Geometric Theory of the Recursion Operators for the Generalized Zakharov-Shabat System in Pole Gauge on the Algebra sl(n,C)

TL;DR

This paper explores the geometric structure of recursion operators for soliton equations linked to the generalized Zakharov-Shabat system on the algebra sl(n,C), highlighting their relation to Poisson-Nijenhuis structures.

Contribution

It introduces the recursion operators for the system and interprets their geometric meaning as conjugates to Nijenhuis tensors within a Poisson-Nijenhuis framework.

Findings

Recursion operators are explicitly constructed.

Geometric interpretation as conjugates to Nijenhuis tensors.

Analysis includes systems with and without reductions.

Abstract

We consider the recursion operator approach to the soliton equations related to the generalized Zakharov-Shabat system on the algebra sl(n,C) in pole gauge both in the general position and in the presence of reductions. We present the recursion operators and discuss their geometric meaning as conjugate to Nijenhuis tensors for a Poisson-Nijenhuis structure defined on the manifold of potentials.