2x

Given the constants of motion for the KdV summarized in (7.25),

it is easy to arrive at their supersymmetric counterparts

1

I1 ’ J1 = dxdθ ¦D¦ (7.91)

2

1

dxdθ 2¦(D¦)2 + D2 ¦D3 ¦

I2 ’ J2 = (7.92)

2

etc.

Consider J2 . In Poisson bracket formulation

¦t = {¦, J2 } = ’D6 ¦ + 4D¦D2 ¦ + 2¦D3 ¦ (7.93)

This picks out a = 2 from the SKdV equation (7.73). While deriving

the above form we have made use of the following de¬nition of the

Poisson bracket

‚ ‚

{¦(x1 , θ1 ), ¦(x2 , θ2 )} = θ1 + ∆ (7.94)

‚x1 ‚θ1

where

∆ ≡ (θ2 ’ θ1 )δ(x2 ’ x1 ) (7.95)

Note that

dx1 dθ1 F (x1 , θ1 ) ∆ = dθ1 (θ2 ’ θ1 ) dx1 F (x1 , θ1 )δ(x2 ’ x1 )

= dθ1 (θ2 ’ θ1 )F (x2 , θ1 )

= F (x2 , θ2 ) (7.96)

Inserting ¦(xi , θi ) = ξi + θi ui (i = 1, 2) in (7.94), it follows that

{ξ1 , ξ2 } = ’δ(x2 ’ x1 )

‚

{u1 , u2 } = [δ(x2 ’ x1 )]

‚x1

{ξ1 , u2 } = 0

{u1 , ξ2 } = 0 (7.97)

© 2001 by Chapman & Hall/CRC

As such we can write

δH

‚t ¦ = D (7.98)

δ¦

where D is diagonal corresponding to the ¬rst two equations of (7.97).

By expanding the right hand side of (7.92) in terms of quantities

independent of θ and those depending on θ and calculating δH along

δu

δH

with δξ and using (7.82), the set of coupled equations (7.74) and

(7.75) are seen to follow.

Corresponding to J1 , equivalence with (7.73) can be established

for a di¬erent value of a namely a = 3 (see [39]). We therefore

conclude that the supersymmetric KdV system does not have a bi-

Hamiltonian structure. Actually, as pointed out by several authors,

SKdV can be given a local meaning only [43-46].

The KdV equation has been extensively studied in relation to

its integrability. Links with Virasoro algebra have been established

through its second Hamiltonian structure [47]. Further the supercon-

formal algebra was found to be related to supersymmetric extension

of the KdV which is integrable [48]. A curious result has also been

obtained concerning a pair of integrable fermionic extensions of the

KdV equation : while one is bi-Hamiltonian but not supersymmetric,

the other turns out to be supersymmetric (the one addressed to in

this section) but not bi-Hamiltonian.

7.8 Conclusion

In this chapter we have discussed the role of SUSY in some nonlin-

ear equations such as the KdV, MKdV, and SG. In the literature

SUSY has also been used to study several aspects of nonlinear sys-

tems. More recently, hierarchy of lower KdV equations has been de-

termined [50-53] which arise as a necessary part of supersymmetric

constructions. It is now known that the supersymmetric structure

of KdV and MKdV hierarchies leads to lower KdV equations and

it becomes imperative to consider Miura™s transformation in super-

symmetric form. A supersymmetric structure has also been found

to hold in Kadomtsev-Petviashvili (KP) hierarchies. In this connec-

tion it is relevant to mention that among various nonlinear evolution

equations, the KdV-MKdV, KP and its modi¬ed partner are gauge

equivalent to one another with the generating function coinciding

© 2001 by Chapman & Hall/CRC

with the equation for the corresponding gauge function [54]. Fi-

nally, we have discussed the SKdV equation and commented upon

its Hamiltonian structures.

7.9 References

[1] D.J. Korteweg and G. de Vries, Phil. Mag., 39, 422, 1895.

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© 2001 by Chapman & Hall/CRC

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© 2001 by Chapman & Hall/CRC

[34] P.B. Abraham and H.E. Moses, Phys. Rev., A22, 1333, 1980.

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© 2001 by Chapman & Hall/CRC

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© 2001 by Chapman & Hall/CRC

CHAPTER 8

Parasupersymmetry

8.1 Introduction

In the literature SUSYQM has been extended [1-16] to what con-

stitutes parasupersymmetric quantum mechanics (PSUSYQM). To

understand its underpinnings we ¬rst must note that the fermionic

operators a and a+ obeying (2.12) and (2.13) also satisfy the com-

mutation condition

11

a+ , a = diag ,’ (8.1)

22

The entries in the parenthesis of the right-hand-side can be looked

upon as the eigenvalues of the 3rd component of the spin 1 operator.

2

The essence of a minimal (that is of order p = 2) PSUSYQM

scheme is to replace the right-hand-side of (8.1) by the eigenvalues of

the 3rd component of the spin 1 operator. Thus in p = 2 PSUSYQM

a new set of operators c and c+ is introduced with the requirement

c+ , c = 2 diag(1, 0, ’1) (8.2)

A plausible set of representations for c and c+ satisfying (8.2) is given

by the matrices

«

0 00

√

21 0 0

c=

0 10

© 2001 by Chapman & Hall/CRC

«