The Painlevé analysis of a generic multiparameter extension of the Korteweg–de Vries (KdV) equation is presented. Unusual aspects of the analysis, pertaining to the presence of two fermionic fields, are emphasized. For the general class of models considered, we find that the only ones which manifestly pass the test are precisely the four known integrable supersymmetric KdV equations, including the case.
REFERENCES
1.
J.
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Still, another integrable system has been found to satisfy the Painlevé test (Ref. 3). However, it is also rather trivial (cf. its precise form given below) and thereby easily proved to be integrable directly [see also,
D.-G.
Zhang
and B.-Z.
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)]. But most importantly for us, it is not supersymmetric.11.
12.
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In addition, the integrability of the bosonic-core version of this system is demonstrated in
P.
Kersten
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,” nlin.SI/0010041. These authors prove the integrability by displaying an infinite series of symmetries.18.
19.
20.
See also
P.
Mathieu
, “Open problems for SuperKdV equations
,” math-ph/0005007, for a pedagogical introduction to the supersymmetric KdV equations.21.
Notice that in Ref. 3, the following system has also been shown to have the Painlevé property However, the change of variable transforms it into [cf.
D.
Depireux
and P.
Mathieu
, Phys. Lett. B
308
, 272
(1993
)]. which is somewhat trivial (but clearly integrable) and manifestly not supersymmetric invariant.22.
Notice that even if the Painlevé analysis breaks the invariance, we are still free to choose the sign of Clearly, in a formulation in terms of the redefined fields we would be free to take either or as an arbitrary function, the other being null.
23.
J. Weiss, in Painlevé Transcendents, edited by D. Levi and P. Winternitz (Plenum, New York, 1992), and references therein.
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© 2001 American Institute of Physics.
2001
American Institute of Physics
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