Multi-solitons of the half-wave maps equation and Calogero-Moser spin-pole dynamics

TL;DR

This paper derives exact multi-soliton solutions for the half-wave maps equation, linking their dynamics to a solvable spin Calogero-Moser system, and extends these results to periodic cases with visualizations.

Contribution

It introduces a novel ansatz for multi-soliton solutions of the HWM equation and connects their evolution to a spin Calogero-Moser system with new constraints.

Findings

Exact multi-soliton solutions for HWM equation derived.

Soliton interactions described by spin Calogero-Moser dynamics.

Generalization to periodic HWM and visualization provided.

Abstract

We consider the half-wave maps (HWM) equation which provides a continuum description of the classical Haldane-Shastry spin chain on the real line. We present exact multi-soliton solutions of this equation. Our solutions describe solitary spin excitations that can move with different velocities and interact in a non-trivial way. We make an ansatz for the solution allowing for an arbitrary number of solitons, each described by a pole in the complex plane and a complex spin variable, and we show that the HWM equation is satisfied if these poles and spins evolve according to the dynamics of an exactly solvable spin Calogero-Moser (CM) system with certain constraints on initial conditions. We also find first order equations providing a B\"acklund transformation of this spin CM system, generalize our results to the periodic HWM equation, and provide plots that visualize our soliton solutions.