Matrix Nonlinear Schr\"odinger Equations and Moment Maps into Loop Algebras${}^\dag$

TL;DR

This paper explores how Darboux coordinates and moment map embeddings can parametrize the phase space of finite gap solutions in matrix nonlinear Schrödinger equations, linking symplectic geometry with loop algebra structures.

Contribution

It introduces a novel parametrization of the phase space using moment maps and Darboux coordinates, connecting finite gap solutions to loop algebra embeddings.

Findings

Phase space parametrized via moment maps into loop algebra duals.

Identification of phase space with rational elements in loop algebra.

Explicit reduced coordinates for classical Hermitian symmetric Lie algebras.

Abstract

It is shown how Darboux coordinates on a reduced symplectic vector space may be used to parametrize the phase space on which the finite gap solutions of matrix nonlinear Schr\"odinger equations are realized as isospectral Hamiltonian flows. The parametrization follows from a moment map embedding of the symplectic vector space, reduced by suitable group actions, into the dual $\tilde\grg^{+*}$ of the algebra $\tilde\grg^+$ of positive frequency loops in a Lie algebra $\grg$. The resulting phase space is identified with a Poisson subspace of $\tilde\grg^{+*}$ consisting of elements that are rational in the loop parameter. Reduced coordinates associated to the various Hermitian symmetric Lie algebras $(\grg,\grk)$ corresponding to the classical Lie algebras are obtained.