Integrable (2+1)-Dimensional Spin Models with Self-Consistent Potentials

TL;DR

This paper introduces three new integrable (2+1)-dimensional spin models with self-consistent potentials, explores their reductions, and connects them to nonlinear Schr"odinger equations, expanding the understanding of integrable systems in higher dimensions.

Contribution

It identifies novel integrable (2+1)D spin systems with self-consistent potentials and derives their Lax pairs and reductions, linking them to nonlinear Schr"odinger equations.

Findings

Three new integrable (2+1)D spin models introduced.

Derived Lax pairs for the new models.

Connected models to nonlinear Schr"odinger family equations.

Abstract

Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schr\"odinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than one scalar potential, or vector potential, or both. We also obtain the associated Lax pairs. We discuss various interesting reductions in (2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear Schr\"odinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schr\"odinger--Hirota--Maxwell--Bloch equations, along with their Lax pairs.