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.

pdf (298 kb)   ps (195 kb)   tex (18 kb)

References

1. Chalykh O.A., The duality of the generalized Calogero and Ruijsenaars problems, Russian Math. Surveys 52 (1997), 1289-1291.
2. Chalykh O.A., Bispectrality for the quantum Ruijsenaars model and its integrable deformation, J. Math. Phys. 41 (2000), 5139-5167.
3. Duistermaat J.J., Grünbaum F.A., Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240.
4. Etingof P., Lectures on Calogero-Moser systems, math.QA/0606233.
5. Grünbaum F.A., Haine L., A theorem of Bochner revisited, in Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman, Editors A.S. Fokas and I.M. Gelfand, Progr. Nonlinear Differential Equations, Vol. 26, Birkhäuser, Boston, MA, 1997, 143-172.
6. Haine L., Iliev P., Commutative rings of difference operators and an adelic flag manifold, Int. Math. Res. Not. 6 (2000), 281-323.
7. Iliev P., Rational Ruijsenaars-Schneider hierarchy and bispectral difference operators, math-ph/0609011.
8. Kasman A., Spectral difference equations satisfied by KP soliton wavefunctions, Inverse Probl. 14 (1998), 1481-1487, solv-int/9811009.
9. Kasman A., Gekhtman M., Solitons and almost-intertwining matrices, J. Math. Phys. 42 (2001), 3540-3551, math-ph/0011011.
10. Kazhdan D., Kostant B., Sternberg S., Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), 481-507.
11. Kuznetsov V.B., Nijhoff F.W., Sklyanin E.K., Separation of variables for the Ruijsenaars system, Comm. Math. Phys. 189 (1997), 855-877, solv-int/9701004.
12. Moser J., Three integrable systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220.
13. Olshanetsky M.A., Perelomov A.M., Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Invent. Math. 37 (1976), 93-108.
14. Reach M., Difference equations for N-soliton solutions to KdV, Phys. Lett. A 129 (1988), 101-105.
15. Ruijsenaars S.N.M., Schneider H., A new class of integrable systems and its relation to solitons, Ann. Physics 170 (1986), 370-405.
16. Ruijsenaars S.N.M., Action-angle maps and scattering theory for some finite-dimensional integrable systems I. The pure soliton case, Comm. Math. Phys. 115 (1988), 127-165.
17. Ruijsenaars S.N.M., Integrable particle systems vs solutions to the KP and 2D Toda equations, Ann. Physics 256 (1997), 226-301.
18. Segal G., Wilson, G., Loop groups and equations of KdV type, Publ. Math. Inst. Hautes Études Sci. 61 (1985), 5-65.
19. Suris Y.B., The problem of integrable discretization: Hamiltonian approach, Progr. Math. Vol. 219, Birkhäuser, Boston, MA, 2003.
20. Wilson G., Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442 (1993), 177-204.
21. Wilson G., Collisions of Calogero-Moser particles and an adelic Grassmannian, Invent. Math. 133 (1998), 1-41.