Effective Generation of Closed-form Soliton Solutions of the Continuous Classical Heisenberg Ferromagnet Equation

TL;DR

This paper develops a rigorous inverse scattering transform for the classical Heisenberg ferromagnet equation, enabling explicit generation and classification of non-topological soliton solutions, including breathers and multipoles.

Contribution

It introduces a new triangular representation for Jost solutions and a general multi-soliton formula using the matrix triplet method, advancing soliton solution theory.

Findings

Explicit multi-soliton solutions including breathers and multipoles

Improved asymptotic analysis of scattering data

A comprehensive classification of soliton solutions

Abstract

The non-topological, stationary and propagating, soliton solutions of the classical continuous Heisenberg ferromagnet equation are investigated. A general, rigorous formulation of the Inverse Scattering Transform for this equation is presented, under less restrictive conditions than the Schwartz class hypotheses and naturally incorporating the non-topological character of the solutions. Such formulation is based on a new triangular representation for the Jost solutions, which in turn allows an immediate computation of the asymptotic behaviour of the scattering data for large values of the spectral parameter, consistently improving on the existing theory. A new, general, explicit multi-soliton solution formula, amenable to computer algebra, is obtained by means of the matrix triplet method, producing all the soliton solutions (including breather-like and multipoles), and allowing their classification and description.