On a class of exact solutions of the Ishimori equation

TL;DR

This paper discovers a new class of exact solutions for the Ishimori equation, a 2D Heisenberg-type model relevant in ferromagnet theory, by linking it to Toda-type lattices and dressing chains.

Contribution

It introduces an innovative method to generate solutions of the Ishimori equation using integrable reductions of Toda-type lattices and dressing chains.

Findings

Identified a class of particular solutions for the Ishimori equation.

Linked the Ishimori equation to a Toda-type lattice as a dressing chain.

Developed new solutions through integrable reductions of the dressing chain.

Abstract

In this paper, a class of particular solutions of the Ishimori equation is found. This equation is known as the spatially two-dimensional version of the Heisenberg equation, which has important applications in the theory of ferromagnets. It is shown that the two-dimensional Toda-type lattice found earlier by Ferapontov, Shabat and Yamilov is a dressing chain for this equation. Using the integrable reductions of the dressing chain, the authors found an essentially new class of solutions to the Ishimori equation.