Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau-Lifschitz equation

TL;DR

This paper explores the gauge equivalence between the Hirota and extended continuous Heisenberg equations, deriving nonlocal multi-soliton solutions and analyzing their properties, including a nonlocal Landau-Lifschitz equation.

Contribution

It introduces a method to construct nonlocal multi-soliton solutions for both equations using gauge and Darboux transformations, revealing inheritance of nonlocality properties.

Findings

Derived closed-form nonlocal multi-soliton solutions

Established equivalence between auto-gauge and Darboux transformations

Analyzed properties of nonlocal Landau-Lifschitz equation

Abstract

We exploit the gauge equivalence between the Hirota equation and the extended continuous Heisenberg equation to investigate how nonlocality properties of one system are inherited by the other. We provide closed generic expressions for nonlocal multi-soliton solutions for both systems. By demonstrating that a specific auto-gauge transformation for the extended continuous Heisenberg equation becomes equivalent to a Darboux transformation, we use the latter to construct the nonlocal multi-soliton solutions from which the corresponding nonlocal solutions to the Hirota equation can be computed directly. We discuss properties and solutions of a nonlocal version of the nonlocal extended Landau-Lifschitz equation obtained from the nonlocal extended continuous Heisenberg equation or directly from the nonlocal solutions of the Hirota equation.