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

SIGMA 1 (2005), 010, 12 pages      nlin.SI/0511039      https://doi.org/10.3842/SIGMA.2005.010

A Gentle (without Chopping) Approach to the Full Kostant-Toda Lattice

Pantelis A. Damianou a and Franco Magri b
a) Department of Mathematics and Statistics, University of Cyprus, 1678, Nicosia, Cyprus
b) Department of Mathematics, University of Milano Bicocca, Via Corsi 58, I 20126 Milano, Italy

Received September 22, 2005, in final form October 24, 2005; Published online October 25, 2005

Abstract
In this paper we propose a new algorithm for obtaining the rational integrals of the full Kostant-Toda lattice. This new approach is based on a reduction of a bi-Hamiltonian system on gl(n, R). This system was obtained by reducing the space of maps from Zn to GL(n, R) endowed with a structure of a pair of Lie-algebroids.

Key words: full Kostant-Toda lattice; integrability; bi-Hamiltonian structure.

pdf (198 kb)   ps (158 kb)   tex (13 kb)

References

  1. Bogoyavlensky O.I., On perturbations of the periodic Toda lattice, Comm. Math. Phys., 1976, V.51, 201-209.
  2. Constantinides K., Generalized full Kostant-Toda lattices, Master Thesis, Department of Mathematics and Statistics, University of Cyprus, 2003.
  3. Damianou P.A., Paschalis P., Sophocleous C., A tri-Hamiltonian formulation of the full Kostant-Toda lattice, Lett. Math. Phys., 1995, V.34, 17-24.
  4. Damianou P.A., Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices, Reviews Math. Phys., 2004, V.16, 175-241.
  5. Deift P.A., Li L.C., Nanda T., Tomei C., The Toda lattice on a generic orbit is integrable, Comm. Pure Appl. Math., 1986, V.39, 183-232.
  6. Ercolani N.M., Flaschka H., Singer S., The Geometry of the full Kostant-Toda lattice, in Colloque Verdier, Progress in Mathematics Series, Birkhaeuser Verlag, 1994, 181-225.
  7. Falqui G., Magri F., Pedroni M., Bi-Hamiltonian geometry, Darboux coverings, and linearization of the KP hierarchy, Comm. Math. Phys., 1998, V.197, 303-324.
  8. Faybusovich L., Gekhtman M., Elementary Toda orbits and integrable lattices, J. Math. Phys., 2000, V.41, 2905-2921.
  9. Flaschka H., On the Toda lattice, Phys. Rev., 1974, V.9, 1924-1925.
  10. Flaschka H., On the Toda lattice II. Inverse-scattering solution, Progr. Theor. Phys., 1974, V.51, 703-716.
  11. Flaschka H., Integrable systems and torus actions, Lecture Notes, University of Arizona, 1992.
  12. Fokas A.S., Fuchssteiner B., The Hierarchy of the Benjamin-Ono equations, Phys. Lett. A, 1981, V.86, 341-345.
  13. Fuchssteiner B., Master symmetries and higher order time-dependent symmetries and conserved densities of nonlinear evolution equations, Progr. Theor. Phys., 1983, V.70, 1508-1522.
  14. Gelfand I.M., Zakharevich I., On the local geometry of a bi-Hamiltonian structure, in The Gelfand Mathematical Seminars 1990-1992, Editors L. Corvin et al., Boston, Birkhauser, 1993, 51-112.
  15. Henon M., Integrals of the Toda lattice, Phys. Rev. B, 1974, V.9, 1921-1923.
  16. Kostant B., The solution to a generalized Toda lattice and representation theory, Adv. Math., V.34, 195-338.
  17. Kupershmidt B., Discrete Lax equations and differential-difference calculus, Asterisque, 1985, V.123, 1-212.
  18. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys., 1978, V.19, 1156-1162.
  19. Manakov S., Complete integrability and stochastization of discrete dynamical systems, Zh. Exp. Teor. Fiz., 1974, V.67, 543-555.
  20. Meucci A., Compatible Lie algebroids and the periodic Toda lattice, J. Geom. Phys., 2000, V.35, 273-287.
  21. Meucci A., Toda equations, bi-Hamiltonian systems, and compatible Lie algebroids, Math. Phys. Analysis Geom., 2001, V.4, 131-146.
  22. Moser J., Finitely many mass points on the line under the influence of an exponential potential - an integrable system, Lect. Notes Phys., 1976, V.38, 97-101.
  23. Moser J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., 1975, V.16, 197-220.
  24. Singer S., The geometry of the full Toda lattice, Ph.D. Dissertation, Courant Institute, 1991.
  25. Toda M., One-dimensional dual transformation, J. Phys. Soc. Japan, 1967, V.22, 431-436.

Previous article   Next article   Contents of Volume 1 (2005)