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D(2) = D, I2 = dx u3 + u2 (7.90)
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)


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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


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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.

[2] R. Rajaraman, Solitons and Instantons. North-Holland Pub-
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[3] J. Scott-Russell, Rep. 14th Meeting British Assoc Adv. Sci.,
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[4] C.S. Gardner and G.K. Morikawa, Courant Inst. Math. Sc.
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[5] N.J. Zabusky, Mathematical Models in Physical Sciences, S.
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[7] M.D. Kruskal, Proc IBM Scienti¬c Computing Symposium on
Large-scale Problems in Physics (IBM Data Processing Divi-
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[8] E. Fermi, J.R. Pasta, and S.M. Ulam, Los Alamos. Report
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[10] M.C. Shen, Siam J. Appl. Math., 17, 260, 1969.

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[13] P.D. Lax, Comm. Pure Appl. Math., 21, 467, 1968.

[14] V.E. Zakharov and A.B. Shabat, Func. Anal. Appl., 8, 226,
1974.


© 2001 by Chapman & Hall/CRC
[15] A.V. B¨cklund, Math. Ann., 19, 387, 1882.
a

[16] G.L. Lamb Jr., J. Math. Phys., 15, 2157, 1974.

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[18] H.C. Morris, J. Math. Phys., 18, 530, 1977.

[19] M. Wadati, H. Sanuki, and K. Konno, Prog. Theor. Phys., 53,
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[20] A. Chodos, Phys. Rev., D21, 2818, 1980.

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[24] B. Bagchi, Int. J. Mod. Phys., A5, 1763, 1990.

[25] A.A. Stahlhofen and A.J. Schramm, Phys. Scripta., 43, 553,
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[26] J. Hruby, J. Phys. A. Math. Gen., 22, 1802, 1989.

[27] G. Darboux, C.R. Acad Sci. Paris, 92, 1456, 1882.

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[31] D.L. Pursey, Phys. Rev., D33, 2267, 1986.

[32] M.M. Nieto, Phys. Lett., B145, 208, 1984.

[33] I.M. Gel™fand and B.M. Levitan, Am. Math. Soc. Trans., 1,
253, 1955.


© 2001 by Chapman & Hall/CRC
[34] P.B. Abraham and H.E. Moses, Phys. Rev., A22, 1333, 1980.

[35] B. Bagchi, A. Lahiri, and P.K. Roy, Phys. Rev., D39, 1186,
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[36] F. Lund and T. Regge, Phys. Rev., D14, 1524, 1976.

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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

21 0 0
c=
0 10


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« 

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