Riemann–Hilbert Factorizations and Inverse Scattering for the AKNS-Equation with $L^1$-Potentials I

Abstract

A detailed group theoretic approach is developed for the $2\times 2$ AKNS system with potentials supported on the half-line. This approach uses consistently Riemann–Hilbert splittings interpreted as factorizations in certain Banach Lie groups. In particular, it is shown how meromorphic splittings introduced by Beals and Coifman can be replaced with regular Riemann–Hilbert splittings leading to the group theoretic notion of a scattering data.