We are motivated by the result for the classical KdV equation (see Section 5

of Chapter 3) and our search is for a nonlocal vector ¬eld ¦ of the following

form

¦ = C1 t¦4 + C2 x¦2 + C3 p1 ¦1 + ¦— , (6.89)

whereas in (6.89) C1 , C2 , C3 are constants and ¦— is a two-component

function to be determined.

We now apply the symmetry condition resulting from the augmented

system (6.87) (compare with (6.80)):

3

Dt (¦v ) ’ 12¦v vv1 ’ 6v 2 Dx (¦v ) + Dx (¦v ) ’ Dx (¦ψ )ψ2

3

4

3 3 3 3 3

’ ψ1 Dx (¦ψ ) ’ ¦ψ ψ3 ’ ψ Dx (¦ψ ) ’ Dx (¦v )ψψ1 ’ v1 ¦ψ ψ1

2 3

4 4 4 2 2

3 3v 3ψ 3

’ v1 ψ Dx (¦ψ ) ’ ¦ ψψ2 ’ v¦ ψ2 ’ vψ Dx (¦ψ ) = 0,

2

2 2 2 2

Dt (¦ψ ) ’ (12v¦v ’ 6Dx (¦v ))ψ1 ’ (6v 2 ’ 6v1 )Dx (¦ψ )

280 6. SUPER AND GRADED THEORIES

’ (6¦v v1 + 6v Dx (¦v ) ’ 3Dx (¦v ))ψ ’ (6vv1 ’ 3v2 )¦ψ + 4Dx (¦ψ ) = 0.

2 3

(6.90)

Condition (6.90) leads to an overdetermined system of partial di¬erential

equations for the functions ¦—u , ¦—ψ , whose dependency on the internal

variables is induced by the scaling of the super mKdV equation, which means

that we are in e¬ect searching for a vector ¬eld ¦ , where ¦v , ¦ψ are of

degree 3 and 2 1 respectively.

2

Solving the overdetermined system of equations leads to the following

result.

The vector ¬eld ¦ with ¦ de¬ned by

3 1

¦ = ’ t¦4 ’ x¦2 + p1 ¦1 + ¦— , (6.91)

2 2

where ¦4 , ¦2 , ¦1 are de¬ned by (6.81) and

3 7

¦— = (’ v2 + 2v 3 + vψψ1 + ψψ2 , ’5ψ2 ’ vψ + 4v 2 ψ ’ 4v1 ψ), (6.92)

2 8

is a nonlocal higher symmetry of the super mKdV equation (6.71). In e¬ect,

the function ¦ is the shadow of the associated symmetry of (6.87).

The ‚/‚p0 - and ‚/‚p1 -components of the symmetry ¦ can be com-

puted from the invariance of the equations

(p0 )x = v,

1

(p1 )x = v 2 + ψψ1 , (6.93)

4

but considered in a once more augmented setting. The reader is referred to

the construction of nonlocal symmetries for the classical KdV equation for

the details of this calculation.

6. Supersymmetric KdV equation

In this section we shall discuss symmetries and conservation laws of the

supersymmetric extension of the KdV equation as it was proposed by several

authors [68, 74, 87].

We shall construct a supersymmetry transforming odd variables into

even variables and vice versa. We shall also construct a nonlocal symmetry of

the supersymmetric KdV equation, which together with the already known

supersymmetry generates a graded Lie algebra of symmetries, comprising a

hierarchy of bosonic higher symmetries and a hierarchy of nonlocal higher

fermionic (or super) symmetries. The well-known supersymmetry is just

the ¬rst term in this hierarchy.

Moreover, higher even and odd conservation laws and conserved quan-

tities arise in a natural and elegant way in the construction of the in¬nite

dimensional graded Lie algebra of symmetries. The construction of higher

even symmetries is given in Subsection 6.1, while the construction of the

above mentioned nonlocal symmetry together with the graded Lie algebra

structure is given in Subsection 6.2.

6. SUPERSYMMETRIC KDV EQUATION 281

6.1. Higher symmetries. The existence of higher even symmetries of

the supersymmetric extension of KdV equation

‚3u

‚u ‚u

= ’ 3 + 6u (6.94)

‚t ‚x ‚x

shall be discussed here. We start at the supersymmetric extension given by

Mathieu [74], i.e.,

ut = ’u3 + 6uu1 ’ a••2 ,

•t = ’•3 + (6 ’ a)•1 u + a•u1 . (6.95)

In (6.95), integer indices refer to di¬erentiation with respect to x, i.e., u 3 =

‚ 3 u/‚x3 ; x, t, u are even, while • is odd ; the parameter a is real. Taking

• ≡ 0, we get (6.94).

For internal local coordinates on the in¬nite jet bundle J ∞ (π; •) we

choose the functions x, t, u, •, u1 , •1 , . . . The total derivative operators

Dx , Dt are de¬ned by

‚ ‚ ‚ ‚ ‚

+ ··· ,

Dx = + u1 + •1 + u2 + •2

‚x ‚u ‚• ‚u1 ‚•1

‚ ‚ ‚ ‚ ‚

+ ···

Dt = + ut + •t + Dx (ut ) + Dx (•t ) (6.96)

‚t ‚u ‚• ‚u1 ‚•1

The vertical vector ¬eld V , the representation of which is given by

∞ ∞

‚ ‚

Dx (¦u )

i

Dx (¦• )

i

V= + , (6.97)

‚ui ‚•i

i=0 i=0

with generating function ¦ = (¦u , ¦• ), is a symmetry of (6.95), if the

following conditions are satis¬ed

Dt (¦u ) = ’Dx (¦u ) + ¦u 6u1 + Dx (¦u )6u ’ a¦• •2 + aDx (¦• )•,

3 2

Dt (¦• ) = ’Dx (¦• ) + (6 ’ a)Dx (¦• )u + (6 ’ a)¦u •1 + a¦• u1

3

+ aDx (¦u )•. (6.98)

In (6.98), ¦u , ¦• are functions depending on a ¬nite number of jet variables.

We restrict our search for higher symmetries at this moment to even vec-

tor ¬elds, moreover our search is for vector ¬elds, whose generating function

¦ = (¦u , ¦• ) depends on x, t, u, •, u1 , •1 , . . . , u5 , •5 . More speci¬cally,

¦u = f1 + f2 ••1 + f3 ••2 + f4 ••3 + f5 ••4 + f6 •1 •2 + f7 •1 •3 ,

¦ • = g 1 • + g 2 •1 + g 3 •2 + g 4 •3 + g 5 •4 + g 6 •5 , (6.99)

whereas in (6.99) f1 , . . . , f7 , g1 , . . . , g6 are dependent on the even variables

x, t, u, . . . , u5 . Formula (6.99) is motivated by the standard grading in the

classical case of (6.94),

3

deg(x) = ’1, deg(t) = ’3, deg(u) = 2, deg(•) = . (6.100)

2

and results for other problems.

282 6. SUPER AND GRADED THEORIES

In e¬ect, this means that we are not only searching for ¦u and ¦• in

the appropriate jet bundle but also restricted to a certain maximal degree.

In this case we assume the vector ¬eld to be of degree less than or equal to

1

5, which means that ¦u , ¦• are of degree at most 7 and 6 2 respectively.

Substitution of (6.99) into (6.98) does lead to an overdetermined system

of partial di¬erential equations for the functions f1 , . . . , f7 , g1 , . . . , g6 . The

solution of this overdetermined system of equations leads to the following

result

Theorem 6.33. For a = 3, there are four vector ¬elds ¦1 , . . . , ¦4

satisfying the higher symmetry condition (6.98), i.e.,

¦1 = (u1 , •1 ),

¦2 = (u3 ’ 6u1 u + 3••2 , •3 ’ 3•1 u ’ 3•u1 ),

¦3 = ’(u5 ’ 10u3 u ’ 20u2 u1 + 30u1 u2 + 5••4 + 5•1 •3

’ 20u••2 ’ 20u1 ••1 , •5 ’ 5u•3 ’ 10u1 •2 ’ 10u2 •1 + 10u2 •1

+ 20u1 u• ’ 5u3 •),

3

¦4 = ’3t¦2 + x¦1 + (2u, •). (6.101)

2

If a = 3, then ¦3 is not a symmetry of (6.95).

Next, our search is for odd vector ¬elds (6.97) satisfying (6.98); the

assumption on the generating function ¦ = (¦u , ¦• ) is

¦ u = f 1 • + f 2 •1 + f 3 •2 + f 4 •3 + f 5 •4 + f 6 •5 + f 7 •6 ,

¦• = g1 + g2 ••1 + g3 ••2 + g4 ••3 + g5 ••4 + g6 ••5 + g7 •1 •2

+ g8 •1 •3 + g9 •1 •4 + g10 •2 •3 , (6.102)

where f1 , . . . , f7 , g1 , . . . , g10 are dependent on x, t, u, . . . , u5 .

Solving the resulting overdetermined system of partial di¬erential equa-

tions leads to:

Theorem 6.34. There exists only one odd symmetry Y 1 of (6.95), i.e.,

2

¦Y 1 = (•1 , u). (6.103)

2

In order to obtain the Lenard recursion operator we did proceed in a way

similar to that discussed in Section 5 of Chapter 3, but unfortunately we were

not successful. We shall discuss a recursion for higher symmetries, resulting

from the graded Lie algebra structure in the next subsection, while the

construction of the recursion operator for the supersymmetric KdV equation

is discussed in Chapter 7.

6.2. Nonlocal symmetries and conserved quantities. By the in-

troduction of nonlocal variables, we derive here a nonlocal even symmetry

for the supersymmetric KdV equation in the case a = 3

ut = ’u3 + 6u1 u ’ 3••2 ,

6. SUPERSYMMETRIC KDV EQUATION 283

•t = ’•3 + 3•1 u + 3•u1 , (6.104)

which together with the supersymmetry generates two in¬nite hier-

Y1

2

archies of higher symmetries. The even and odd nonlocal variables and

conserved quantities arise in a natural way.

We start with the observation that

Dt (•) = Dx (’•2 + 3•u) (6.105)

is a conservation law for (6.104), or equivalently,

x

q1 = • dx, (6.106)

2

’∞

is a potential of (6.104), i.e.,

(q 1 )x = •, (q 1 )t = ’•2 + 3•u. (6.107)

2 2

The quantity Q 1 de¬ned by

2

∞

Q1 = • dx (6.108)

2

’∞

is a conserved quantity of the supersymmetric KdV equation (6.104).

We now make the following observation:

Theorem 6.35. The nonlocal vector ¬eld Z1 , whose generating func-

tion ¦Z1 is

¦Z1 = (q 1 •1 , q 1 u ’ •1 ) (6.109)

2 2

is a nonlocal symmetry of the KdV equation (6.104). Moreover, there is

no nonlocal symmetry linear with respect to q 1 which satis¬es (6.95) with

2

a = 3.

The function ¦Z1 is in e¬ect the shadow of a nonlocal symmetry of the

augmented system of equations

ut = ’u3 + 6u1 u ’ 3••2 ,

•t = ’•3 + 3•1 u + 3•u1 ,

(q 1 )x = •,

2

(q 1 )t = ’•2 + 3•u. (6.110)

2

Total partial derivative operators Dx and Dt are given here by

‚

Dx = D x + • ,

‚q 1

2

‚

Dt = Dt + (’•2 + 3•u) ,

‚q 1

2

284 6. SUPER AND GRADED THEORIES

and the generating function ¦Z1 satis¬es the invariance of the ¬rst and the

second equation in (6.110), i.e.,

Dt (¦u 1 ) + Dx (¦u 1 ) ’ ¦u 1 6u1 ’ Dx (¦u 1 )6u + 3¦•1 •2 ’ 3Dx (¦•1 )• = 0,

3 2

Z Z Z Z Z Z

Dt (¦•1 ) + Dx (¦•1 ) ’ 3Dx (¦•1 )u ’ 3¦u 1 •1 ’ 3¦•1 u1 ’ 3Dx (¦u 1 )• = 0.

3

Z Z

Z Z Z Z

The vector ¬eld together with the vector ¬eld play a fundamental

Z1 Y1

2

role in the construction of the graded Lie algebra of symmetries of (6.104).

From now on, for obvious reasons, we shall restrict ourselves to (6.104),

i.e., to the case a = 3.

Remark 6.14. All odd variables •0 , •1 , . . . , q 1 are, with respect to the

2

grading (6.100), of degree n/2, where n is odd. The vector ¬eld Z1 is even,

while Y 1 is odd.

2

We now want to compute the graded Lie algebra with and as

Z1 Y1

2

“seed elements”.

In order to do so, we have to prolong the vector ¬eld towards the

Y1

2

nonlocal variable q 1 , or by just writing for this prolongation, we have

Y1

2 2

to calculate the component ‚/‚q 1 , in the augmented setting (6.110)).

2

q1

The calculation is as follows. The coe¬cient Y 1 2 has to be such that the

2

vector ¬eld leaves invariant (6.105), i.e., the Lie derivative of (6.105)

Y1

2

with respect to is to be zero.

Y1

2

Since

u u

Y 1 1 = Dx (Y 1 ) = Dx (•1 ) = •2 ,

2

2

•1 •

Y 1 = Dx (Y 1 ) = Dx (u) = u1 ,

2 2

•2 •

Y 1 = Dx (Y 1 1 ) = Dx (u1 ) = u2 , (6.111)

2 2

the invariance of the third and fourth equation in (6.110) leads to

q1

Dx (Y 1 2 ) ’ u = 0,

2

q1 2

• • u

Dt (Y 1 2 ) + Dx (Y 1 ) ’ 3Y 1 u + 3•Y 1 = 0, (6.112)

2

2 2 2

from which we have

q1

Dx (Y 1 2 ) ’ u = 0,

2

q1

Dt (Y 1 2 ) + u2 ’ 3u2 + 3••1 = 0. (6.113)

2

By (6.109), (6.111), (6.113) we are led in a natural and elegant way to the

introduction of a new nonlocal even variable p1 , de¬ned by

x

p1 = u dx (6.114)

’∞

6. SUPERSYMMETRIC KDV EQUATION 285

and satisfying the system of equations

(p1 )x = u,

(p1 )t = ’u2 + 3u2 ’ 3••1 , (6.115)

i.e., p1 is a potential of the supersymmetric KdV equation (6.104); the com-

patibility conditions being satis¬ed, while the associated conserved quantity

is P1 .

Now the vector ¬eld Y 1 is given in the setting (6.110) by

2

‚ ‚ ‚

= •1 +u + p1 + ... (6.116)

Y1

‚u ‚• ‚q 1

2

2

Computation of the graded commutator [ Z1 , Y1 ] of and leads

Z1 Y1

2 2

us to a new symmetry of the KdV equation, given by

=[ Z1 , Y 1 ], (6.117)

Y3

2 2

where the generating function is given by

¦Y 3 = (2q 1 u1 ’ p1 •1 + u• ’ •2 , 2q 1 •1 ’ p1 u + u1 ). (6.118)

2 2

2

This symmetry is a new nonlocal odd symmetry of (6.104) and is of degree

3

2.

Note that as polynomials in q 1 and p1 , the coe¬cients in (6.118) just

2

constitute the generating functions of the symmetries 2 X1 and ’ Y 1 re-

2

spectively, i.e.,

¦Y 3 = 2q 1 ¦1 ’ p1 ¦Y 1 + (u• ’ •2 , u1 ). (6.119)

2

2 2

We now proceed by induction.

In order to compute the graded Lie bracket [ Z1 , Y 3 ], we ¬rst have

2

to compute the prolongation of Z1 towards the nonlocal variables p1 and

q 1 , which is equivalent to the computation of the ‚/‚p1 - and the ‚/‚q 1 -

2 2

components of the vector ¬eld Z1 , again denoted by the same symbol Z1 .

It is perhaps illustrative to mention at this stage that we are in e¬ect

considering the following augmented system of graded partial di¬erential

equations

ut = ’u3 + 6u1 u ’ 3••2 ,

•t = •3 + 3•1 u + 3•u1 ,

(q 1 )x = •,

2

(q 1 )t = ’•2 + 3•u,

2

(p1 )x = u,

(p1 )t = ’u2 + 3u2 ’ 3••1 , (6.120)

286 6. SUPER AND GRADED THEORIES

We now consider the invariance of the ¬fth equation in (6.120), i.e., of

(p1 )x = u, by the vector ¬eld Z1 which leads to the condition

p

Dx (Z1 1 ) = q 1 •1 , (6.121)

2

from which we have

p

Z1 1 = q 1 •. (6.122)

2

q1

The ‚/‚q 1 -component of Z1 , i.e., Z1 2 has to satisfy the invariance of the

2

third equation in (6.120), i.e., (q 1 )x = • by the vector ¬eld Z1 which leads

2

to the condition

q1

•

Dx (Z1 2 ) ’ Z1 = 0,

i.e.,

q1

Dx (Z1 2 ) ’ q 1 u + •1 = 0, (6.123)

2

from which we derive

x

q1

Z1 = q 1 p1 ’ • ’