The generalised Zakharov–Shabat system and the gauge group action

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 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 in the example of the multi-component nonlinear Schrödinger equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic type models, related to \documentclass[12pt]{minimal}\begin{document}$so(5,{\mathbb C})$\end{document}so(5,C) algebra.