Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part 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. The results are formulated in terms of sheaves to allow for singularities due to a degenerate spectrum. 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, reduced by the group of constant loops. A similar construction involving a framing at infinity is given for the nonreduced orbits. The general construction is given for g = gl(r) or sl(r), with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. As illustrative examples, the case g = sl(2), together with its real forms, is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, with the Liouville generating function expressed in hyperellipsoidal coordinates, as well as the quasi-periodic solutions of the cubically nonlinear Schrödinger equation. For g = sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation. This case requires a further symplectic constraining procedure in order to deal with singularities in the spectral data at ∞.

No abstract

No abstract

No abstract

No abstract