Scattering of Rational Solutions to the Half-Wave Maps Equation

TL;DR

This paper analyzes the scattering behavior of rational solutions to the Half-Wave Maps equation, characterizing conditions for scattering, providing explicit formulas, and constructing global solutions with prescribed spectra.

Contribution

It introduces a characterization of scattering behavior, proves that solutions with non-singular spectrum scatter, and constructs solutions with spectra close to any target spectrum.

Findings

Solutions with non-singular spectrum scatter in Sobolev norm.

The scattering map is shown to be the identity.

Explicit formulas for scattering solutions are provided.

Abstract

This article studies the rational solutions of the Half-Wave Maps equation (HWM) in the non-singular spectrum case. We first provide characterizations to what we call \emph{scattering behavior}, and show that they imply scattering in Sobolev norm. We then provide a local condition implying \emph{scattering behavior}. Building on this, we show that any solution with non-singular spectrum scatters and give an explicit formula for the function to which the solution is scattering. This allows us to show that the scattering map is the identity. Additionally, we create, for any given number of spins and any target non-singular spectrum, global solutions of (HWM) with a spectrum arbitrarily close to the target. Finally, using a diagonal characterization of traveling waves, we show that if a wave scatters to a traveling wave, it is a scattering wave.