 ‚• 8 ‚•
the symmetries of degree 1/2 by
‚ ‚ 1‚ 1‚
Y 1 = •1 + ψ1 +u +v ,
‚u ‚v 2 ‚• 2 ‚ψ
2

‚ ‚ 1 ‚
¯ + (q 1 ψ ’ k ’1 u)
’ uq 1
Y 1 = vq 1
2 ‚u 2 ‚v 4 ‚•
2 2

1 ‚
+ (’q 1 • ’ k ’1 v) (6.203)
4 ‚ψ
2

the symmetries of degree 1 by
‚ ‚ ‚ ‚
X1 = u 1 + v1 + •1 + ψ1
‚u ‚v ‚• ‚ψ

¯
X1 = (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 ψp
‚•
2


+ ’ 4kvq 1 ’ 2•1 ’ 2k(c ’ 4k)¯1 •p ; (6.204)
‚ψ
2

the symmetries of degree 3/2 by
1 1 ‚
Y 3 = vq 3 + u1 q 1 ’ k ’1 ψ2 + u2 ψ ’ uv• ’ k ’1 c•ψ•1
2 2 ‚u
2 2 2

1 1 ‚
+ ’ uq 3 + v1 q 1 + k ’1 •2 ’ v 2 • + uvψ ’ k ’1 c•ψψ1
2 2 ‚v
2 2

‚ ‚
+ ’ q 3 • + q 1 ψ1
+ q 3 ψ + q 1 •1 ,
‚ψ ‚ψ
2 2 2 2


¯
Y 3 = 4kcu1 q 1 ’ (c ’ 4k)(ψ2 + c•ψ•1 ) + k(c ’ 12k)(u2 + v 2 )ψ
‚u
2 2
306 6. SUPER AND GRADED THEORIES


+ 4kcv1 q 1 + (c ’ 4k)(•2 ’ c•ψψ1 ) ’ k(c ’ 12k)(u2 + v 2 )•
‚u
2


+ 4kcq 1 •1 + 2k(c ’ 12k)u•ψ + 4kv1
‚•
2


+ 4kcq 1 ψ1 + 2k(c ’ 12k)v•ψ ’ 4ku1 , (6.205)
‚ψ
2

and ¬nally the symmetries of degree 2 by
X2 = v2 ’ kv(u2 + v 2 ) + (c ’ 4k)u1 •ψ + 4k(vψψ1 ’ u•ψ1 )

+ (c + 8k)uψ•1 ’ cv••1
‚u
’ u2 + kv(u2 + v 2 ) + (c ’ 4k)v1 •ψ ’ 4k(u••1 ’ vψ•1 )
+

’ (c + 8k)v•ψ1 + cu••1
‚v

+ ψ2 ’ k(3u2 + v 2 )ψ + (c ’ 4k)•ψ•1 + 2kuv•
‚•

’ •2 ’ k(u2 + 3v 2 )• + (c ’ 4k)•ψψ1 ’ 2kuvψ
+ ,
‚ψ
¯
X2 = (c + 4k)(’kψq 5 + ψ2 q 1 ’ 3ku2 ψq 1 + c•ψ•1 q 1 )
2 2 2 2

+ (c ’ 4k)(4k p2 v + vψ•1 ’ v•ψ1 )
¯

+ 16k 2 vq 1 q 3 + (c ’ 12k)kv 2 ψq 1 + 4ckuv•q 1
‚u
2 2 2 2

+ (c + 4k)(k•q 5 ’ •2 q 1 + 3kv 2 •q 1 + c•ψψ1 q 1 )
2 2 2 2

+ (c ’ 4k)(’4k p2 u ’ uψ•1 + u•ψ1 )
¯

’ 16k 2 uq 1 q 3 ’ (c ’ 12k)ku2 •q 1 ’ 4ckuvψq12
‚v
2 2 2

’ 2•2 + (c ’ 4k)(4k p2 ψ + 2•ψψ1 ) ’ 2(c ’ 12k)ku•ψq 1
+ ¯
2


+ (4ku + 16k 2 ψq 1 )q 3 ’ 4kv1 q 1 ’ 4kuvψ + 4kv 2 •
‚•
2 2 2


’ 2ψ2 ’ (c ’ 4k)(4k p2 • + 2•ψ•1 ) ’ 2(c ’ 12k)kv•ψq 1
+ ¯
2


+ (4kv ’ 16k 2 •q 1 )q 3 + 4ku1 q 1 ’ 4kuv• + 4kv 2 ψ . (6.206)
‚ψ
2 2 2


Analogously to case A, we have the following result.
¯
Theorem 6.44. The nonlocal even symmetry X1 given by (6.204) acts
as a recursion symmetry on the hierarchies of odd symmetries.
¯ ¯¯
In order to compute the graded commutators [X1 , Y 1 ] and [X1 , Y 1 ], we
2 2
¯
have to compute the components of Y 1 and Y 1 with respect to the nonlocal
2 2
9. CONCLUDING REMARKS 307

variables q 1 , q 3 and p1 . Analogously to the computations in Subsection 8,
¯
2 2
we ¬nd
q1
Y 1 2 = •ψ,
2
q3 1
Y 1 2 = (u2 + v 2 ),
4
2
1 1
Y 1¯1 = •ψq 1 + k ’1 (u• + vψ) ’ k ’1 q 3
p
(6.207)
4 2
2 2
2

and
q1
Y 1 2 = 0,
2
q3 1
Y 1 2 = ’ k ’1 (u2 + v 2 ),
8
2
1 1
Y 1¯1 = ’ k ’2 (u• + vψ) + k ’2 q 3 .
p
(6.208)
8 4 2
2

¯¯ ¯
Moreover, the computation of [X1 , Y 1 ] requires the ‚/‚ q 1 -component of X1
¯
2 2
which is given to be
q1
¯
X1 2 = ’(c + 4k)•ψq 1 ’ 2q 3 . (6.209)
2 2

Now the computation of the commutators leads to
1 1
¯ ¯
[X1 , Y 1 ] = (c ’ 12k)Y 3 ’ k ’1 Y 3 ,
2 4
2 2 2

1 1
¯¯ ¯
[X1 , Y 1 ] = (’ k ’1 c + 1)Y 3 + k ’2 Y 3 , (6.210)
4 8
2 2 2

¯ ¯
indicating that X1 acts as a recursion operator on the Y , Y hierarchies.
¯
It is our conjecture that X1 is a Hamiltonian symmetry for equation
(6.172). We refer to the concluding remarks for more comments on this
issue.

9. Concluding remarks
In the previous sections we proposed a construction for supersymmetric
generalizations of the cubic nonlinear Schr¨dinger equation (6.160) and dis-
o
cussed symmetries, conserved quantities for the resulting interesting cases
A and B. In both cases we found an in¬nite set of (higher order) local and
nonlocal symmetries. These facts indicate the complete integrability of both
systems.
It is possible to transform the results obtained thus far in the super¬eld
formulation. Namely, if we introduce the odd quantity ¦ by
¦ = ω + θq, (6.211)
where θ is an additional odd variable, and put
1‚
Dθ = + θDx , (6.212)
2 ‚θ
308 6. SUPER AND GRADED THEORIES

then
[Dθ , Dθ ] = Dx
and it is clear that Dθ corresponds to the supersymmetry Y 1 given by
2
1
(6.203). Notice that our de¬nition of Dθ di¬ers a factor 2 in the ‚/‚θ
term. This is caused by our requirement that [Dθ , Dθ ] = Dx , whereas the
operator Dθ introduced by Mathieu satis¬es [Dθ , Dθ ] = 2Dx . In this setting
the general complex equation (6.174) takes the form
i¦t = ’4Dθ ¦ + 2(c1 ’ c2 )¦¦— Dθ ¦
4 2

+ 2(c2 + 2k)¦Dθ ¦Dθ ¦— ’ 2c2 ¦— (Dθ ¦)2 (6.213)
Our hypothesis is that there exist Hamiltonian structures of the systems of

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