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].
Keywords:
Calogero–Moser type systems; bispectral problems.
Received:November 22, 2006; in final form January 5, 2007; Published online January 27, 2007
\Bibitem{Hai07}
\by Luc Haine
\paper KP Trigonometric Solitons and an Adelic Flag Manifold
\jour SIGMA
\yr 2007
\vol 3
\papernumber 015
\totalpages 15
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This publication is cited in the following 12 articles:
V. V. Prokofev, A. V. Zabrodin, “Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model”, Theoret. and Math. Phys., 208:2 (2021), 1093–1115
Prokofev V., Zabrodin A., “Elliptic Solutions to Matrix Kp Hierarchy and Spin Generalization of Elliptic Calogero-Moser Model”, J. Math. Phys., 62:6 (2021), 061502
Prokofev V., Zabrodin A., “Elliptic Solutions to the Kp Hierarchy and Elliptic Calogero-Moser Model”, J. Phys. A-Math. Theor., 54:30 (2021), 305202
V. V. Prokofev, A. V. Zabrodin, “Matrix Kadomtsev–Petviashvili Hierarchy and Spin Generalization of Trigonometric Calogero–Moser Hierarchy”, Proc. Steklov Inst. Math., 309 (2020), 225–239
Zabrodin A., “Kp Hierarchy and Trigonometric Calogero-Moser Hierarchy”, J. Math. Phys., 61:4 (2020)
Prokofev V., Zabrodin A., “Toda Lattice Hierarchy and Trigonometric Ruijsenaars?Schneider Hierarchy”, J. Phys. A-Math. Theor., 52:49 (2019), 495202
Alex Kasman, “Bispectrality of N-Component KP Wave Functions: A Study in Non-Commutativity”, SIGMA, 11 (2015), 087, 22 pp.
Haine L., Horozov E., Iliev P., “The trigonometric Grassmannian and a difference W-algebra”, Transform. Groups, 15:1 (2010), 92–114
Fehér L., Klimčík C., “On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models”, J. Phys. A, 42:18 (2009), 185202, 13 pp.
Bergvelt M., Gekhtman M., Kasman A., “Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy”, Math. Phys. Anal. Geom., 12:2 (2009), 181–200
Haine L., Horozov E., Iliev P., “Trigonometric Darboux transformations and Calogero–Moser matrices”, Glasg. Math. J., 51:A (2009), 95–106
Mukhin E., Tarasov V., Varchenko A., “Bispectral and (glN,glM) dualities, discrete versus differential”, Adv. Math., 218:1 (2008), 216–265
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