Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 015, 15 pages      nlin.SI/0701054      http://dx.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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