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SIGMA 3 (2007), 015, 15 pages nlin.SI/0701054
https://doi.org/10.3842/SIGMA.2007.015
Contribution to the Vadim Kuznetsov Memorial Issue
KP Trigonometric Solitons and an Adelic Flag Manifold
Luc Haine
Department of Mathematics, Université catholique de Louvain,
Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
Received November 22, 2006, in final form January 5, 2007; Published online January 27, 2007
Abstract
We show that the trigonometric solitons of the KP
hierarchy enjoy a differential-difference bispectral property,
which becomes transparent when translated on two suitable spaces
of pairs of matrices satisfying certain rank one conditions. The
result can be seen as a non-self-dual illustration of Wilson's
fundamental idea [Invent. Math. 133 (1998), 1-41]
for understanding the (self-dual) bispectral property of the
rational solutions of the KP hierarchy. It also gives a bispectral
interpretation of a (dynamical) duality between the hyperbolic
Calogero-Moser system and the rational Ruijsenaars-Schneider
system, which was first observed by Ruijsenaars [Comm.
Math. Phys. 115 (1988), 127-165].
Key words:
Calogero-Moser type systems; bispectral problems.
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