The non-chiral intermediate Heisenberg ferromagnet equation

TL;DR

This paper introduces and solves the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation, a soliton equation describing coupled spin densities, and explores its relation to hyperbolic spin Calogero-Moser systems and integrability via a Lax pair.

Contribution

The paper presents the ncIHF equation, establishes its integrability through a Lax pair, and connects it to spin Calogero-Moser systems, including multi-soliton solutions and a periodic variant.

Findings

ncIHF reduces to decoupled half-wave maps as δ→∞

Multi-soliton solutions derived from a spin-pole ansatz

Established integrability via Lax pair construction

Abstract

We present and solve a soliton equation which we call the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation. This equation, which depends on a parameter $\delta >0$, describes the time evolution of two coupled spin densities propagating on the real line, and in the limit $\delta \to \infty$ it reduces to two decoupled half-wave maps (HWM) equations of opposite chirality. We show that the ncIHF equation is related to the A-type hyperbolic spin Calogero-Moser (CM) system in two distinct ways: (i) it is obtained as a particular continuum limit of a Inozemtsev-type spin chain related to this CM system, (ii) it has multi-soliton solutions obtained by a spin-pole ansatz and with parameters satisfying the equations of motion of a complexified version of this CM system. The integrability of the ncIHF equation is shown by constructing a Lax pair. We also propose a periodic variant of the ncIHF equation related to the A-type elliptic spin CM system.