ψ1 is given by (7.48) and K is a constant. In this way the hierarchy

of potentials Q(3) , Q(4) , . . . can be determined.

7.6 SUSY and Conservation Laws in the KdV-

SG Systems

Supersymmetric transformations can also be used [35] to connect

the conserved quantities of the KdV to those of the SG equation. As

stated earlier while the KdV equation arises in the context of water

wave problems, a natural place for the SG equation to exist is in the

motion of a closed string under an external ¬eld [36]. However, the

SG equation can also be recognized as the evolution equation for a

scalar ¬eld in 1 + 1 dimensions in the presence of highly nonlinear

self-interactions.

A remarkable property of the Lax form for the SG equation is

that it is endowed with a supersymmetric structure corresponding

to the eigenvalue problem of the L operator. At ¬rst sight it is

not obvious why a link should exist between the KdV which is a

nonrelativistic equation and SG which is a relativistic one. However

what emerges is that supersymmetric transformations do not map the

SG equation as a whole into the KdV. Only the eigenvalue equation

of the L operator and as a consequence the conserved quantities for

the two are transformed to each other.

© 2001 by Chapman & Hall/CRC

Consider the spectral problem for the SG equation, which reads

Lχ = ξχ (7.55)

where ξ is a constant, L is given by (7.33), and χ is the column

matrix

χ1

χ= (7.56)

χ2

Then (7.55) translates to

2χ1 ’ iψ χ2 = ξχ1 (7.57a)

iψ χ1 ’ 2χ2 = ξχ2 (7.57b)

De¬ning the quantities χ1 = χ1 ± χ2 , (7.57a) and (7.57b) can be

put in the form

ξ2

1 1

W2 ’ W

’ χ+ + χ+ = ’ χ+ (7.58a)

2 2 8

ξ2

1 1

W2 + W

’ χ’ + χ’ = ’ χ’ (7.58b)

2 2 8

with W = ’ iψ . The supersymmetric Hamiltonian Hs which acts

2

χ’

on the two-component column , may be expressed as Hs =

χ+

d2

diag H’ , H+ ) similar to (2.28) where H± = ’ 1 dx2 ± V± and V±

2

1 2 “ W . We thus see that a natural

denoting the combinations 2 W

embedding of SUSY to be present in the eigenvalue problem of the

L operator for the SG equation.

It is straightforward to relate the conservation laws in the KdV

and SG systems. The eigenvalue problem of the L operator for the

KdV system being given by (7.34), it follows that either of Equations

(7.58) is identical to (7.34) if u transforms as

1

W2 “ W

u± ’ (7.59a)

2

» ’ ’(ξ 2 /4) (7.59b)

E¬ectively, this implies that given the set of conserved quantities

I0 , I1 , I2 , etc. of the KdV system, the corresponding ones for the SG

system may be obtained through the mapping (7.59) and using the

© 2001 by Chapman & Hall/CRC

relation between W and ψ given earlier. Indeed this turns out to

be so (upto an overall constant in the coe¬cients of the integrals) as

can be veri¬ed from (7.25) and comparing the transformed quantities

with (7.27). It should be emphasized that the I™s of SG equation do

not depend on the arbitrariness of the sign in (7.59a), (7.21) being

invariant under ψ ’ ’ψ.

Let us distinguish the two cases in (7.59) by u+ and u’ . Our

discussion in the preceding section now tells us that if suitable bound-

ary conditions are prescribed to ψ then u+ may be interpreted as an

(N + 1) soliton solution if u’ corresponds to the N soliton solution.

A relation between u+ and u’ can be obtained on eliminating W

(u+ + u’ )1/2 = (u’ ’ u+ ) (7.60)

7.7 Supersymmetric KdV

In recent times a number of attempts have been made to seek su-

persymmetric extensions of the KdV equation [37-39]. This may be

done in a superspace formalism (discussed in Chapter 2) replacing

the coordinate x by the set (x, θ), θ being Gransmannian (N = 1

SUSY).

Let us consider a super¬eld ¦ given by

¦ = ξ(x) + θu(x) (7.61)

where ξ is anticommuting in nature. Then the character of such a ¦

is fermionic. Further, we de¬ne the covariant derivative to be

D = θ‚x + ‚θ (7.62)

The anticommuting nature of ξ and θ implies

D2 = ‚x

{D, θ} = 0 (7.63)

where Q is the supersymmetric generator

Q = ‚θ ’ θ‚x (7.64)

© 2001 by Chapman & Hall/CRC

Note that the supersymmetric transformations are realized ac-

cording to

x ’ x ’ ·θ

θ ’ θ+· (7.65)

where · is anticommuting. Indeed if we write

δ¦ = ·Q¦

= ·u + θ·ξ (7.66)

we can deduce from (7.61)

δξ = ·u(x)

δu = ·ξ (x) (7.67)

To have invariance under supersymmetric transformations (7.67)

we need to work with quantities involving the covariant derivative

and the super¬eld. So, in terms of these, the D and ¦, we can

construct a supersymmetric version of the KdV equation, called the

SKdV equation.

To derive SKdV, we observe that by multiplying both sides of

(7.2) by the anticommuting variable θ we get the form

θut = ’θu + 6θuu (7.68)

Hence we can think of transforming θut ’ ¦t since ¦ contains a θu

part

¦t = ξt + θut (7.69)

We also have the relations

D¦ = θξx + u

D2 ¦ = θux + ξx

D4 ¦ = θuxx + ξxx

D6 ¦ = θuxxx + ξxxx (7.70)

So the dispersive term of (7.68) is seen to reside in D6 ¦. However,

the nonlinear term can be traced in the following two expressions

(D¦)(D2 ¦) = θξx + uξx + θuux

2

(7.71)

D2 (¦D¦) = ’θξx ’ θξξxx + ux ξ + uξx + 2θuux (7.72)

2

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We therefore write the SKdV equation in a one-parameter rep-

resentation

¦t = ’D6 ¦ + aD2 (¦D¦) + (6 ’ 2a)D¦D2 ¦ (7.73)

where a is arbitrary. We need to mention here that had we considered

a bosonic extension of the KdV induced by a super¬eld χ = u(x) +

θ±(x) in place of (7.61), the linearity of the fermionic ¬eld ±(x)

would have ensured that the resulting system yields the trivial case

that KdV is self-generate with no in¬‚uence from the quantity ±(x).

However, (7.73) is nontrivial. We get, componentwise, for u(x)

and ξ(x), the evolutions

ut = ’uxxx + 6uux ’ aξξxx (7.74)

ξt = ’ξxxx + aux ξ + (6 ’ a)uξx (7.75)

Thus the process of supersymmetrization a¬ects the KdV equation

in that it appears coupled with the ξ ¬eld.

Kupershmidt [38] also obtained a coupled set of equations in-

volving the KdV

ut = ’uxxx + 6uux ’ 3ξξxx (7.76)

ξt = ’4ξxxx + 6ξx u + 3ξux (7.77)

and showed that superconformal algebra is related to it. But the

above equations are not invariant under supersymmetric transfor-

mations.

We now turn to the Hamiltonian structure of the SKdV equa-

tion (7.73). In this regard, let us ¬rst of all demonstrate that the

KdV is a bi-Hamiltonian system. We have already given the asso-

ciated Hamiltonian structures for the conservation laws of the KdV

equation in (7.25). Such a correspondence implies that the conserved

quantities are in e¬ect a sequence of Hamiltonians each generating

its own evolution equation [40,41].

We now note that the KdV equation (7.2) can be expressed in

‚

two equivalent ways as follows D ≡ ‚x

‚u

= Ou, O = ’D3 + 2(Du + uD) (7.78)

‚t

© 2001 by Chapman & Hall/CRC

and

‚u

= D 3u2 ’ D2 u (7.79)

‚t

Introducing the functional forms of the Poisson bracket

δA[u] δB[u]

{A[u], B[u]}1 = d„ O (7.80)

δu(„ ) δu(„ )

δA[u] δB[u]

{A[u], B[u]}2 = d„ D (7.81)

δu(„ ) δu(„ )

and using the standard de¬nitions of functional di¬erentiation [42]

δF [X] 1

= lim F [X] ’ F [X] (7.82)

δX(y) ’0

where

X(x) = X(x) + δ(x ’ y) (7.83)

it follows that

{u(x), u(y)}1 = ’δ (x ’ y) + 4uδ (x ’ y)

+2u δ(x ’ y) (7.84)

{u(x), u(y)}2 = δ (x ’ y) (7.85)

The de¬nitions (7.80) and (7.81) then lead to

‚u

= {u(x), I1 }1

‚t

= right hand side of (7.2) (7.86)

and

‚u

= {u(x), I2 }2

‚t

= right hand side of (7.2) (7.87)

That the KdV equation is bi-Hamiltonian follows from the fact

that it can be written in the form

δIi

ut = D(i) (7.88)

δu

© 2001 by Chapman & Hall/CRC

in two di¬erent ways

D(1) = ’D3 + 2(Du + uD), I1 = dxu2 (7.89)

1