Darboux coordinates and Liouville-Arnold integration in loop algebras

• DOI: 10.1007/bf02097398
• Publication Date: 2005-10-29T00:50:37Z
• Authors: M. R. Adams; J. Harnad; J. Hurtubise
• Keywords: Linearization; Hamiltonian (control theory); Matrix (chemical analysis); Matrix representation; Monodromy matrix

61 pgs<br/>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 Schr��dinger equation. For $\frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schr��dinger equation.<br/>