+ (4u3 + u2 ’ 2uu2 + 12u••1 + 8•1 •2 ’ 4••3 ) . (6.65)

1

‚p3

We are motivated by the result for the classical KdV equation (see Sec-

tion 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 + p3 ¦— + ¦—— , (6.66)

whereas in (6.66) C1 , C2 , C3 are constants and ¦— = (¦—u , ¦—• ), ¦—— =

(¦——u , ¦——• ) are functions to be determined.

We now apply the symmetry condition resulting from the augmented

system (6.64), compare with (6.57)

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

3 2

Dt (¦• ) ’ 3Dx (¦u )• ’ 3u1 ¦• ’ 6¦u •1 ’ 6uDx (¦• ) + 4Dx (¦• ) = 0.

3

(6.67)

Condition (6.67) leads to an overdetermined system of partial di¬erential

equations for the functions ¦—u , ¦—• , ¦——u , ¦——• , whose dependency on the

internal variables is induced by the scaling of the super KdV equation, which

means that we are in e¬ect searching for a vector ¬eld ¦ , where ¦u , ¦•

are of degree 4 and 3 1 respectively. Solving the overdetermined system of

2

equations leads to the following result.

The vector ¬eld ¦ with ¦ de¬ned by

3 1 1

¦ = ’ t¦5 ’ x¦2 ’ p1 ¦1 + ¦—— , (6.68)

4 4 2

where ¦5 , ¦2 , ¦1 are de¬ned by (6.58) and

3 7

¦—— = (u2 ’ 2u2 ’ ••1 , •2 ’ 3u•), (6.69)

2 2

is a nonlocal higher symmetry of the super KdV equation (6.48). In e¬ect,

the function ¦ is the shadow of the associated symmetry of (6.64).

5. THE KUPERSHMIDT SUPER MKDV EQUATION 275

The ‚/‚p1 - and ‚/‚p3 -components of the symmetry can be com-

¦

puted from the invariance of the equations (6.70),

(p1 )x = u,

(p3 )x = u2 + 3••1 , (6.70)

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.

It would be possible to describe the recursion here, but we prefer to

postpone it to the chapter devoted to the deformations of the equation

structure (see Chapter 7), from which the recursion operator can be obtained

rather easily and straightforwardly.

5. The Kupershmidt super mKdV equation

As a second application of the graded calculus for symmetries of graded

partial di¬erential equations, we discuss the symmetry structure of the so-

called Kupershmidt super mKdV equation, which is an extension of the

classical mKdV equation to the graded setting [24].

The super mKdV equation is given as the following system of graded

partial di¬erential equations E for an even function v and an odd function

ψ on J 3 (π; ψ) (see the notation in the previous section),

3 3 3 3

vt = 6v 2 vx ’ vxxx + ψx ψxx + ψψxxx + vx ψψx + vψψxx ,

4 4 2 2

ψt = (6v 2 ’ 6vx )ψx + (6vvx ’ 3vxx )ψ ’ 4ψxxx , (6.71)

where subscripts denote partial derivatives with respect to x, t. Here t is the

time variable and x is the space variable, v, x, t, v, vx , vt , vxx , vxxx are even

(commuting) variables, while ψ, ψx , ψxx , ψxxx are odd (anticommuting)

variables.

We introduce the total derivative operators Dx , Dt on J ∞ (π; ψ) by

‚ ‚ ‚ ‚ ‚

+ ··· ,

Dx = + vx + ψx + vxx + ψxx

‚x ‚v ‚ψ ‚vx ‚ψx

‚ ‚ ‚ ‚ ‚

+ ···

Dt = + vt + ψt + vtx + ψtx (6.72)

‚t ‚v ‚ψ ‚vx ‚ψx

The in¬nite prolongation E ∞ is the submanifold of J ∞ (π; ψ) de¬ned by

the graded system of partial di¬erential equations

3 3 3 3

Dx Dt (vt ’ 6v 2 vx + vxxx ’ ψx ψxx ’ ψψxxx ’ vx ψψx ’ vψψxx ) = 0,

nm

4 4 2 2

Dx Dt (ψt ’ (6v 2 ’ 6vx )ψx ’ (6vvx ’ 3vxx )ψ + 4ψxxx ) = 0,

nm

(6.73)

where n, m ∈ N.

We choose internal coordinates on E ∞ as x, t, v, ψ, v1 , ψ1 , . . . , where

we use a notation

vx = v 1 , ψx = ψ 1 , vxx = v2 , ψxx = ψ2 , . . . (6.74)

276 6. SUPER AND GRADED THEORIES

The restriction of the total derivative operators Dx , Dt to E ∞ , again denoted

by the same symbols, are then given by

‚ ‚ ‚

Dx = + vn+1 + ψn+1 ,

‚x ‚vn ‚ψn

n≥0

‚ ‚ ‚

Dt = + (vn )t + (ψn )t . (6.75)

‚t ‚vn ‚ψn

n≥0

We note that (6.71) admits a scaling symmetry, which leads to the assigning

a degree to each variable,

deg(x) = ’1, deg(t) = ’3,

deg(v) = 1, deg(v1 ) = 2, . . . ,

1 3

deg(ψ) = , deg(ψ1 ) = , . . . (6.76)

2 2

From this we see that each term in (6.71) is of degree 4 and 3 1 respectively.

2

5.1. Higher symmetries. We start the discussion of searching for

(higher) symmetries at the representation of vertical vector ¬elds,

‚ ‚ ‚ ‚

= ¦v + ¦ψ Dx (¦v )

n

+ Dx (¦ψ )

n

+ , (6.77)

¦

‚v ‚ψ ‚vn ‚ψn

n>0

where ¦ = (¦v , ¦ψ ) is the generating function of the vertical vector ¬eld ¦ .

We restrict our search for higher symmetries to even vector ¬elds, meaning

that ¦v is even, while ¦ψ is odd. Moreover we restrict our search for higher

symmetries to vector ¬elds ¦ whose generating function ¦ = (¦v , ¦ψ )

depends on the variables x, t, v, ψ, . . . , v5 , ψ5 . The above mentioned re-

quirements lead to a representation of the function ¦ = (¦v , ¦ψ ) in the

following form

¦v = f0 + f1 ψψ1 + f2 ψψ2 + f3 ψψ3 + f4 ψψ4 + f5 ψψ5 + f6 ψ1 ψ2

+ f7 ψ1 ψ3 + f8 ψ1 ψ4 + f9 ψ1 ψ5 + f10 ψ2 ψ3 + f11 ψ2 ψ4 + f12 ψ2 ψ5

+ f13 ψ3 ψ4 + f14 ψ3 ψ5 + f15 ψ4 ψ5 + f16 ψψ1 ψ2 ψ3 + f17 ψψ1 ψ2 ψ4

+ f18 ψψ1 ψ2 ψ5 + f19 ψψ1 ψ3 ψ4 + f20 ψψ1 ψ3 ψ5 + f21 ψψ1 ψ4 ψ5

+ f22 ψψ2 ψ3 ψ4 + f23 ψψ2 ψ3 ψ5 + f24 ψψ2 ψ4 ψ5 + f25 ψψ3 ψ4 ψ5

+ f26 ψ1 ψ2 ψ3 ψ4 + f27 ψ1 ψ2 ψ3 ψ5 + f28 ψ2 ψ3 ψ4 ψ5

+ f29 ψψ1 ψ2 ψ3 ψ4 ψ5 ,

¦ψ = g 1 ψ + g 2 ψ1 + g 3 ψ2 + g 4 ψ3 + g 5 ψ4 + g 6 ψ5

+ g7 ψψ1 ψ2 + g8 ψψ1 ψ3 + g9 ψψ1 ψ4 + g10 ψψ1 ψ5 + g11 ψψ2 ψ3

+ g12 ψψ2 ψ4 + g13 ψψ2 ψ5 + g14 ψψ3 ψ4 + g15 ψψ3 ψ5 + g16 ψψ4 ψ5

+ g17 ψ1 ψ2 ψ3 + g18 ψ1 ψ2 ψ4 + g19 ψ1 ψ2 ψ5 + g20 ψ1 ψ3 ψ4 + g21 ψ1 ψ3 ψ5

+ g22 ψ1 ψ4 ψ5 + g23 ψ2 ψ3 ψ4 + g24 ψ2 ψ3 ψ5 + g25 ψ2 ψ4 ψ5 + g26 ψ3 ψ4 ψ5

5. THE KUPERSHMIDT SUPER MKDV EQUATION 277

+ g27 ψψ1 ψ2 ψ3 ψ4 + g28 ψψ1 ψ2 ψ3 ψ5 + g29 ψψ1 ψ2 ψ4 ψ5 + g30 ψψ1 ψ3 ψ4 ψ5

+ g31 ψψ2 ψ3 ψ4 ψ5 + g32 ψ1 ψ2 ψ3 ψ4 ψ5 , (6.78)

where f0 , . . . , f29 , g1 , . . . , g32 are functions depending on the even variables

x, t, v, v1 , . . . , v5 . We have to mention here that we are constructing generic

elements, even and odd explicitly, of the exterior algebra C ∞ (x, t, v, . . . , v5 )—

Λ(ψ, . . . , ψ5 ), where Λ(ψ, . . . , ψ5 ) is the exterior algebra generated by the

elements ψ, . . . , ψ5 . The symmetry condition (6.37) reads in this case

3 3 3 3

Dt (¦v ) = ¦ (6v 2 v1 ’ v3 + ψ1 ψ2 + ψψ3 + v1 ψψ1 + vψψ2 ),

4 4 2 2

ψ 2

Dt (¦ ) = ¦ ((6v ’ 6v1 )ψ1 + (6vv1 ’ 3v2 )ψ ’ 4ψ3 ), (6.79)

which results in equations

3

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

3

4

3 3 3

’ ψ1 Dx (¦ψ ) ’ ¦ψ ψ3 ’ ψDx (¦ψ )

2 3

4 4 4

3 3 3

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

2 2 2

3 3 3

’ ¦v ψψ2 ’ v¦ψ ψ2 ’ vψDx (¦ψ ) = 0,

2

2 2 2

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

’ (6¦v v1 + 6vDx (¦v ) ’ 3Dx (¦v ))ψ

2

’ (6vv1 ’ 3v2 )¦ψ + 4Dx (¦ψ ) = 0.

3

(6.80)

Substitution of the representation (6.78) for ¦ = (¦v , ¦ψ )), leads to

an overdetermined system of classical partial di¬erential equations for the

coe¬cients f0 , . . . , f26 , g1 , . . . , g32 which are as mentioned above functions

depending on the variables x, t, v, v1 , . . . , v5 .

The general solution of equations (6.80) and (6.78) is generated by the

functions

¦1 = (v1 , ψ1 ),

3 3 3 3

¦2 = (’v3 + 6v 2 v1 + v1 ψψ1 + vψψ2 + ψψ3 + ψ1 ψ2 ,

2 2 4 4

2

’ 4ψ3 + (6v ’ 6v1 )ψ1 + (6vv1 ’ 3v2 )ψ),

¦3 = ’2x¦1 ’ 6t¦2 + (’2v, ’ψ),

15 25 5

¦v = v5 ’ 10v3 v 2 ’ 40v2 v1 v ’ 10v1 + 30v1 v 4 ’

3

ψψ5 ’ ψ1 ψ4 ’ ψ2 ψ3

4

4 4 2

15 15 15

’ vψψ4 ’ 5vψ1 ψ3 + ( v 2 ’ 15v1 )ψψ3 + ( v 2 ’ 5v1 )ψ1 ψ2

2 2 2

15

+ (15v 3 + 15v1 v ’ 15v2 )ψψ2 + (45v1 v 2 ’ v3 )ψψ1 ,

2

¦ψ = 16ψ5 + (40v1 ’ 40v 2 )ψ3 + (60v2 ’ 120v1 v)ψ2

4

278 6. SUPER AND GRADED THEORIES

+ (50v3 ’ 100v2 v ’ 60v1 v 2 ’ 70v1 + 30v 4 )ψ1

2

+ (15v4 ’ 30v3 v ’ 30v2 v 2 ’ 60v2 v1 ’ 60v1 v + 60v1 v 3 )ψ.

2

(6.81)

We note that the vector ¬elds ¦1 , ¦2 , ¦3 are equivalent to the classical

symmetries

‚

S1 = ,

‚x

‚

S2 = ,

‚t

‚ ‚ ‚ ‚

+ 6t ’ 2v ’ψ

S4 = 2x . (6.82)

‚x ‚t ‚v ‚ψ

In (6.82), S1 , S2 re¬‚ect space and time translation, while S3 re¬‚ects the scal-

ing as mentioned already, (6.73). The ¬eld ¦4 is the ¬rst higher symmetry

of the super mKdV equation and reduces to the evolutionary vector ¬eld

‚

(v5 ’ 10v3 v 2 ’ 40v2 v1 v ’ 10v1 + 30v1 v 4 )

3

+ ..., (6.83)

‚v

in the absence of odd variables ψ, ψ1 , . . . , being then just the classical ¬rst

higher symmetry of the mKdV equation.

vt = 6v 2 vx ’ vxxx . (6.84)

Remark 6.13. It should be noted that this section is just a copy of

the previous one concerning the Kupershmidt super KdV equation, except

for the speci¬c results! This demonstrates the algorithmic structure of the

symmetry computations.

5.2. A nonlocal symmetry. In this subsection we demonstrate the

existence and construction of nonlocal higher symmetries for the super

mKdV equation (6.71). The construction runs exactly along the same lines

as it is for the classical equations.

So we start at the construction of conservation laws, conserved densities

and conserved quantities as discussed in Section 2. According to this con-

struction, we arrive, amongst others, at the following two conservation laws,

i.e.,

3 3

Dt (v) = Dx (2v 3 ’ v2 + ψψ2 + vψψ1 ),

4 2

1

Dt (v 2 + ψψ1 ) = Dx (3v 4 ’ 2v2 v + v1 ’ ψψ3 + 2ψ1 ψ2

2

4

3 9

+ vψψ2 ’ 3v1 ψψ1 + v 2 ψψ1 ), (6.85)

2 2

from which we obtain the nonlocal variables

x

p0 = v dx,

’∞

x

1

(v 2 + ψψ1 ) dx.

p1 = (6.86)

4

’∞

5. THE KUPERSHMIDT SUPER MKDV EQUATION 279

Now using these new nonlocal variables p0 , p1 we de¬ne the augmented

system E of partial di¬erential equations for the variables v, p0 , p1 , ψ,

where v, p0 , p1 are even and ψ is odd,

3 3 3 3

vt = 6v 2 vx ’ vxxx + ψx ψxx + ψψxxx + vx ψψx + vψψxx ,

4 4 2 2

2

ψt = (6v ’ 6vx )ψx + (6vvx ’ 3vxx )ψ ’ 4ψxxx ,

(p0 )x = v,

3 3

(p0 )t = 2v 3 ’ v2 + ψψ2 + vψψ1 ,

4 2

1

(p1 )x = v 2 + ψψ1 ,

4

3 9

(p1 )t = 3v 4 ’ 2v2 v + v1 ’ ψψ3 + 2ψ1 ψ2 + vψψ2 ’ 3v1 ψψ1 + v 2 ψψ1 .

2

2 2

(6.87)

Internal coordinates for the in¬nite prolongation E ∞ of this augmented

system (6.87)) are given as x, t, v, p0 , p1 , ψ, v1 , ψ1 , . . . . The total derivative

operators Dx and Dt on E ∞ are given by

‚ 1 ‚

+ (v 2 + ψψ1 )

Dx = D x + v ,

‚p0 4 ‚p1

3 3 ‚

Dt = Dt + (2v 3 ’ v2 + ψψ2 + vψψ1 )

4 2 ‚p0

3 9 ‚

+ (3v 4 ’ 2v2 v + v1 ’ ψψ3 + 2ψ1 ψ2 + vψψ2 ’ 3v1 ψψ1 + v 2 ψψ1 )

2

.

2 2 ‚p1

(6.88)

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 ’ • ’

2

p1 • dx. (6.124)

2

’∞

So prolongation of Z1 towards the nonlocal variable q 1 , or equivalently,

2

computation of the ‚/‚q 1 -component of the vector ¬eld Z1 , requires formal

2

introduction of a new odd nonlocal variable q 3 de¬ned by

2

x

q3 = p1 • dx, (6.125)

2

’∞

where

(q 3 )x = p1 •,

2

(q 3 )t = p1 (’•2 + 3u•) ’ u1 • + u•1 , (6.126)

2

while the compatibility condition on (6.126) is satis¬ed; so q 3 is a new odd

2

potential, Q 3 being the new odd conserved quantity.

2

The vector ¬eld Z1 is now given in the augmented setting (6.120) by

‚ ‚ ‚ ‚

+ (q 1 u ’ •1 ) + (q 1 p1 ’ • ’ q 3 )

= q 1 •1 + q1 • .

Z1

‚u ‚• 2 ‚q 1 ‚p1

2 2 2 2

2

(6.127)

The system of graded partial di¬erential equations under consideration is

now the once more augmented system (6.120):

ut = ’u3 + 6u1 u ’ 3••2 ,

•t = ’•3 + 3•1 u + 3•u1 ,

(q 1 )x = •,

2

(q 1 )t = ’•2 + 3•u,

2

6. SUPERSYMMETRIC KDV EQUATION 287

(p1 )x = u,

(p1 )t = ’u2 + 3u2 ’ 3••1 ,

(q 3 )x = p1 •,

2

(q 3 )t = p1 (’•2 + 3u•) ’ u1 • + u•1 . (6.128)

2

The prolongation of the vector ¬eld towards the nonlocal variables q 1 ,

Y3

2

2

p1 is now constructed from the respective equations for (q 1 )x and (p1 )x ,

2

(6.128) resulting in

q1 1

Y 3 2 = 2q 1 • ’ p2 + u,

21

2

2

p

Y 3 1 = 2q 1 u ’ p1 • ’ •1 . (6.129)

2

2

Computation of the graded Lie bracket [ Z1 , Y3 ] leads to

2

1 ‚

= (’2q 3 u1 + p2 •1 + p1 (•2 ’ u•) ’ 4u1 • ’ 3u•1 + •3 )

Y5 1

2 ‚u

2

2

1 ‚

+ (’2q 3 •1 + p2 u ’ p1 u1 + u2 ’ 2u2 + ••1 )

21 ‚•

2

1 ‚

+ (’2q 3 u + p2 • + p1 •1 ’ 4u• + •2 )

21 ‚p1

2

1 1 ‚

+ (’2q 3 p1 • + p4 ’ p2 u + p1 u1 ’ u2 ’ ••1 ) + · · · , (6.130)

81 1

2 ‚q 3

2

2

whereas the ‚/‚p1 - and ‚/‚q 3 -components of are obtained by the in-

Y5

2 2

variance of the associated di¬erential equations for these variables in (6.128).

In order to obtain the ‚/‚q 1 -component of Y 5 , we have to require the

2 2

invariance of the equation (q 1 )x ’ • = 0, which results in the following

2

condition

q1 1

Dx (Y 5 2 ) = ’2q 3 •1 + p2 u ’ p1 u1 + u2 ’ 2u2 + ••1 , (6.131)

21

2

2

from which we have

x

q1 13

(u2 + 2(p1 •)• + ••1 ’ 2u2 ) dx

Y 5 = p1 ’ p1 u + u1 ’ 2q 3 • +

2

6 2

’∞

2

x

13

(u2 ’ ••1 ) dx.

= p1 ’ p1 u + u1 ’ 2q 3 • ’ (6.132)

6 2

’∞

So expression (6.132) requires in a natural way the introduction of the even

nonlocal variable p3 , de¬ned by

x

(u2 ’ ••1 ) dx,

p3 = (6.133)

’∞

where

(p3 )x = u2 ’ ••1 ,

288 6. SUPER AND GRADED THEORIES

(p3 )t = 4u3 ’ 2u2 u + u2 ’ 9u••1 + ••3 ’ 2•1 •2 . (6.134)

1

Here p3 is a well-known potential, P3 being the associated conserved quan-

tity.

Finally, the commutator Y 7 = [ Z1 , Y 5 ] requires the prolongation

2 2

of the vector ¬eld Z1 towards the nonlocal variable q 3 , obtained by the

2

invariance of the condition (q 3 )x ’ p1 • = 0 by Z1 , so

2

q3

= (q 1 •)• + p1 (q 1 u ’ •1 ).

Dx (Z1 2 ) = Z1 (p1 •) (6.135)

2 2

Integration of (6.135) leads to

x

q3 1 1

= p2 q 1 ’ p 1 • ’ ( p2 • ’ u•) dx.

2

Z1 (6.136)

212 1

’∞ 2

The new odd nonlocal variable q 5 is, due to (6.136), formally de¬ned by

2

x

1

( p2 • ’ u•) dx.

q5 = (6.137)

1

’∞ 2

2

Here q 5 is a nonlocal odd potential of the supersymmetric KdV equation

2

(6.104),

1

(q 5 )x = p2 • ’ u•,

21

2

1

(q 5 )t = p2 (’•2 + 3u•) + p1 (’u1 • + u•1 ) + u2 • ’ u1 •1 ’ 4u2 • + u•2 .

21

2

(6.138)

Proceeding in this way, we obtain a hierarchy of nonlocal higher supersym-

metries by induction,

n ∈ N.

=[ Z1 , Yn’ 1 ], (6.139)

Yn+ 1

2 2

The higher even potentials p1 , p3 , . . . arise in a natural way in the prolon-

gation of the vector ¬elds Y2n+ 1 towards the nonlocal variable q 1 , whereas

2

2

the higher nonlocal odd potentials q 1 , q 3 , q 5 , . . . are obtained in the pro-

2 2 2

longation of the recursion symmetry Z1 .

To obtain the graded Lie algebra structure of symmetries we calculate

the graded Lie bracket of the vector ¬elds derived so far. The result is

remarkable and fascinating:

[ Y1 , Y1 ] =2 X1 ,

2 2

[ Y3 , Y3 ] =2 X3 ,

2 2

[ Y5 , Y5 ] =2 X5 , (6.140)

2 2

so the “squares” of the supersymmetries Y1 , Y3 , are just the “clas-

Y5

2 2 2

sical” symmetries 2 X1 , 2 X3 , 2 obtained previously (see (6.101)). The

X5

6. SUPERSYMMETRIC KDV EQUATION 289

other commutators are

[ Y1 , Y3 ] = 0,

2 2

= ’2

[ Y1 , Y5 ] X3 ,

2 2

[ Y3 , Y5 ] = 0,

2 2

[ X1 , X3 ] =[ X1 , X5 ] =[ X3 , X5 ] = 0,

[ Z1 , X1 ] =[ Z1 , X3 ] =[ Z1 , X5 ] = 0,

[ Yn+ 1 , X2m+1 ] = 0, (6.141)

2

where n = 0, 1, 2, m = 0, 1, 2. We conjecture that in this way we obtain

an in¬nite hierarchy of nonlocal odd symmetries Yn+ 1 , n ∈ N, and an in¬-

2

nite hierarchy of ordinary even higher symmetries X2n+1 , n ∈ N, while the

even and odd nonlocal variables p2n+1 , qn+ 1 and the associated conserved

2

quantities P2n+1 , Qn+ 1 are obtained by the prolongation of the vector ¬elds

2

Yn+ 1 and Z1 respectively.

2

We ¬nish this section with a lemma concerning the Lie algebra structure

of the symmetries.

Lemma 6.36. Let X2n+1 , n ∈ N, be de¬ned by

1

X2n+1 = [Yn+ 1 , Yn+ 1 ], (6.142)

2 2 2

and assume that

n ∈ N.

[Z1 , X2n+1 ] = 0, (6.143)

Then

(’1)m’n 2Xn+m+1 m ’ n is even,

1. [Yn+ 1 , Ym+ 1 ] =

m ’ n is odd.

0

2 2

2. [Yn+ 1 , X2m+1 ] = 0, n, m ∈ N.

2

3. [X2n+1 , X2m+1 ] = 0, n, m ∈ N.

Proof. The proof of (1) is by induction on k = m ’ n. First consider

the cases k = 1 and k = 2:

0 = [Z1 , [Yn+ 1 , Yn+ 1 ]] = [Yn+1+ 1 , Yn+ 1 ] + [Yn+ 1 , Yn+1+ 1 ]

2 2 2 2 2 2

= 2[Yn+ 1 , Yn+1+ 1 ],

2 2

0 = [Z1 , [Yn+ 1 , Yn+1+ 1 ]] = [Yn+1+ 1 , Yn+1+ 1 ] + [Yn+ 1 , Yn+2+ 1 ], (6.144)

2 2 2 2 2 2

so

[Yn+ 1 , Yn+2+ 1 ] = ’2X2n+3 . (6.145)

2 2

For general k, the result is obtained from the identity

0 = [Z1 , [Yn+ 1 , Yn+k+ 1 ]] = [Yn+1+ 1 , Yn+k+ 1 ] + [Yn+ 1 , Yn+k+1+ 1 ], (6.146)

2 2 2 2 2 2

290 6. SUPER AND GRADED THEORIES

i.e.,

[Yn+ 1 , Yn+k+1+ 1 ] = ’[Yn+1+ 1 , Yn+k+ 1 ]. (6.147)

2 2 2 2

The proof of (3) is a consequence of (2) by

[X2n+1 , X2m+1 ] = [[Yn+ 1 , Yn+ 1 ], X2m+1 ] = 2[Yn+ 1 , [Yn+ 1 , X2m+1 ]] = 0.

2 2 2 2

(6.148)

So we are left with the proof of statement (2), the proof of which is by

induction too. Let us prove the following statement:

E(n) : for all i ¤ n, j ¤ n one has [Yi+ 1 , X2j+1 ] = 0.

2

One can see that E(0) is true for obvious reasons: [Y 1 , X1 ] = 0. The

2

induction step is in three parts,

(b1): [Yn+1+ 1 , X2n+3 ] = 0;

2

(b2): [Yn+1+ 1 , X2j+1 ] = 0, j ¤ n;

2

(b3): [Yi+ 1 , X2n+3 ] = 0, i ¤ n.

2

The proof of (b1) is obvious by means of the de¬nition of X2n+3 .

The proof of (b2) follows from

[Yn+1+ 1 , X2j+1 ]] = [[Z1 , Yn+ 1 ], X2j+1 ]

2 2

= [Z1 , [Yn+ 1 , X2j+1 ]] + [[Z1 , X2j+1 ], Yn+ 1 ] = 0, (6.149)

2 2

while both terms in the right-hand side are equal to zero by assumption and

(6.144) respectively.

Finally, the proof of (b3) follows from

1

[Yi+ 1 , X2n+3 ] = [Yi+ 1 , [Yn+1+ 1 , Yn+1+ 1 ]] = [[Yi+ 1 , Yn+1+ 1 ], Yn+1+ 1 ] = 0,

2

2 2 2 2 2 2 2

(6.150)

by statement 1 of Lemma 6.36, which completes the proof of this lemma.

7. Supersymmetric mKdV equation

Since constructions and computations in this section are completely sim-

ilar to those carried through in the previous section, we shall here present

just the results for the supersymmetric mKdV equation (6.151)

vt = ’v3 + 6v 2 v1 ’ 3•(v•1 )1 ,

•t = ’•3 + 3v(v•)1 . (6.151)

Note that the supersymmetric mKdV equation (6.151) is graded

deg(x) = ’1, deg(t) = ’3,

1

deg(v) = 1, deg(•) = . (6.152)

2

7. SUPERSYMMETRIC MKDV EQUATION 291

The supersymmetry Y of (6.151) is given by

1

2

‚ ‚

Y = •1 +v . (6.153)

1

‚v ‚•

2

The associated nonlocal variable q 1 and the conserved quantity Q 1 are given

2 2

by

∞

x

q1 = (v•) dx, Q1 = (v•) dx, (6.154)

2 2

’∞ ’∞

where

(q 1 )x = v•,

2

(q 1 )t = ’v2 • + v1 •1 ’ v•2 + 3v 3 •. (6.155)

2

The nonlocal symmetry Z 1 is given by

‚ ‚

+ (q 1 v ’ •1 ) .

Z 1 = (q 1 •1 ) (6.156)

‚v ‚•

2 2

We now present the even nonlocal variables p1 , p3 and the odd nonlocal

variables q 1 , q 3 , q 5 , where

2 2 2

x

(v 2 ’ ••1 ) dx,

p1 =

’∞

x

(’v 4 ’ v1 + 3••1 v 2 + •1 •2 ) dx,

2

p3 =

’∞

x

q1 = (v•) dx,

2

’∞

x

q3 = (p1 v• + v•1 ) dx,

2

’∞

x

(’p2 v• ’ 2p1 v•1 + v 3 • ’ 2v•2 ) dx.

q5 = (6.157)

1

2

’∞

The x-derivatives of these nonlocal variables are just the integrands in

(6.157), while the t-derivatives are given by

(p1 )t = 3v 4 + v1 ’ 2vv2 ’ 9v 2 ••1 + ••3 ’ 2•1 •2 ,

2

(p3 )t = ’4v 6 + 4v2 v 3 ’ v2 + 2v1 v3 ’ 12v 2 v1 + 21v 4 ••1

2 2

’ 9vv2 ••1 + 3v1 ••1 + 12vv1 ••2 ’ 3v 2 ••3

2

+ 9v 2 •1 •2 ’ •1 •4 + 2•2 •3 ,

(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 ,