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

J. Leon·A. Spire
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Abstract

The inverse spectral transform for the Zakharov-Shabat equation on the semi-line is reconsidered as a Hilbert problem. The boundary data induce an essential singularity at large k to one of the basic solutions. Then solving the inverse problem means solving a Hilbert problem with particular prescribed behavior. It is demonstrated that the direct and inverse problems are solved in a consistent way as soon as the spectral transform vanishes with 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.

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J. Leon, A. Spire. 2001-05-30. The Zakharov-Shabat spectral problem on the semi-line: Hilbert formulation and applications. https://doi.org/10.1088/0305-4470%2F34%2F36%2F316

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