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298 6. SUPER AND GRADED THEORIES

q 1 ,x = uψ ’ v•,
2
q 1 ,t = u•1 + vψ1 ’ u1 • ’ v1 ψ,
2
q 1 ,x = u• + vψ,
¯
2
q 1 ,t = ’uψ1 + v•1 + u1 ψ ’ v1 •,
¯
2

p1,x = u2 + v 2 ’ 2••1 ’ 2ψψ1 ,
p1,t = 2u1 v ’ 2uv1 ’ 4•1 ψ1 + 2•2 ψ + 2•ψ2 ,
p2,x = uv1 + 2•1 ψ1 ,
1 1
p2,t = u2 u ’ (u2 + v1 ) + k(v 4 ’ 2u2 v 2 ’ 3u4 ) + 2(ψ1 ψ2 + •1 •2 )
2
21 4
2
+ 4ku (••1 + ψψ1 ) + 4kuv1 •ψ + 8k•ψ•1 ψ1 . (6.177)
From the conservation laws given in Theorem 6.39 we can introduce
nonlocal variables by formally de¬ning
’1
p0 = Dx p0,x ,
’1
q 1 = Dx q 1 ,x ,
2 2
’1
q1 =
¯ Dx q 1 ,x ,
¯
2 2
’1
p1 = Dx p1,x ,
’1
p2 = Dx p2,x . (6.178)
Using these nonlocal variables one can try to ¬nd a nonlocal generalized
symmetry, which might be used in the construction of an in¬nite hierarchy
of symmetries and conserved quantities for (6.170). From the associated
computations we arrive at the following theorem.
Theorem 6.40. The supersymmetric NLS equation given by (6.170) ad-
mits a nonlocal symmetry of degree 1 of the form
‚ ‚ 1‚ 1‚
’ψ1 ’
Z1 = q 1 + •1 +
‚u ‚v 2 ‚• 2 ‚ψ
2

‚ ‚ 1‚ 1‚
+ q 1 •1
¯ + ψ1 + +
‚u ‚v 2 ‚• 2 ‚ψ
2

‚ ‚ ‚ ‚
+ k ’1 •1 + k ’1 ψ1
’ 2v•ψ + 2u•ψ
‚u ‚v ‚• ‚ψ
¯¯
= q1 Y1 ’ q1 Y1 + B (6.179)
2 2 2 2

where B is given by
‚ ‚ ‚ ‚
+ k ’1 •1 + k ’1 ψ1
B = ’2v•ψ + 2u•ψ . (6.180)
‚u ‚v ‚• ‚ψ
The existence of the symmetry Z1 of the form (6.179) should be com-
pared with the existence of a similar symmetry for the supersymmetric KdV
equation, considered in the previous Sections 4.2 and 6. It should be noted
8. SUPERSYMMETRIC EXTENSIONS OF THE NLS 299

that relation (6.179) just holds for the ‚/‚u-, ‚/‚v-, ‚/‚•- and ‚/‚ψ-
components. Starting from (6.175) and (6.179), we can construct new sym-
metries of (6.170) by using the graded commutator of vector ¬elds
[X, Y ] = X —¦ Y ’ (’1)|X|·|Y | Y —¦ X.
Computing the commutators of (6.175) we get the identities
¯¯
[Y 1 , Y 1 ] = X1 , [Y 1 , Y 1 ] = X 1 ,
2 2 2 2
¯ ¯
[X0 , Y 1 ] = ’Y 1 , [X0 , Y 1 ] = Y 1 ,
2 2 2 2
¯ ¯ ¯¯
[X0 , Y 1 ] = ’Y 1 ,
[X 0 , Y 1 ] = Y 1 , (6.181)
2 2 2 2

all other commutators of (6.175) being zero.
¯
In order to compute the commutators [Z1 , Y 1 ] and [Z1 , Y 1 ], we are forced
2 2
¯ 1 towards the nonlocal
to compute the prolongations of the vector ¬elds Y 1 , Y
2 2
variables q 1 and q 1 . In other words we have to compute the ‚/‚q 1 - and
¯
2 2 2
¯ 1 . These components can be
‚/‚ q 1 -components of the vector ¬eld Y 1 and Y
¯
2 2 2
obtained by requiring the invariance of q 1 ,x and q 1 ,x , i.e.,
¯
2 2
q1
ψ •
u v
Dx (Y 1 2 ) = Y 1 (q 1 ,x ) = Y 1 (uψ ’ v•) = Y 1 ψ + uY 1 ’ Y 1 • ’ vY 1
2 2 2 2 2
2 2 2
q1
¯
• ψ
u v
Dx (Y 1 2 ) = Y 1 (¯1 ,x ) = Y 1 (u• + vψ) = Y 1 • + uY 1 + Y 1 ψ + vY 1 (6.182)
q
2 2 2 2 2
2 2 2
q1 q1
¯
2
and Y 1 2 are the ‚/‚q 1 - and ‚/‚ q 1 -components of Y 1 . Similar
where Y 1 ¯
2 2 2
2 2
q1 q1
¯
¯ ¯
relations hold for Y 1 2 and Y 1 2 .
2 2
A straightforward computation yields
q1 q1
1 ¯ 1 2 = •ψ,
Y 1 = ’ p1 ,
2
Y
2
2 2
q1
¯ q1
¯ 1
¯
Y 1 2 = •ψ, Y 1 2 = p1 ,
2
2 2
p1 p1
¯
Y 1 = ’(uψ ’ v•), Y 1 = u• + vψ. (6.183)
2 2

¯
Now the commutators [Z1 , Y 1 ] and [Z1 , Y 1 ] give the following results:
2 2

1
Y 3 = [Z1 , Y 1 ] = q 1 X1 + p1 Y 1
2
2 2 2 2

1 ‚
+ ’ k ’1 ψ2 + (u2 + 3v 2 )ψ + 3•ψ•1 + uv•
2 ‚u
1 ‚
+ k ’1 •2 + (3u2 + v 2 )• + 3•ψψ1 ’ uvψ
2 ‚v
1 3 ‚ 1 ’1 3 ‚
+ ’ k ’1 v1 + u•ψ + k u1 + v•ψ ,
2 2 ‚• 2 2 ‚ψ

¯ ¯
Y 3 = [Z1 , Y 1 ] = ’¯1 X1 + p1 Y 1
q
2
2 2 2 2
300 6. SUPER AND GRADED THEORIES

1 ‚
+ k ’1 ψ2 ’ (u2 + 3v 2 )• + 3•ψψ1 + uvψ
2 ‚u
1 ‚
+ k ’1 ψ2 ’ (3u2 + v 2 )• ’ 3•ψ•1 ’ uv•
2 ‚v
1 3 ‚ 1 ’1 3 ‚
+ ’ k ’1 u1 ’ v•ψ + k v1 + u•ψ , (6.184)
2 2 ‚• 2 2 ‚ψ
¯
i.e., Y 3 and Y 3 are two new higher order supersymmetries of (6.170). In ef-
2 2
fect, we are here considering the supersymmetric NLS equation in the graded
Abelian covering by the variables p1 , q 1 , q 1 , where the following system of
¯
2 2
di¬erential equations holds

ut = ’v2 + kv(u2 + v 2 ) + 4ku1 •ψ ’ 4kv(••1 + ψψ1 ),
vt = u2 ’ ku(u2 + v 2 ) + 4kv1 •ψ + 4ku(••1 + ψψ1 ),
•t = ’ψ2 + k(u2 + v 2 )ψ + 4k•ψ•1 ,
ψt = •2 ’ k(u2 + v 2 )• + 4k•ψψ1 ,
q 1 ,x = uψ ’ v•,
2
q 1 ,t = u•1 + vψ1 ’ u1 • ’ v1 ψ,
2
q 1 ,x = u• + vψ,
¯
2
q 1 ,t = ’uψ1 + v•1 + u1 ψ ’ v1 •,
¯
2

p1,x = u2 + v 2 ’ 2••1 ’ 2ψψ1 ,
p1,t = 2u1 v ’ 2uv1 ’ 4•1 ψ1 + 2•2 ψ + 2•ψ2 . (6.185)

We are now able to prove the following lemma.

Lemma 6.41. By de¬ning

Yn+ 1 = [Z1 , Yn’ 1 ],
2 2
¯ ¯
Yn+ 1 = [Z1 , Yn’ 1 ], (6.186)
2 2


n = 1, 2, . . . , we obtain two in¬nite hierarchies of nonlocal supersymmetries
of equation (6.170).

Proof. First of all note, that the vector ¬eld ‚/‚p1 is a nonlocal sym-
metry of (6.170) and an easy computation shows that

[ , Z1 ] = 0,
‚p1
‚ 1
[ , Y3 ] = Y1 ,
‚p1 2 22
‚¯ 1¯
[ , Y3 ] = Y1 . (6.187)
‚p1 2 22
8. SUPERSYMMETRIC EXTENSIONS OF THE NLS 301

Secondly, note that by an induction argument and using the Jacobi identity
and (6.187) it is easy to prove that
‚ ‚ 1 1
[ , Yn+ 1 ] = [ , [Z1 , Yn’ 1 ]] = [Z1 , Yn’ 3 ] = Yn’ 1 ,
‚p1 ‚p1 2 2
2 2 2 2

‚¯ 1¯
[ , Yn+ 1 ] = Yn’ 1 . (6.188)
‚p1 2
2 2

From (6.188) it immediately follows that the assumption Yn+ 1 = 0 leads to
2
the conclusion that also Yn’ 1 = 2[‚/‚p1 , Yn+ 1 ] = 0, which proves that the
2 2
¯
hierarchies {Y 1 }n∈N , {Y 1 }n∈N are in¬nite.
n+ 2 n+ 2

Higher order conservation laws arise in the construction of prolongation
¯
of the vector ¬elds Y 1 , Y 1 and Z1 towards nonlocal variables, the ¬rst of
2 2
which resulted in (6.183).
q1
In order to compute the Z1 2 component of the vector ¬eld Z1 we have
to require the invariance of the equation q 1 ,x = uψ ’ v•, i.e.,
2

q1
¯¯
Dx (Z1 2 ) = Z1 (q 1 ,x ) = q 1 Y 1 (q 1 ,x ) ’ q 1 Y 1 (q 1 ,x ) + B(q 1 ,x )
2 2 2 2 2 2 2 2
1
= q 1 (’ p1,x ) ’ q 1 (•ψ)1 + B(q 1 ,x )
¯
2
2 2 2

due to (6.177) and (6.179), from which we obtain
q1 1
Z1 2 = ’ p1 q 1 ’ q 1 (•ψ) + k ’1 (uψ ’ v•)
¯
2 2 2
x
1
p1 (uψ ’ v•) ’ k ’1 (u1 ψ ’ v1 •) dx (6.189)
+
’∞ 2
and in a similar way
q1
¯
¯¯ q
Dx (Z1 2 ) = Z1 (¯1 ,x ) = q 1 Y 1 (¯1 ,x ) ’ q 1 Y 1 (¯1 ,x ) + B(¯1 ,x )
q q q
2 2 2 2 2 2 2 2
1
= q 1 (•ψ)1 ’ q 1 ( p1,x ) + B(¯1 ,x ),
¯ q
22
2 2

yielding
q1
¯ 1
Z1 2 = q 1 (•ψ) ’ p1 q 1 + k ’1 (u• + vψ)
¯
2
2 2
x
1
p1 (u• + vψ) ’ k ’1 (u1 • + v1 ψ)) dx. (6.190)
+
’∞ 2
So the prolongation of Z1 towards the nonlocal variables q 1 , q 1 requires the
¯
2 2
introduction of two additional nonlocal variables
x
1
p1 (uψ ’ v•) ’ k ’1 (u1 ψ ’ v1 •) dx,
q3 =
’∞ 2
2
x
1
p1 (u• + vψ) ’ k ’1 (u1 • + v1 ψ) dx.
q3 =
¯ (6.191)
’∞ 2
2
302 6. SUPER AND GRADED THEORIES

It is a straightforward check that q 3 , q 3 are associated to nonlocal conserved
¯
2 2
¯
quantities Q 3 , Q 3 . Thus we have found two new nonlocal variables q 3 and
2 2 2
q 3 with
¯
2

1
q 3 ,x = p1 (uψ ’ v•) ’ k ’1 (u1 ψ ’ v1 •),
2
2

1
q 3 ,x = p1 (u• + vψ) ’ k ’1 (u1 • + v1 ψ).
¯ (6.192)
2
2

¯
From this we proceed to construct the nonlocal components of Y 1 and Y 1
2 2
with respect to q 3 , q 3 , which can be obtained by requiring the invariance of
¯
2 2
q 3 ,x and q 3 ,x .
¯
2 2
In this way we ¬nd
q3
Dx (Y 1 2 ) = Y 1 (q 3 ,x )
2 2
2
1 1
= Y 1 (p1 )(uψ ’ v•) + p1 Y 1 (uψ ’ v•)
22 2 2

’ k ’1 Y 1 (u1 ψ ’ v1 •)
2
1 1 1
= p1 (’ψ1 ψ ’ u2 ’ •1 • ’ v 2 )
2 2 2
1 1
’ k ’1 (’ψ2 ψ ’ uu1 ’ •2 • ’ vv1 ) (6.193)
2 2
yielding
q3
1 1
Y 1 2 = p2 ’ k ’1 (u2 + v 2 ) + 4(••1 + ψψ1 ) (6.194)
81 4
2

In similar way we ¬nd
x
q3
¯ 1 1
’1 ’1
Y 1 = p1 •ψ ’ k (•1 ψ + •ψ1 + uv) ’ k
2
(uv1 + 2•1 ψ1 ) dx,
2 2 ’∞
2
x
q3
¯ 1 2 = 1 p1 •ψ ’ k ’1 (•1 ψ + •ψ1 + 1 uv) ’ k ’1
Y (uv1 + 2•1 ψ1 ) dx,
2 2 ’∞
2
q3
¯ 1 1
¯
Y 1 2 = p2 ’ k ’1 (u2 + v 2 ) ’ 4(••1 + ψψ1 ). (6.195)
1
8 4
2

Hence we see from (6.193) that the computation of the nonlocal components
q3 q3
¯
¯
Y 1 2 and Y 1 2 requires the introduction of a new nonlocal variable
2 2
x
p2 = (uv1 + 2•1 ψ1 ) dx. (6.196)
’∞
It is easily veri¬ed that p2 is associated to a conserved quantity P2 . In
arriving at the previous results, (6.195), we are working in a covering of the
supersymmetric NLS equation with nonlocal variables p1 , q 1 , q 1 , q 3 , q 3 ,
¯ ¯
2 2 2 2
p2 ; i.e., we consider system (6.185), together with the di¬erential equations,
de¬ning q 3 , q 3 , p2 .
¯
2 2
8. SUPERSYMMETRIC EXTENSIONS OF THE NLS 303

Summarizing the results obtained so far, we see that the odd potentials
¯ ¯
Q 1 , Q 1 , Q 3 and Q 3 enter in a natural way in the prolongation of Z1 ,
2 2 2 2
whereas the even potentials P1 and P2 enter in the prolongation of Y 1 and
2
¯ 1 . This situation is similar to that arising in the supersymmetric KdV
Y
2
equation treated in Section 6.
8.2.2. Case B. In order to gain insight in the structure of the super-
symmetric NLS equation (6.172), we start with the computation of (x, t)-
independent conserved quantities of degree ¤ 3. We arrive at the following
result.
Theorem 6.42. The supersymmetric NLS equation (6.172) admits the
following set of local even and odd conserved quantities of degree ¤ 3:

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
Cases A and B in this setting. Due to the conjecture that the nonlocal recur-
¯
sion symmetry X1 given by (6.204) is a Hamiltonian symmetry associated to
¯
a linear combination of P2 and P2 we hope to prove the formal construction
and the Lie superalgebra structure of the local and nonlocal symmetries and
the Poisson structure of the associated hierarchies of conserved quantities.
Remark 6.16. The contents of this section clearly indicates how to con-
struct supersymmetric extensions of classical integrable systems, which can
be termed completely integrable by the existence of in¬nite hierarchies of
local and/or nonlocal symmetries and conservation Laws.
CHAPTER 7


Deformations of supersymmetric equations

We shall illustrate the developed theory of deformations of supersym-
metric equations and systems through a number of examples.
First of all we shall continue the theory for the supersymmetric extension
of the KdV equation [35, 72, 74, 87] started in Section 6 of the previous
chapter. We shall construct the recursion operator for symmetries, which is
just realized by the contraction of a symmetry and the deformation. More-
over we construct a new hierarchy of conserved quantities and a hierarchy
of (x, t)-dependent symmetries.
As a second application, we consider the two supersymmetric extensions
of the nonlinear Schr¨dinger equation (Section 2) leading to the recursion
o
operators for symmetries and new hierarchies of odd and even symmetries.
We shall also construct a supersymmetric extension of the Boussinesq
equation, construct deformations for this system and eventually arrive at
the recursion operator for symmetries and at hierarchies of odd and even
symmetries and conservation laws.
Finally, we construct two-dimensional supersymmetric extensions (i.e.,
extensions including two odd dependent variables) of the KdV and study
their symmetries, conservation laws, and deformations, obtaining recursion
operators and hierarchies of symmetries.

1. Supersymmetric KdV equation
We start at the supersymmetric extension of the KdV equation [72, 74]
and restrict our considerations to the case a = 3 in the system
ut = ’u3 + 6uu1 ’ a••2 ,
•t = ’•3 + (6 ’ a)•1 u + a•u1 (7.1)
(see Section 6 of Chapter 6).
Features and properties of the equation were discussed in several papers,
cf. [35, 87].
1.1. Nonlocal variables. In order to construct a deformation of (7.1),
we have to construct an appropriate covering by the introduction of a num-
ber of nonlocal variables. These nonlocal variables, which arise classically
from conserved densities related to conservation laws, have been computed
to be
q 1 =D’1 (•),
2

309
310 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

q 3 =D’1 (p1 •),
2

12
q 5 =D’1 p • ’ u• (7.2)
21
2

and
p1 =D’1 (u),
p0 =D’1 (p1 ),
p1 =D’1 (•q 1 ),
2
’1 2
(u ’ ••1 ),
p3 =D
p3 =D’1 (u2 ’ 2u•q 1 + uq 1 q 3 ), (7.3)
2 2 2

where D = Dx .
Odd nonlocal variables will be denoted by q, while even nonlocal vari-
ables will be denoted by p and p. We mention that, in e¬ect, the total
derivative operator Dx should be lifted to an appropriate covering, where it
is denoted by the same symbol Dx , i.e.,
‚ ‚ ‚ ‚
Dx = + u1 + u2 + u3 + ...
‚x ‚u ‚u1 ‚u2
‚ ‚ ‚
+ (q 1 )x + (q 3 )x + (q 5 )x
‚q 1 ‚q 3 ‚q 5
2 2 2
2 2 2
‚ ‚ ‚
+ (p0 )x+ (p1 )x + (p3 )x
‚p0 ‚p1 ‚p3
‚ ‚
+ (p1 )x + (p3 )x . (7.4)
‚p1 ‚p3
Other odd nonlocal variables, q 7 and q 9 , are given by
2 2

12
q 7 = D’1 p3 • + q 1 p1 u ’ p 1 u 1 + u 2 ’ u 2 ,
2
2 2

q 9 = D’1 6p3 p1 • + q 1 p3 u ’ 3p2 u1 + 6p1 u2 ’ 6p1 u2
1 1
2 2

+ 36uu1 ’ 6u3 . (7.5)
Note that the variables q 3 , q 5 , q 7 , q 9 , p0 , p1 , p3 contain higher nonlocalities.
2 2 2 2

1.2. Symmetries. For hierarchies {Y 2n+1 }, {X2n+1 }, n ∈ N, of sym-
2
metries of equation (7.1) we refer to 4 of Chapter 6. Recall that
‚ ‚
Y 1 =•1 +u ,
‚u ‚•
2

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

‚ ‚
X1 =u1 + •1 ,
‚u ‚•
1. SUPERSYMMETRIC KDV EQUATION 311

‚ ‚
X3 = ’ u t ’ •t ,
‚u ‚•
X5 = ’ (u5 ’ 10u3 u ’ 20u2 u1 + 30u1 u2 + 5••4 + 5•1 •3

’ 20u••2 ’ 20u1 ••1 )
‚u

’ (•5 ’ 5u•3 ’ 10u1 •2 ’ 10u2 •1 + 10u2 •1 + 20u1 u• ’ 5u3 •) .
‚•
(7.6)
Moreover we found the supersymmetric analogue of the (x, t)-dependent
symmetry which acts as recursion on the even hierarchy {X2n+1 }, n ∈ N,
i.e.,
V2 = ’6tX5 ’ 2xX3 + H2 , (7.7)
where
H2 = ’ q 1 (•2 + p1 •1 ’ •u) + 3q 3 •1 ’ 13••1
2 2

+ 4p1 u1 ’ 2p1 u1 ’ 8u2 + 16u2
‚u
+ ’ q 1 (p1 u ’ u1 ) + 3q 3 u
2 2

+ 2p1 •1 ’ 2p1 •1 ’ 7•2 + 14•u . (7.8)
‚•
It should be noted that the vector ¬elds
‚ ‚ ‚
’ q1 + (p1 q 3 ’ 2q 5 )
Y’ 1 = ,
‚q 1 2 ‚p 2 ‚p
2 2
1 3
2
‚ ‚ ‚ ‚
X ’1 = + q1 + q3 +x ,
‚p1 2 ‚q 3 2 ‚q 5 ‚p0
2 2

X’1 = = (7.9)
‚p1
are symmetries of equation (7.1) in the covering de¬ned by (7.2), (7.3).
These symmetries are vertical in the covering under consideration.
Computation of graded Lie brackets leads to the identities
[Y’ 1 , V2 ] =Y 3 ,
2 2

[X’1 , V2 ] =2Z1 + 4X1 ,
[X ’1 , V2 ] = ’ 2X1 , (7.10)
where Z1 is the nonlocal symmetry of degree 1 (cf. 4 of Chapter 6), which
acts, by its Lie bracket, as a recursion operator on the odd hierarchy {Y n+ 1 },
2
n ∈ N. Recall that
‚ ‚
+ (q 1 u ’ •1 )
Z1 = (q 1 •1 ) + ... (7.11)
‚u ‚•
2 2
312 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

1.3. Deformations. In order to construct a deformation of (7.1), we
formally construct the in¬nite-dimensional Cartan covering (see Subsection
3.5 of Chapter 6) over the in¬nite covering of (7.1) by (7.2), (7.3).
In the setting under consideration, the Cartan covering is described by
the Cartan forms ω0 , . . . , ωk , . . . on the in¬nite prolongation of the super-
symmetric KdV equation together with the forms corresponding to the non-
local variables (7.2), (7.3):

ωq 1 , ωq 3 , ωq 5 , ωp0 , ωp1 , ωp1 , ωp3 , ωp3 , (7.12)
2 2 2


where ωf = LU• (f ) denotes the Cartan form corresponding to the potential
f (see (2.13) on p. 66). According to (7.12), we search for a generalized
vector ¬eld which is linear with respect to the Cartan forms. Applying the
deformation condition on this vector ¬eld and taking into account the grad-
ing of (7.1), (7.2), (7.3), and (7.12), we arrive at the following deformation

U1 = ωu2 + ωu (’4u) + ω•1 (’2•) + ω• (•1 )
+ ωq 1 (q 1 u1 + p1 •1 + •2 ’ u•)
2
2

+ ωp1 (’2u1 ) + ωp1 (u1 ) + ωq 3 (’•1 )
‚u
2

+ ω•2 + ω• (’2u) + ωu (’2•)
+ ωq 1 (’q 1 •1 + p1 u ’ u1 )
2
2

+ ωp1 (’•1 ) + ωp1 (•1 ) + ωq 3 (’u) . (7.13)
‚•
2


Similar to the results of Subsection 2.8 of Chapter 6, the element U1 satis¬es
the identity

[[U1 , U1 ]]fn = 0, (7.14)

which means that U1 is a graded Nijenhuis operator in the sense [49].
We now rede¬ne our hierarchies in the following way. First we put

‚ ‚
Y 1 =•1 +u ,
‚u ‚•
2

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

‚ ‚
X1 =u1 + •1 ,
‚u ‚•
‚ ‚
+ (q 1 u ’ •1 )
X 1 =(q 1 •1 ) = Z1 ,
‚u ‚•
2 2

‚ 3 ‚
V0 =(2u + xu1 + 3tut ) + ( • + x•1 + 3t•t ) (7.15)
‚u 2 ‚•
1. SUPERSYMMETRIC KDV EQUATION 313

and de¬ne the odd and even hierarchies of symmetries by
n
Y2n+ 1 = ((. . . (Y 1 U1 ) U1 ) . . . ) U1 ) = Y 1 U1 ,
2 2 2

n times
n
Y2n+ 3 = Y 3 U1 ,
2 2
n
X2n+1 = X1 U1 ,
n
X 2n+1 = X 1 U1 ,
n
V2n = V0 U1 . (7.16)
1.4. Passing from deformations to “classical” recursion oper-
ators. Here we rewrite the main result of the previous subsection in more
conventional terms, i.e., as formal matrix integro-di¬erential operators. We
shall see that this representation is far less “economical” than representation
(7.13). Moreover, if one uses conventional left action of di¬erential opera-
tors, additional parasitic signs arise, which makes this representation even
more cumbersome.
Let X = (F,G) be a nonlocal symmetry of (7.1) in the covering de¬ned
by (7.2), (7.3) with 2-component generating function (F, G) and let |X| be
the degree of X; then one has |F | = |X| and |G| = |X| + 1.
It means that X is of the form

‚ ‚
Di (F )) + Di (G)
X= , (7.17)
‚ui ‚•i
i=0
where F and G satisfy the shadow equation for the covering in question and
D denotes the extension of the total derivative Dx onto the covering. Then
one has
iX (ωui ) = Di (F ),
iX (ω•i ) = Di (G) (7.18)
for all i = 0, 1, . . . From the de¬nition of nonlocal variables (see (7.2) and
(7.3)) one also has
iX (ωp1 ) =D’1 (F ),
iX (ωq 1 ) =D’1 (G),
2

iX (ωp0 ) =D’1 (D’1 (F )),
iX (ωq 3 ) =D’1 (D’1 (F )• + Gp1 ),
2

iX (ωp1 ) =D’1 (Gq 1 ’ D’1 (G)•),
2
1
iX (ωq 5 ) =D’1 (D’1 (F )p1 • + Gp2 ’ F • ’ Gu),
21
2

iX (ωp3 ) =D’1 (2uF ’ G•1 + D(G)•), (7.19)
while
314 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS


iX (ωp3 ) = D’1 2F u ’ 2F •q 1 ’ 2Guq 1 + 2D ’1 (G)u•
2 2

+ F q 1 q 3 + D’1 (G)uq 3 ’ D’1 D’1 (F )• + Gp1 uq 1 (7.20)
2 2 2 2

(the last equality is given for reasons of completeness only and will not be
used below).
Then the recursion operator R corresponding to the deformation U 1 ,
(7.13) acts as
R(X) = iX (U1 ) (7.21)
and is of the form
R(F, G) = (F1 , G1 ), (7.22)
where
F1 = D2 (F ) + F (’4u) + D(G)(’2•) + G(•1 )
+ D’1 (G)(’q 1 u1 + p1 •1 + •2 ’ u•) + D ’1 (F )(’2u1 )
2

+ D’1 (Gq 1 ’ D’1 (G)•)u1 + D’1 (D’1 (F )• + Gp1 )(’•1 ),
2

G1 = D2 (G) + G(’2u) + F (’2•)
+ D’1 (G)(’q 1 •1 + p1 u ’ u1 ) + D’1 (F )(’•1 )
2
’1
(Gq 1 ’ D’1 (G)•)(•1 ) + D’1 (D’1 (F )• + Gp1 )(’u).
+D (7.23)
2

Due to the relations
D’1 (Gq 1 ) = D’1 (G)q 1 ’ D’1 (D’1 (G)•)
2 2

= ’(’1)|X| q 1 D’1 (G) + (’1)|X| D’1 (•D’1 (G)),
2

D’1 (Gp1 ) = p1 D’1 (G) ’ D ’1 (uD’1 (G)), (7.24)
we rewrite F1 , G1 in a left action notation as
F1 = D2 (F ) ’ 4uF + (’1)|X| 2•D(G) ’ (’1)|X| •1 G
’ (’1)|X| (’q 1 u1 + p1 •1 + •2 ’ u•)D ’1 (G) ’ 2u1 D’1 (F )
2
|X|
u1 q 1 D’1 (G) + (’1)|X| u1 D’1 (•D’1 (G))
’ (’1)
2

+ (’1)|X| u1 D’1 (•D’1 (G)) + •1 D’1 (•D’1 (F )
+ (’1)|X| •1 p1 D’1 (G) ’ (’1)|X| •1 D’1 (uD’1 (G))),
G1 = D2 (G) ’ 2uG ’ (’1)|X| 2•F
+ (’q 1 •1 + p1 u ’ u1 )D’1 (G) ’ (’1)|X| •1 D’1 (F )
2

+ (’1)|X| •1 ((’1)|X|+1 q 1 D’1 (G) ’ (’1)|X|+1 D’1 (•D’1 (G)))
2
’1 ’1
(G)) ’ (’1)|X| uD’1 (•D’1 (F ))
2|X|
+ (’1) •1 D (•D
’ up1 D’1 (G) + uD ’1 (uD’1 (G)). (7.25)
2. SUPERSYMMETRIC EXTENSIONS OF NLS 315

From this we ¬nally arrive at
F1 = D2 (F ) ’ 4uF ’ 2u1 D’1 (F ) + •1 D’1 (•D’1 (F ))
+ (’1)|X| 2•D(G) ’ (’1)|X| •1 G ’ (’1)|X| (•2 ’ u•)D ’1 (G)
+ (’1)|X| 2u1 D’1 (•D’1 (G))
’ (’1)|X| •1 D’1 (uD’1 (G)),
G1 = ’(’1)|X| 2•F ’ (’1)|X| •1 D’1 (F )
’ (’1)|X| uD’1 (•D’1 (F ))
+ D2 (G) ’ 2uG ’ u1 D’1 (G) + 2•1 D’1 (•D’1 (G))
+ uD’1 (uD’1 (G)), (7.26)
or
F1 = D2 (F ) ’ 4uF ’ 2u1 D’1 (F ) + •1 D’1 (•D’1 (F ))
+ (’1)|X| 2•D(G) ’ •1 G + (’•2 + u•)D ’1 (G)
+ 2u1 D’1 (•D’1 (G)) ’ •1 D’1 (uD’1 (G)) ,
G1 = (’1)|X| ’ 2•F ’ •1 D’1 (F ) ’ uD ’1 (•D’1 (F ))
D2 (G) ’ 2uG ’ u1 D’1 (G) + 2•1 D’1 (•D’1 (G)) + uD ’1 (uD’1 (G)),
(7.27)
leading to the recursion operator R = Rij , where
R11 = D2 ’ 4u ’ 2u1 D’1 + •1 D’1 •D’1 ,
R12 = (’1)|X| (2•D ’ •1 ’ •2 D’1 + u•D ’1 + 2u1 D’1 •D’1
’ •1 D’1 uD’1 ),
R21 = (’1)|X| (’2• ’ •1 D’1 ’ uD’1 •D’1 ),
R22 = D2 ’ 2u ’ u1 D’1 + 2•1 D’1 •D’1 + uD’1 uD’1 . (7.28)
Note that the classical recursion operator for the KdV equation is just the
•-independent part of R11 :
R0 = D2 ’ 4u ’ 2u1 D’1 . (7.29)
From the above representation it becomes clear that the action of the re-
cursion operator considered as action from the left, requires introduction of
the sign (’1)|X| , which makes the operation not natural. Therefore we shall
restrict ourselves to representations similar to (7.13).

2. Supersymmetric extensions of the NLS equation
In this section, we shall discuss deformations and recursion operators for
the two supersymmetric extensions of the nonlinear Schr¨dinger equation
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

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. 13
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