Hermitian and Pseudo-Hermitian Reduction of the GMV Auxiliary System. Spectral Properties of the Recursion Operators

TL;DR

This paper investigates the spectral properties of recursion operators in the Zakharov-Shabat spectral problem, focusing on Hermitian and pseudo-Hermitian reductions, with implications for soliton equations related to GMV systems.

Contribution

It introduces a method to construct eigenfunction expansions considering different reductions and the discrete spectrum, enhancing understanding of spectral symmetries in integrable systems.

Findings

Constructed eigenfunction expansions for reduced spectral problems.

Analyzed the impact of symmetries on the expansions.

Connected spectral properties to soliton equations in GMV systems.

Abstract

We consider simultaneously two different reductions of a Zakharov-Shabat's spectral problem in pole gauge. Using the concept of gauge equivalence, we construct expansions over the eigenfunctions of the recursion operators related to the afore-mentioned spectral problem with arbitrary constant asymptotic values of the potential functions. In doing this, we take into account the discrete spectrum of the scattering operator. Having in mind the applications to the theory of the soliton equations associated to the GMV systems, we show how these expansions modify depending on the symmetries of the functions we expand.