<< . .

. 45
( : 58)



. . >>

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:

<< . .

. 45
( : 58)



. . >>