On the Birkhoff factorization problem for the Heisenberg magnet and nonlinear Schroedinger equations

TL;DR

This paper provides a geometric framework for the Heisenberg magnet equation using Birkhoff factorization on an infinite-dimensional Lie group, linking it to the nonlinear Schrödinger equation through algebraic transformations and explicit multisoliton solutions.

Contribution

It introduces a geometric description of the Heisenberg magnet equation via Birkhoff factorization and establishes a connection to the nonlinear Schrödinger equation with explicit multisoliton solutions.

Findings

Heisenberg magnet flows are induced by an $R^2$ action on a quotient space.

The HM equation can be integrated through Birkhoff factorization.

Explicit multisoliton solutions are constructed using Baker functions.

Abstract

A geometrical description of the Heisenberg magnet (HM) equation with classical spins is given in terms of flows on the quotient space $G/H_+$ where $G$ is an infinite dimensional Lie group and $H_+$ is a subgroup of $G$. It is shown that the HM flows are induced by an action of $\mathbb{R}^2$ on $G/H_+$, and that the HM equation can be integrated by solving a Birkhoff factorization problem for $G$. For the HM flows which are Laurent polynomials in the spectral variable we derive an algebraic transformation between solutions of the nonlinear Schroedinger (NLS) and Heisenberg magnet equations. The Birkhoff factorization for $G$ is treated in terms of the geometry of the Segal-Wilson Grassmannian $Gr(H)$. The solution of the problem is given in terms of a pair of Baker functions for special subspaces of $Gr(H)$. The Baker functions are constructed explicitly for subspaces which yield multisoliton solutions of NLS and HM equations.