Toda-Heisenberg chain: interacting sigma-fields in two dimensions

TL;DR

This paper investigates a two-dimensional Toda-Heisenberg chain model, reducing it to an integrable system related to known equations, and derives dark-soliton solutions demonstrating its integrability and solution structure.

Contribution

It introduces a novel (2+1)-dimensional model of coupled sigma-fields and establishes its connection to integrable systems, providing explicit dark-soliton solutions.

Findings

Model reduces to an integrable system related to Toda and Ablowitz-Ladik equations

Derivation of explicit dark-soliton solutions

Establishment of integrability in a higher-dimensional sigma-field system

Abstract

We study a (2+1)-dimensional system that can be viewed as an infinite number of O(3) sigma-fields coupled by a nearest-neighbour Heisenberg-like interaction. We reduce the field equations of this model to an integrable system that is closely related to the two-dimensional relativistic Toda chain and the Ablowitz-Ladik equations. Using this reduction we obtain the dark-soliton solutions of our model.