Abstract
It is shown how to construct infinitely many conserved quantities for the classical non-linear Schrödinger equation associated with an arbitrary Hermitian symmetric spaceG/K. These quantities are non-local in general, but include a series of local quantities as a special case. Their Poisson bracket algebra is studied, and is found to be a realization of the “half” Kac-Moody algebrak R ⊗ ℂ [λ], consisting of polynomials in positive powers of a complex parameter λ which have coefficients in the compact real form ofk (the Lie algebra ofK).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fordy, A.P., Kulish, P.P.: Nonlinear Schrödinger equations and simple Lie algebras. Commun. Math. Phys.89, 427–443 (1983)
Faddeev, L.D. In: Recent advances in field theory and statistical mechanics, pp. 563–608. Les Houches, Session 39. North-Holland: Elsevier 1984
Olive, D., Turok, N.: Local conserved densities and zero curvature conditions for Toda lattice field theories. Nucl. Phys. B257 [FS14], 277 (1985)
Dolan, L.: Kac-Moody algebras and exact solvability in hadronic physics. Phys. Rep.109, No. 1, 1–94 (1984)
Goddard, P., Olive, D.: Kac-Moody and Virasoro algebras in relation to quantum physics (to appear)
Manakov, S.V.: On the theory of two dimensional stationary self-focusing of electromagnetic waves. Sov. Phys. JETP38, No. 2, 248–253 (1974)
Helgason, S.: Differential geometry, Lie groups and symmetric spaces, 2nd ed. New York: Academic Press 1978
Author information
Authors and Affiliations
Additional information
Communicated by K. Osterwalder
Rights and permissions
About this article
Cite this article
Crumey, A.D.W.B. Local and non-local conserved quantities for generalized non-linear Schrödinger equations. Commun.Math. Phys. 108, 631–646 (1987). https://doi.org/10.1007/BF01214421
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01214421