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B = c2 c4 /c1 (7.54)

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


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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)


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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


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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


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in two di¬erent ways

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

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