Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras

TL;DR

This paper constructs Darboux coordinates on rational coadjoint orbits of loop algebras, providing a Liouville-Arnold integration framework that linearizes Hamiltonian flows and connects spectral invariants with classical integrable systems.

Contribution

It introduces a new method for Liouville-Arnold integration on loop algebra orbits using spectral parameters and Serre duality, generalizing classical integrable system techniques.

Findings

Darboux coordinates are explicitly constructed on rational coadjoint orbits.

Liouville generating functions are obtained in separated form, enabling linearization of flows.

The approach reproduces classical solutions for sl(2) and extends to quasi-periodic solutions of nonlinear Schrödinger equations.

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 Schr\"odinger equation. For $\frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schr\"odinger equation.