Darboux coordinates and Liouville-Arnold integration in loop algebras

Abstract

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part $$\tilde{\mathfrak{g}}^+$$ 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 Liouvile-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. As illustrative examples, the caseg =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. Forg =sl(3), the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schrödinger equation, a case which requires further symplectic constraints in order to deal with singularities in the spectral data at ∞.