Abstract
We define Poisson structures which lead to a tri-Hamiltonian formulation for the full Kostant-Toda lattice. In addition, a hierarchy of vector fields called master symmetries are constructed and they are used to generate the nonlinear Poisson brackets and other invariants. Various deformation relations are investigated. The results are analogous to results for the finite nonperiodic Toda lattice.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Damianou, P.: Nonlinear Poisson brackets, PhD dissertation, University of Arizona (1989).
Damianou, P.: Master symmetries and R-matrices for the Toda Lattice,Lett. Math. Phys. 20, 101–112 (1990).
Damianou, P.:B n Toda systems,Lie Groups Appl. 1, 71–78 (1994).
Das, A. and Okubo, S.: A systematic study of the Toda lattice,Ann. Phys. 190, 215–232 (1989).
Deift, P. A., Li, L. C., Nanda, T. and Tomei, C.: The Toda lattice on a generic orbit is integrable,Comm. Pure Appl. Math. 39, 183–232 (1986).
Ercolani, N. M., Flaschka, H. and Singer, S.: The geometry of the full Kostant-Toda lattice,Colloque Verdier, Progress in Math. series, Birkhaeuser Verlag, Basel, 1994, pp. 181–225.
Fernandes, R. L.: On the Mastersymmetries and bi-Hamiltonian structure of the Toda lattice,J. Phys. A. 26, 3797–3803 (1993).
Flaschka, H.: On the Toda lattice,Phys. Rev. 9, 1924–1925 (1974).
Flaschka, H.: Integrable systems and torus actions, Lecture notes, University of Arizona (1992).
Fokas, A. S. and Fuchssteiner, B.: The hierarchy of the Benjamin-Ono equations,Phys. Lett. 86A, 341–345 (1981).
Fuchssteiner, B.: Mastersymmetries and higher order time-dependent symmetries and conserved densities of nonlinear evolution equations.Progr. Theor. Phys. 70, 1508–1522 (1983).
Kostant, B.: The solution to a generalized Toda lattice and representation theory,Adv. Math. 34, 195–338 (1979).
Li, L. C. and Parmentier, S.: Nonlinear Poisson structures and r-matrices,Comm. Math. Phys. 125, 545–563 (1989).
Magri, F.: A simple model of the integrable Hamiltonian equation,J. Math. Phys. 19, 1156–1162 (1978).
Oevel, W.:Topics in Soliton Theory and Exactly Solvable Non-linear Equations, World Scientific, Singapore, 1987.
Oevel, W. and Ragnisco O.: R-matrices and higher Poisson brackets for integrable systems,Physica A,161, 181–220 (1989).
Olver, P. J.: Evolution equations possessing infinitely many symmetrices,J. Math. Phys. 18, 1212–1215 (1977).
Ratiu, T.: Involution theorems, inLecture Notes in Math. 775, Springer-Verlag, New York, 219–257, (1980).
Singer, S.: The geometry of the full Toda lattice, PhD dissertation, Courant Institute, 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Damianou, P.A., Paschalis, P. & Sophocleous, C. A tri-Hamiltonian formulation of the full Kostant-Toda lattice. Lett Math Phys 34, 17–24 (1995). https://doi.org/10.1007/BF00739371
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00739371