Abstract
We associate a system of integrable, generalised nonlinear Schrödinger (NLS) equations with each Hermitian symmetric space. These NLS equations are considered as reductions of more general systems, this time associated with a reductive homogeneous space. The nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space. The Hamiltonian structure is investigated using “r-matrix” techniques and shown to be “canonical” for all these equations. Throughout the reduction procedure this Hamiltonian structure does not degenerate. Each of the above systems of equations is gauge equivalent to a generalised ferromagnet. Reductions of the latter are discussed in terms of the corresponding NLS type equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bogoyavlensky, O.I.: On perturbations of the periodic Toda lattice. Commun. Math. Phys.51, 201–209 (1976)
Mikhailov, A.V.: Integrability of a two-dimensional generalization of the Toda chain. JETP Letts.30, 414–418 (1979)
Fordy, A.P., Gibbons, J.: Integrable nonlinear Klein-Gordon equations and Toda lattices. Commun. Math. Phys.77, 21–30 (1980)
Mikhailov, A.V., Olshanetsky, M.A., Perelomov, A.M.: Two-dimensional generalized Toda lattice. Commun. Math. Phys.79, 473–488 (1981)
Fordy, A.P., Gibbons, J.: Nonlinear Klein Gordon equations and simple Lie algebras. In proceedings of the Dublin Workshop on Nonlinear Evolution Equations, May 1980. Proc. R. Irish Acad.83A, 33–45 (1983)
Kulish, P.P., Sklyamin, E.K.:O(N)-Invariant nonlinear Schrödinger equation — a new completely integrable system. Phys. Lett.84A, 349–352 (1981)
Kulish, P.P., Sklyamin, E.K.: On solutions of the Yang-Baxter equation. Zap. Nauch. Semin. LOMI95, 129 (1980)
Kulish, P.P., Sklyamin, E.K.: Quantum spectral transform method. Recent developments. In: Lecture Notes in Physics, Vol. 151, pp. 61–119. Berlin, Heidelberg, New York: Springer 1982
Zakharov, V.E., Takhtadzhyan, L.A.: Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet. Theor. Math. Phys.38, 17 (1979)
Helgason, S.: Differential geometry, Lie groups and symmetric spaces, 2nd ed. New York: Academic Press 1978
Humphreys, J.E.: Introduction to Lie algebras and representation theory. New York: Springer 1972
Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vol. II. New York: Intersceince, John Wiley 1969
Kulish, P.P.: Quantum difference nonlinear Schrödinger equation. Lett. Math. Phys.5, 191–197 (1981)
Kulish, P.P.: On action-angle variable for the multicomponent nonlinear Schrödinger equation. Zap. Nauch. Semin. LOMI115, 126–136 (1982) (in Russian)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
On leave from the Steklov Mathematical Institute, Leningrad, USSR
Rights and permissions
About this article
Cite this article
Fordy, A.P., Kulish, P.P. Nonlinear Schrödinger equations and simple Lie algebras. Commun.Math. Phys. 89, 427–443 (1983). https://doi.org/10.1007/BF01214664
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01214664