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The second Hamiltonian structure for the periodic Toda lattice

Published online by Cambridge University Press:  14 November 2011

Gloria Marí Beffa
Gloria Marí Beffa
Affiliation:
Department of Mathematics, University of Wisconsin at Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706, U.S.A.
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Abstract

In this paper we study the Poisson geometry of the second Hamiltonian structure for the periodic N Toda lattice, around a certain family of singularities. We show that their singular leaves are not isolated and that the regular codimension of the leaves at points of this kind is always equal to three. This result is based on a rather unexpected result about a certain Toeplitz matrix.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 3 , 1997 , pp. 547 - 566
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Adler, M.. On a trace functional for formal pseudo-differential operators and the symplectic structure of the KdV. Invent. Math. 50 (1979), 291–48.Google Scholar
2Adler, M. and van Moerbeke, P.. Compatible Poisson structures and the Virasoro algebra. Comm. Pure Appl. Math. 47 (1994), 5–37.CrossRefGoogle Scholar
3Antonowicz, M. and Fordy, A. P.. Hamiltonian structure of nonlinear evolution equations. In Soliton theory: a survey of results (New York: Manchester University Press, 1990).Google Scholar
4Faddeev, L. D. and Takhtajan, L. A.. Hamiltonian Methods in the Theory of Solitons (Berlin: Springer, 1987).Google Scholar
5Kirillov, A. A.. Infinite dimensional groups: their representation, orbits, invariants. In Proceedings of International Congress of Mathematics Helsinki, 1978, Vol. 2, ed. Olli Lehto, , 705–8 (Academia Scientiarum Fennica, 1980).Google Scholar
6Lax, P. D.. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21 (1968), 467–90.CrossRefGoogle Scholar
7Lazutkin, V. F. and Pankratova, T. F.. Normal forms and versal deformations for Hill's equations. Funktional Anal, i Prilozhen 9(4) (1976), 41–8.Google Scholar
8Mari Beffa, G.. A transverse structure for the Lie-Poisson bracket on the dual of the Virasoro algebra. Pacific J. Math. 163 (1994), 43–72.CrossRefGoogle Scholar
9Marsden, J. and Harnad, J. (eds). Proceedings of the CRM workshop on Hamiltonian systems, Transformation groups and Spectral transform methods (Montreal: CRM, 1990).Google Scholar
10Olver, P.. Applications of Lie Groups to differential equations (Berlin: Springer, 1986).CrossRefGoogle Scholar
11Saitoh, N.. J. Phys. Soc. Japan 49 (1980), 409; 54 (1985), 3261.Google Scholar
12Segal, G.. Unitary representations of some infinite dimensional groups. Comm. Math. Phys. 80 (1981), 301–42.Google Scholar
13Toda, M.. Theory of Nonlinear Lattices, Springer Series in Solid-State Sciences 20 (Berlin: Springer, 1989).Google Scholar
14Weinstein, A.. The local structure of Poisson manifolds. J. Differential Geom. 18 (1983), 523–57.Google Scholar
15Witten, E.. Coadjoint orbits of the Virasoro group. Comm. Math. Phys. 114 (1988), 1–53.Google Scholar