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On a class of exact solutions of the Ishimori equation

Rustem N. Garifullin, Ismagil T. Habibullin

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.

On a class of exact solutions of the Ishimori equation

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.

Paper Structure

This paper contains 8 sections, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Coupled systems associated with lattice \ref{['fsy']}
  3. What is the coupled system associated with a lattice?
  4. A coupled system symmetric with respect to $x$ and $y$
  5. On the relation between the Ishimori equation \ref{['ishim']} and the lattice \ref{['fsy']}
  6. Exact particular solutions of the dressing chain
  7. Solutions to the Ishimiri equation
  8. Numerical simulation

Figures (1)

  • Figure 1: $S^{1}$ for $t=0$ and for $t=10$