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Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras | | M.R. Adams
; J. Harnad
; J. Hurtubise
; | | Date: |
16 Oct 1992 | | Journal: | Commun. Math. Phys. 155, 385-413 (1993) | | Subject: | hep-th | | Abstract: | Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part $wt{frak{g}}^+$ of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general construction is given for $frak{g}=frak{gl}(r)$ or $frak{sl}(r)$, with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. The case $frak{g=sl}(2)$ is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, as well as the quasi-periodic solutions of the cubically nonlinear Schrödinger equation. For $frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation. | | Source: | arXiv, hep-th/9210089 | | Services: | Forum | Review | PDF | Favorites | |
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