The Zakharov-Shabat spectral problem on the semi-line: Hilbert formulation and applications

Abstract

The inverse spectral transform for the Zakharov-Shabat equation on the semi-line x>0 is reconsidered as a Hilbert problem. The boundary data induce an essential singularity as k→∞ to one of the basic solutions. Then solving the inverse problem means solving a Hilbert problem with particular prescribed behaviour. It is demonstrated that the direct and inverse problems are solved in a consistent way as soon as the spectral transform vanishes as (1/k) at infinity in the whole upper half-plane (where it may possess single poles) and is continuous and bounded on the real k axis. The method is applied to stimulated Raman scattering and sine-Gordon (light cone) for which it is demonstrated that time evolution conserves the properties of the spectral transform.