The Generalised Zakharov-Shabat System and the Gauge Group Action

TL;DR

This paper explores the generalized Zakharov-Shabat systems with complex Cartan elements, analyzing their analytical solutions, scattering data, and Hamiltonian hierarchies, with applications to multi-component nonlinear Schrödinger and Heisenberg ferromagnetic models.

Contribution

It introduces a comprehensive analysis of the gauge-equivalent CBC systems, including their solutions, scattering data, and integrable hierarchies, extending the understanding of multi-component integrable models.

Findings

Properties of fundamental analytical solutions are characterized.

A minimal set of scattering data is identified.

Hierarchies of Hamiltonian structures are constructed.

Abstract

The generalized Zakharov-Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schrodinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type models, related to so(5;C) algebra.