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Bihamiltonian manifolds and the τ-function
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Proceedings of the 1991 Joint Summer Research Conference on Mathematical Aspects of Classical Field Theory, edited by M. Gotai, J. Marsden, and V. Moncrief
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P. Casati, F. Magri, and M. Pedroni, "Bihamiltonian manifolds and the τ-function," in Proceedings of the 1991 Joint Summer Research Conference on Mathematical Aspects of Classical Field Theory, edited by M. Gotai, J. Marsden, and V. Moncrief, Contemporary Math. 132, 213-234 (1992).
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Casati, P.1
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Algebraic structures connected with the Yang-Baxter equation
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Sklyanin, E.K.1
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R-matrices and higher Poisson brackets for integrable systems
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Oevel, W.1
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0001306018
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Nonlinear Poisson structures and r-matrices
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L. C. Li and S. Parmentier, "Nonlinear Poisson structures and r-matrices," Commun. Math. Phys. 125, 545-563 (1989).
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Li, L.C.1
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34250286595
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On a trace functional for formal pseudo-differential operators and the symplectic structure for Korteweg-de Vries type equations
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M. Adler, "On a trace functional for formal pseudo-differential operators and the symplectic structure for Korteweg-de Vries type equations," Invent. Math. 50, 219-248 (1979).
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Adler, M.1
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0001043985
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Quantization and representation theory
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Representation theory of Lie groups, Proc. SRC/LMS Res. Symp., Oxford 1977
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B. Kostant, "Quantization and representation theory," in Representation theory of Lie groups, Proc. SRC/LMS Res. Symp., Oxford 1977, Lond. Math. Soc. Lect. Notes 34, 287-316 (1979).
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Kostant, B.1
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9
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34250248977
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Systems of Toda type, inverse spectral problems, and representation theory
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W. Symes, "Systems of Toda type, inverse spectral problems, and representation theory," Invent. Math. 59, 13-51 (1980).
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Symes, W.1
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21844519645
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Multiple Hamiltonian structures for Toda-type systems
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P. A. Damianou, "Multiple Hamiltonian structures for Toda-type systems," J. Math. Phys. 35, 5511-5541 (1994).
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Damianou, P.A.1
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11
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21844491841
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R-matrix approach to lattice integrable systems
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M. Blaszak and K. Marciniak, "R-matrix approach to lattice integrable systems," J. Math. Phys. 35, 4661-4682 (1994).
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Blaszak, M.1
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Completely integrable systems, Kac-Moody Lie algebras and curves
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M. Adler and P. van Moerbeke, "Completely integrable systems, Kac-Moody Lie algebras and curves," Adv. Math. 38, 267-317 (1980).
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Adler, M.1
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13
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0002506343
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Discrete Lax equations and differential-difference calculus
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Société Mathématique de France
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B. A. Kupershmidt, "Discrete Lax equations and differential-difference calculus," Astérisque 123 (1985), Société Mathématique de France.
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Kupershmidt, B.A.1
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14
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0002190849
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Soliton equations as dynamical systems on infinite dimensional Grassmann manifold
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(US-Japan Seminar, Tokyo), edited by P. Lax and H. Fujita North-Holland, Amsterdam, The application discussed in this article is of a different kind and concerns the iterative biHamiltonian schemes
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The ordinary Schur polynomials were employed by the Kyoto school to construct the soliton solutions of the KP hierarchy: see M. Sato, Y. Sato, "Soliton equations as dynamical systems on infinite dimensional Grassmann manifold," in Nonlinear PDEs in Applied Sciences (US-Japan Seminar, Tokyo), edited by P. Lax and H. Fujita (North-Holland, Amsterdam, 1982), pp. 259-271. The application discussed in this article is of a different kind and concerns the iterative biHamiltonian schemes.
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Nonlinear PDEs in Applied Sciences
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Sato, M.1
Sato, Y.2
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15
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0000876641
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The solution to a generalized Toda lattice and representation theory
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B. Kostant, "The solution to a generalized Toda lattice and representation theory," Adv. Math. 34, 195-338 (1979).
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Kostant, B.1
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16
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0040225698
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Integrability condition and finite-periodic Toda lattice
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This author constructs a recursion operator and claims that an ad hoc extension of the biHamiltonian formalism is necessary for the treatment of the periodic case
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For a different approach, using directly the physical coordinates (positions and momenta of the particles), see S. Okubo, "Integrability condition and finite-periodic Toda lattice," J. Math. Phys. 31. 1919-1928 (1990). This author constructs a recursion operator and claims that an ad hoc extension of the biHamiltonian formalism is necessary for the treatment of the periodic case.
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Okubo, S.1
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17
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27644437798
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On the bi-Hamiltonian structure of Toda and relativislic Toda lattices
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Yu. B. Suris, "On the bi-Hamiltonian structure of Toda and relativislic Toda lattices," Phys. Lett. A 180, 419-429 (1993).
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Suris, Yu.B.1
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18
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84972540666
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Les variétés de poisson et leurs algèbres de lie associées
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A. Lichnérowicz, "Les variétés de Poisson et leurs algèbres de Lie associées," J. Diff. Geom. 12, 253-300 (1977).
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Lichnérowicz, A.1
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84972531238
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The local structure of Poisson manifold
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A. Weinstein, "The local structure of Poisson manifold," J. Diff. Geom. 18, 523-557 (1983).
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Weinstein, A.1
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21
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36749117832
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A simple model of the integrable Hamiltonian equation
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F. Magri, "A simple model of the integrable Hamiltonian equation," J. Math. Phys. 19, 1156-1162 (1978).
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Magri, F.1
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22
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49049150360
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Symplectic structures, their Backlund transformations and hereditary symmetries
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B. Fuchssteiner and A. S. Fokas, "Symplectic structures, their Backlund transformations and hereditary symmetries," Physica D 4, 47-66 (1981).
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Fuchssteiner, B.1
Fokas, A.S.2
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23
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0001082070
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Reduction techniques for infinite-dimensional Hamiltonian systems: Some ideas and applications
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F. Magri, C. Morosi, and O. Ragnisco, "Reduction techniques for infinite-dimensional Hamiltonian systems: some ideas and applications," Commun. Math. Phys. 99, 115-140 (1985).
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Magri, F.1
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24
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0000757211
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Reduction of Poisson manifold
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J. E. Marsden and T. Ratiu, "Reduction of Poisson manifold," Lett. Math. Phys. 11, 161-169 (1986).
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Lett. Math. Phys.
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Marsden, J.E.1
Ratiu, T.2
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25
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0003270559
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Group theoretical methods in the theory of finite dimensional integrable systems
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edited by V. I. Arnol'd and S. P. Novikov Springer, Berlin
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A. G. Reyman and M. A. Semenov-Tian-Shansky, "Group theoretical methods in the theory of finite dimensional integrable systems," in Dynamical systems VII, edited by V. I. Arnol'd and S. P. Novikov (Springer, Berlin, 1994), pp. 116-225.
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Reyman, A.G.1
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25344451956
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On the Toda lattice. I
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H. Flaschka, "On the Toda lattice. I," Phys. Rev. B 9, 1924-1925 (1974).
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Flaschka, H.1
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0001600405
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Integrals of the Toda lattice
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M. Hénon, "Integrals of the Toda lattice," Phys. Rev. B 9, 1921-1923 (1974).
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Hénon, M.1
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30
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0040820387
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-
note
-
Here, and in the rest of the paper, we say that some property ℬ(σ), depending on an integer σ, holds for σ≫0 if there is κ∈ℤ such that ℬ(σ) is satisfied for each σ≥κ similarly, ℬ(σ) holds for σ≪0 if it is true for each σ below some κ. Other statements of a similar kind will appear in the sequel; for example, we will say that some property ℬ(α,β), depending on two integers, holds for β-α≫0 or for |β-α|≫0, with an obvious meaning.
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31
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0040225701
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-
note
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The symplectic leaves are, by definition, the integral submanifolds of the distribution Im Q; as it is well known, a Poisson tensor can be properly restricted to anyone of its symplectic leaves, and the restriction is kernel free (see Ref. 20).
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34
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0040225695
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Sur les variations séculaires des éléments des orbites pour les sept planètes principales
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U. J. J. Leverrier, "Sur les variations séculaires des éléments des orbites pour les sept planètes principales," J. Math 5, 230 (1840).
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J. Math
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Leverrier, U.J.J.1
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35
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0039633770
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-
note
-
k are not zero for k≥;n + 2 (even though they are not functionally independent of the first n integrals and vector fields).
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-
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36
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0039041767
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-
note
-
α= H′ for α odd. The condition on the average of r fixes the sum H + H′.
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-
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37
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0040820384
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-
note
-
0υ = 1. The sequence υ-a/K satisfies this condition due to the choice made for K.
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38
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85045502624
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On the continuous limit of integrable lattices I. The Kac-Moerbeke system and KdV theory
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to appear
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C. Morosi and L. Pizzocchero, "On the continuous limit of integrable lattices I. The Kac-Moerbeke system and KdV theory," to appear in Commun. Math. Phys.
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Commun. Math. Phys.
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Morosi, C.1
Pizzocchero, L.2
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39
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34250120087
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Lie algebras and equations of the Korteweg-de Vries type
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V. G. Drinfeld and V. V. Sokolov, "Lie algebras and equations of the Korteweg-de Vries type," J. Sov. Math. 30, 1975-2036 (1985).
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Drinfeld, V.G.1
Sokolov, V.V.2
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