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

SIGMA 2 (2006), 097, 15 pages      nlin.SI/0701004
Contribution to the Vadim Kuznetsov Memorial Issue

On the Darboux-Nijenhuis Variables for the Open Toda Lattice

Yuriy A. Grigoryev and Andrey V. Tsiganov
St.Petersburg State University, St.Petersburg, Russia

Received November 17, 2006; Published online December 30, 2006

We discuss two known constructions proposed by Moser and by Sklyanin of the Darboux-Nijenhuis coordinates for the open Toda lattice.

Key words: bi-Hamiltonian systems; Toda lattice.

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  1. Atiyah M., Hitchin N., The geometry and dynamics of magnetic monopoles, M.B. Porter Lectures, Princeton Univ. Press, 1988.
  2. Das A., Okubo S., A systematic study of the Toda lattice, Ann. Phys., 1989, V.30, 215-232.
  3. Falqui G., Magri F., Pedroni M., Bihamiltonian geometry and separation of variables for Toda lattices, J. Nonlinear Math. Phys., 2001, V.8, 118-127, nlin.SI/0002008.
  4. Falqui G., Pedroni M., Separation of variables for bi-Hamiltonian systems, Math. Phys. Anal. Geom., 2003, V.6, 139-179, nlin.SI/0204029.
  5. Faybusovich L., Gekhtman M., Poisson brackets on rational functions and multi Hamiltonian structures for integrable lattices, Phys. Lett. A, 2000, V.272, 236-244, nlin.SI/0006045.
  6. Fernandes R.L., On the master symmetries and bi-Hamiltonian structure of the Toda lattice, J. Phys. A: Math. Gen., 1993, V.26, 3797-3803.
  7. Flaschka H., McLaughlin D.W., Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys., 1976, V.55, 438-456.
  8. Fokas A., Fuchssteiner B., Symplectic structures, Bäcklund transformations and hereditary symmetries, Phys. D, 1981, V.4, 47-66.
  9. Frölicher A., Nijenhuis A., Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms, Nederl. Akad. Wetensch. Proc. Ser. A, 1956, V.59, 338-359.
  10. Gelfand I.M., Dorfman I.Ya., Hamiltonian operators and algebraic structures associated with them, Funktsional. Anal. i Prilozhen., 1979, V.13, N 4, 13-30.
  11. Gelfand I.M., Zakharevich I., On the local geometry of a bi-Hamiltonian structure, in The Gelfand Mathematical Seminars 1990-1992, Editors L. Corwin et al., Boston, Birkhauser, 1993, 51-112.
  12. Krein M.G., Naimark M.A., The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations, Linear and Multilinear Algebra, 1981, V.10, N 4, 265-308 (translated from the Russian by O. Boshko and J.L. Howland).
  13. Kosmann-Schwarzbach Y., Magri F., Poisson-Nijenhuis structures, Ann. Inst. Poincaré (Phys. Theor.), 1990, V.53, 35-81.
  14. Komarov I.V., Tsiganov A.V., Two particle quantum periodic Toda lattice, Vestnik Leningrad Univ., 1988, V.2, 69-71.
  15. Levi-Civita T., Integrazione delle equazione di Hamilton-Jacobi per separazione di variabili, Math. Ann., 1904, V.24, 383-397.
  16. Magri F., Geometry and soliton equations, in La Mécanique Analytique de Lagrange et son héritage, Atti Acc. Sci. Torino Suppl., 1990, V.24, 181-209.
  17. Magri F., Eight lectures on integrable systems, in Integrability of Nonlinear Systems (1996, Pondicherry), Lecture Notes in Phys., Vol. 495, Berlin, Springer, 1997, 256-296.
  18. Moser J., Finitely many mass points on the line under the influence of an exponential potential - an integrable system, in Dynamical Systems, Theory and Applications (1974, Rencontres, BattelleRes. Inst., Seattle, Wash.), Lecture Notes in Phys., Vol. 38, Berlin, Springer, 1975, 467-497.
  19. Sklyanin E.K., The quantum Toda chain, in Nonlinear Equations in Classical and Quantum Field Theory (1983/1984, Meudon/Paris), Lecture Notes in Phys., Vol. 226, Berlin, Springer, 1985, 196-293.
  20. Sklyanin E.K., Separation of variables - new trends, in Quantum Field Theory, Integrable Models and Beyond (1994, Kyoto), Progr. Theoret. Phys. Suppl., 1995, V.118, 35-60, solv-int/9504001.
  21. Smirnov F.A., Structure of matrix elements in quantum Toda chain, J. Phys. A: Math. Gen., 1998, V.31, 8953-8971, math-ph/9805011.
  22. Stieltjes T., Recherches sur les fractions continues, Ann. de Toulouse, VIII-IX, 1894-1895.
  23. Tsiganov A.V., On the invariant separated variables, Regul. Chaotic Dyn., 2001, V.6, 307-326.
  24. Tsiganov A.V., On the Darboux-Nijenhuis coordinates for the generalizaed open Toda lattices, Theoret. and Math. Phys., submitted.
  25. Vaninsky K.L., The Atiyah-Hitchin bracket and the open Toda Lattice, J. Geom. Phys., 2003, V.46, 283-307, math-ph/0202047.

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