Half-Wave Maps: Explicit Formulas for Rational Functions with Simple Poles

TL;DR

This paper derives explicit formulas for the evolution of rational functions with simple poles under the Half-Wave maps equation, utilizing Lax pairs, half-spin formulations, and Toeplitz operators to describe pole dynamics.

Contribution

It introduces a new explicit formula for Half-Wave maps with rational functions, connecting Lax pairs, half-spin formulations, and Toeplitz operators for pole evolution.

Findings

Explicit formula for Half-Wave maps with rational functions

Description of pole evolution via Lax pair and half-spin formulation

Representation of the evolution using Toeplitz operators

Abstract

We establish an explicit formula for the Half-Wave maps equation for rational functions with simple poles. The Lax pair provides a description of the evolution of the poles. By considering a half-spin formulation, we use linear algebra to derive a time evolution equation followed by the half-spins, in the moving frame provided by the Lax pair. We then rewrite this formula using a Toeplitz operator and $G$, the adjoint of the operator of multiplication by $x$ on the Hardy space $L_+^2(\mathbb{R})$.