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The inverse spectral transform for the Zakharov-Shabat equation on thesemi-line is reconsidered as a Hilbert problem. The boundary data induce anessential singularity at large k to one of the basic solutions. Then solvingthe inverse problem means solving a Hilbert problem with particular prescribedbehavior. It is demonstrated that the direct and inverse problems are solved ina consistent way as soon as the spectral transform vanishes with 1/k atinfinity in the whole upper half plane (where it may possess single poles) andis continuous and bounded on the real k-axis. The method is applied tostimulated Raman scattering and sine-Gordon (light cone) for which it isdemonstrated that time evolution conserves the properties of the spectraltransform.Comment: LaTex file, 1 figure, submitted to J. Phys.

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