Two Extended Versions of the Continuous 2D-Heisenberg Model

TL;DR

This paper explores two extended versions of the (2+1)-dimensional Heisenberg model, deriving new exact solutions and analyzing their properties within the complex Hirota scheme, revealing diverse phenomenological behaviors.

Contribution

It introduces new exact solutions for the Ishimori and modified Ishimori models, expanding understanding of their integrability and solution space.

Findings

New static solutions related to Painlevé equations

Time-dependent solutions linked to Painlevé transcendent

Elliptic function solutions and double sine-Gordon type solutions

Abstract

We analyze two extended versions [the Ishimori model (IM) and a related system, which will be called modified Ishimori model (mIM)] of the continuous Heisenberg model in (2+1)-dimensions within the complex Hirota scheme. The IM, proposed in 1984, is an integrable (2+1)-dimensional topological spin field model which has been studied in many theoretical frameworks. The mIM has been introduced quite recently by some of the present authors [Phys. Rev. {\bf B 49}, 12915 (1994)]. Using the same stereographic variable in the Hirota formulation, we build up some new exact solutions both for the IM and the mIM in the compact and noncompact case. For the IM new configurations are a class of static solutions related to a special third Painlev\'e equation, time-dependent solutions linked to an other kind of the third Painlev\'e transcendent, and asympotic time-dependent solutions whose energy density behaves as Yukawa potential. For the mIM, new configurations are a class of exact solutions expressed in terms of elliptic functions, and a class of time-dependent solutions related to a particular form of the double sine-Gordon and the double sinh-Gordon equations with variable coefficients. We discuss the new configurations and certain known solutions which clarify the different possible phenomenological role played by the considered topological spin field models.