Quasi-Periodic Solutions for matrix nonlinear Schroedinger Equations

TL;DR

This paper develops a method to construct quasi-periodic solutions for matrix nonlinear Schrödinger equations using algebraic geometric techniques related to integrable systems and loop algebras.

Contribution

It introduces a novel approach linking the Adler-Kostant-Symes theorem with finite-gap solutions expressed via theta functions for matrix NLS equations.

Findings

Finite-gap solutions expressed through quotients of theta functions.

Application of Lie algebraic methods to matrix nonlinear Schrödinger equations.

Connection between integrable flows and algebraic geometric solutions.

Abstract

The Adler-Kostant-Symes theorem yields isospectral hamiltonian flows on the dual $\tilde\grg^{+*}$ of a Lie subalgebra $\tilde\grg^+$ of a loop algebra $\tilde\grg$. A general approach relating the method of integration of Krichever, Novikov and Dubrovin to such flows is used to obtain finite-gap solutions of matrix Nonlinear Schr\"odinger Equations in terms of quotients of $\tet$-functions.