Riemann-Hilbert factorizations and inverse scattering for the AKNS- equation with \(L^ 1\)-potentials. I (Q1322399)

Summary: 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.