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(q 1 )t = ’v2 • + v1 •1 ’ v•2 + 3v 3 •,
2

(q 3 )t = p1 (’v2 • + v1 •1 ’ v•2 + 3v 3 •) + 2v 2 v1 •
2

’ v2 •1 + 4v 3 •1 + v1 •2 ’ v•3 ,
(q 5 )t = ’p2 (’v2 • + v1 •1 ’ v•2 + 3v 3 •)
1
2
292 6. SUPER AND GRADED THEORIES

+ p1 (2v•3 ’ 2v1 •2 ’ 8v 3 •1 + 2v2 •1 ’ 4v 2 v1 •)
+ 2v•4 ’ 2v1 •3 ’ 9v 3 •2 + 2v2 •2 ’ 13v 2 v1 •1
+ 4v••1 •2 + 5v 5 • ’ 9v 2 v2 •. (6.158)

The resulting symmetries are given here by
‚ ‚ ‚
+ (q 1 v ’ •1 )
Z 1 = (q 1 •1 ) + (q 1 v• + ••1 )
‚v ‚• ‚p1
2 2 2


+ (q 1 p1 ’ q 3 ) ,
2 ‚q 1
2
2
‚ ‚ ‚ ‚
Y = •1 +v + v• + p1 ,
1
‚v ‚• ‚p1 ‚q 1
2
2
‚ ‚
= (2q 1 v1 ’ p1 •1 + v 2 • ’ •2 ) + (2q 1 •1 ’ p1 v + v1 )
Y 3
‚v ‚•
2 2 2


+ (2q 1 (v 2 ’ ••1 ) ’ p1 v• ’ 2v•1 + v1 •)
‚p1
2

1 1 ‚
+ (2q 1 v• ’ p2 + v 2 + ••1 ) ,
21 2 ‚q 1
2
2
1 5 ‚
= ( p2 •1 + p1 (•2 ’ v 2 •) ’ 2q 3 v1 + •3 ’ v 2 •1 ’ 3vv1 •)
Y 5 1
2 2 ‚v
2 2

1 3 ‚
+ ( p2 v ’ p1 v1 ’ 2q 3 •1 + v2 ’ v 3 + 2v••1 )
21 2 ‚•
2

1
+ ( p2 v• + p1 (2v•1 ’ v1 •) ’ 2q 3 (v 2 ’ ••1 ) + 2v•2 ’ 2v1 •1
21 2

7 ‚
+ v2 • ’ v 3 •)
2 ‚p1
1 1 ‚
+ ( p3 ’ 2q 3 v• ’ p1 (••1 + v 2 ) ’ ••2 + vv1 + p3 )
61 2 ‚q 1
2
2
1 1
+ ( p4 ’ p2 (v 2 + 4••1 ) ’ 2p1 q 3 v• ’ p1 ••2
1
41
8 2

11 12 ‚
’ 2q 3 v•1 ’ •1 •2 + v 2 ••1 ’ v 4 + vv2 ’ v1 ) ,
8 2 ‚q 3
2
2
‚ ‚
X 1 = v1 + •1 ,
‚v ‚•
‚ ‚
X 3 = (’v3 + 6v 2 v1 ’ 3v••2 ’ 3v1 ••1 ) + (’•3 + 3v 2 •1 + 3vv1 •) ,
‚v ‚•
X 5 = (v5 ’ 10v3 v 2 ’ 40v2 v1 v ’ 10v1 + 30v1 v 4
3

+ 5v••4 + 10v1 ••3 + 5v••3 + 5v1 •1 •2 ’ 20v 3 ••2

+ 10v2 ••2 + 5v3 ••1 ’ 60v1 v 2 ••1 )
‚v
8. SUPERSYMMETRIC EXTENSIONS OF THE NLS 293

+ (•5 ’ 5v 2 •3 ’ 15v1 v•2 ’ 15v2 v•1 ’ 10v1 •1 + 10v 4 •1
2


’ 5v3 v• ’ 10v2 v1 • + 20v1 v 3 •) . (6.159)
‚•
The graded Lie algebra structure of the symmetries is similar to the structure
of that for the supersymmetric extension of the KdV equation considered in
the previous section.

8. Supersymmetric extensions of the NLS
Symmetries, conservation laws, and prolongation structures of the su-
persymmetric extensions of the KdV and mKdV equation, constructed by
Manin“Radul, Mathieu [72, 74], have already been investigated in previous
sections.
A supersymmetric extension of the cubic Schr¨dinger equation has been
o
constructed by Kulish [15] and has been discussed by Roy Chowdhury [89],
who applied the Painlev´ criterion to it. A simple calculation shows however
e
that the system does not admit a nontrivial prolongation structure. More-
over, as it can readily be seen, the resulting system of equations does not
inherit the grading of the classical NLS equation.
We shall now discuss a formal construction of supersymmetric extensions
of the classical integrable systems, the cubic Schr¨dinger equation being
o
just a very interesting application of this construction, which does inherit
its grading, based on considerations along the lines of Mathieu [74]. This
construction leads to two supersymmetric extensions, one of which contains
a free parameter. The resulting systems are proven to admit in¬nite series
of local and nonlocal symmetries and conservation laws.
8.1. Construction of supersymmetric extensions. We shall dis-
cuss supersymmetric extensions of the nonlinear Schr¨dinger equation
o
iqt = ’qxx + k(q — q)q, (6.160)
where q is a complex valued function. If we put q = u + iv then (6.160)
reduces to a system of two nonlinear equations
ut = ’vxx + kv(u2 + v 2 ),
vt = uxx ’ ku(u2 + v 2 ). (6.161)
Symmetries, conservation laws and coverings for this system were discussed
by several authors, see [88] and references therein.
Now we want to construct a supersymmetric extension of (6.161). This
construction is based on two main principles:
1. The existence of a supersymmetry Y 1 , whose “square”
2

.‚
[Y 1 , Y 1 ] = , (6.162)
‚x
2 2
.
where in (6.162)) “=” refers to equivalence classes of symmetries2 .
2
Recall that by the de¬nition of a higher
294 6. SUPER AND GRADED THEORIES

2. The existence of a higher (third ) order even symmetry X3 , which
reduces to the classical symmetry of (6.162) in the absence of odd
variables.
The technical construction heavily relies on the grading of equations
(6.161) and (6.162),
deg(x) = ’1, deg(t) = ’2, deg(u) = 1, deg(v) = 1,
deg(ux ) = deg(vx ) = 2, deg(ut ) = deg(vt ) = 3,
deg(uxx ) = deg(vxx ) = 3, . . . (6.163)
Condition 1, together with the assumption that the odd variables •, ψ to
be introduced are of degree ≥ 0, immediately leads to two possible choices
for the degree of •, ψ and the supersymmetry Y 1 , namely,
2

1
deg(•) = deg(ψ) = ,
2
‚ ‚ u‚ v‚
Y 1 = •1 + ψ1 + + , (6.164)
‚u ‚v 2 ‚• 2 ‚ψ
2

or
3
deg(•) = deg(ψ) = ,
2
‚ ‚ u1 ‚ v1 ‚
Y1 = • +ψ + + , (6.165)
‚u ‚v 2 ‚• 2 ‚ψ
2


where it should be noted that the presentations (6.164), (6.165) for Y 1 are
2
not unique, but can always be achieved by simple linear transformations
(•, ψ) ’ (• , ψ ). The choice (6.165) leads to just one possible extension of
(6.161), namely,
ut = ’vxx + kv(u2 + v 2 ) + ±•ψ,
vt = uxx ’ ku(u2 + v 2 ) + β•ψ,
•t = f1 [u, v, •, ψ],
ψt = f2 [u, v, •, ψ], (6.166)
where f1 , f2 are functions of degree 7/2 depending on u, v, •, ψ and their
derivatives with respect to x.
A straightforward computer computation, however, shows that there
does not exist a supersymmetric extension of (6.162) satisfying the two
basic principles and (6.164) in this case. Therefore we can restrict ourselves
to the case (6.164) from now on.
For reasons of convenience, we shall use subscripts to denote di¬erenti-
ation with respect to x in the sequel, i.e., u1 = ux , u2 = uxx , etc. In the

symmetry (see Chapter 2), it a coset in the quotient DC (E)/CD. Usually, we choose a
canonical representative of this coset ” the vertical derivation which was proved to be an
evolutionary one. But in some cases it is more convenient to choose other representatives.
8. SUPERSYMMETRIC EXTENSIONS OF THE NLS 295

case of (6.164), a supersymmetric extension of (6.161) by two odd variables
•, ψ is given by
ut = ’v2 + kv(u2 + v 2 ) + f1 [u, v, •, ψ],
vt = u2 + ku(u2 + v 2 ) + f2 [u, v, •, ψ],
•t = f3 [u, v, •, ψ],
ψt = f4 [u, v, •, ψ], (6.167)
where f1 , f2 are functions in of degree 3 and f3 , f4 are functions of degree
5/2. Expressing these functions into all possible terms of appropriate degree
requires the introduction of 72 constants.
Moreover, basic Principle 2 requires the existence of a vector ¬eld
‚ ‚ ‚ ‚
X3 = g1 [u, v, •, ψ] + g2 [u, v, •, ψ] + g3 [u, v, •, ψ] + g4 [u, v, •, ψ]
‚u ‚v ‚• ‚ψ
(6.168)
of degree 3 (i.e., g1 , g2 and g3 , g4 have to be functions of degree 4 and 7/2,
respectively) which is a symmetry of (6.167) and, in the absence of odd
variables, reduces to
‚ ‚
¯
X3 = (u3 ’ 3k(u2 + v 2 )u1 ) + (v3 ’ 3k(u2 + v 2 )v1 ) , (6.169)
‚u ‚v
the classical third order symmetry of (6.161). The condition that (6.168)
is a higher order symmetry of (6.167) gives rise to a large number of equa-
tions for both the 72 constants determining (6.167) and the 186 constants
determining (6.168). Solving this system of equations leads to the following
theorem.
Theorem 6.37. The NLS equation (6.161) admits two supersymmetric
extensions satisfying the basic Principles 1 and 2. These systems are:
Case A. The supersymmetric equation in this case is given by
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 (6.170)
with a third order symmetry

X3 = u3 ’ 3ku1 (u2 + v 2 ) + 6kv2 •ψ + 3ku1 (••1 + ψψ1 )

+ 3kv1 (•ψ1 + •1 ψ) + 3ku(••2 + ψψ2 ) + 3kv(ψ•2 ’ •ψ2 ) + 6kv•1 ψ1
‚u
+ v3 ’ 3kv1 (u2 + v 2 ) ’ 6ku2 •ψ + 3kv1 (••1 + ψψ1 ) ’ 3ku1 (•ψ1 + •1 ψ)

+ 3kv(••2 + ψψ2 ) ’ 3ku(ψ•2 ’ •ψ2 ) ’ 6ku•1 ψ1
‚v
296 6. SUPER AND GRADED THEORIES

3 3 3 ‚
+ •3 + 6k•ψψ2 ’ k(u2 + v 2 )•1 + k(uv1 ’ u1 v)ψ ’ k(uu1 + vv1 )•
2 2 2 ‚•
3 3 3 ‚
+ ψ3 ’ 6k•ψ•2 ’ k(u2 + v 2 )ψ1 ’ k(uv1 ’ u1 v)• ’ k(uu1 + vv1 )ψ .
2 2 2 ‚ψ
(6.171)
Case B. The supersymmetric equation in this case is given by
ut = ’v2 + kv(u2 + v 2 ) ’ (c ’ 4k)u1 •ψ ’ 4kvψψ1 ’ (c + 8k)uψ•1
+ 4ku•ψ1 + cv••1 ,
vt = u2 ’ ku(u2 + v 2 ) ’ (c ’ 4k)v1 •ψ + 4ku••1 + (c + 8k)v•ψ1
’ 4kvψ•1 ’ cuψψ1 ,
•t = ’ψ2 + k(3u2 + v 2 )ψ ’ 2kuv• + (c ’ 4k)•ψ•1 ,
ψt = •2 ’ k(u2 + 3v 2 )• + 2kuvψ ’ (c ’ 4k)•ψψ1 , (6.172)
where c is an arbitrary real constant. This system has a third order
symmetry
3
X3 = u3 ’ 3ku1 (u2 + v 2 ) ’ (c ’ 4k)v2 •ψ + 12kv1 (•ψ1 + •1 ψ)
2
3 3 ‚
’ (c + 4k)uψψ2 + (c + 4k)v•ψ2 + 12kv•1 ψ1
2 2 ‚u
3
+ v3 ’ 3kv1 (u2 + v 2 ) + (c ’ 4k)u2 •ψ ’ 12ku1 (•ψ1 + •1 ψ)
2
3 3 ‚
+ (c + 4k)uψ•2 ’ (c + 4k)v••2 ’ 12ku•1 ψ1
2 2 ‚v
3 ‚
+ •3 ’ (c ’ 4k)•ψψ2 ’ 3k(u2 + v 2 )•1 + 6kv1 (uψ ’ v•)
2 ‚•
3 ‚
+ ψ3 + (c ’ 4k)•ψ•2 ’ 3k(u2 + v 2 )ψ1 ’ 6ku1 (uψ ’ v•) . (6.173)
2 ‚ψ
Equations (6.170) and (6.172) may also be written in complex form.
Namely, if we put q = u + iv and ω = • + iψ, equations (6.170) and (6.172)
are easily seen to originate from the complex equation
iqt = ’q2 + k(q — q)q ’ 2kq(ω — ω1 + ωω1 ) + c2 q(ω — ω1 ’ ωω1 )
— —

+ (c1 + 2k)(qω — ’ q — ω)ω1 + (c1 ’ c2 )q1 ωω — ,
1
iωt = ’ω2 + k(q — q)ω + c2 q(q — ω ’ qω — ) + (c1 ’ c2 )ωω — ω1 , (6.174)
2
where c1 , c2 are arbitrary complex constants.
Now from (6.174), equation (6.170) can be obtained by putting c1 = ’4k
and c2 = 0, while equation (6.172) can be obtained by putting c1 = c,
c2 = 4k.
Hence we have found two supersymmetric extensions of the classical
NLS equation, one of them containing a free parameter. We shall discuss
symmetries of these systems in subsequent subsections.
8. SUPERSYMMETRIC EXTENSIONS OF THE NLS 297

8.2. Symmetries and conserved quantities. Let us now describe
symmetries and conserved densities of equation (6.174).
8.2.1. Case A. In this section we shall discuss symmetries, supersym-
metries, recursion symmetries and conservation laws for case A, i.e., the
supersymmetric extension of the NLS given by equation (6.170).
We searched for higher or generalized local symmetries of this system
and obtained the following result.
Theorem 6.38. The local generalized (x, t)-independent symmetries of
degree ¤ 3 of equation (6.170) are given by
‚ ‚
’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

‚ ‚ ‚ ‚
X1 = u 1 + v1 + •1 + ψ1 ,
‚u ‚v ‚• ‚ψ
.‚
X2 = , (6.175)
‚t
together with X3 as given by (6.171).
Similarly we obtained the following conserved quantities and conserva-
tion laws
Theorem 6.39. All local conserved quantities of degree ¤ 2 of system
(6.170) are given by

P0 = •ψ dx,
’∞

(uψ ’ v•) dx,
Q1 =
2
’∞

¯
Q1 = (u• + vψ) dx,
2
’∞

(u2 + v 2 ’ 2••1 ’ 2ψψ1 ) dx,
P1 =
’∞

P2 = (uv1 + 2•1 ψ1 ) dx, (6.176)
’∞
with the associated conservation laws
p0,x = •ψ,
p0,t = ••1 + ψψ1 ,
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

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