Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. I. Auxiliary System and Fundamental Properties

TL;DR

This paper explores the pseudo-Hermitian reduction of the GMV spectral problem, establishing integrable hierarchies, recursion operators, and eigenfunction expansions, advancing understanding of generalized Heisenberg ferromagnet equations.

Contribution

It introduces the pseudo-Hermitian reduction for the GMV system, constructs recursion operators, and develops eigenfunction expansions, providing new tools for analyzing integrable systems.

Findings

Established integrable hierarchies for GMV systems

Constructed recursion operators for the systems

Derived eigenfunction expansions for arbitrary potentials

Abstract

We consider an auxiliary spectral problem originally introduced by Gerdjikov, Mikhailov and Valchev (GMV system) and its modification called pseudo-Hermitian reduction which is extensively studied here for the first time. We describe the integrable hierarchies of both systems in a parallel way and construct recursion operators. Using the concept of gauge equivalence, we construct expansions over the eigenfunctions of recursion operators. This permits us to obtain the expansions for both GMV systems with arbitrary constant asymptotic values of the potential functions in the auxiliary linear problems.