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

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

2001-05-30

arxiv.org/abs/nlin/0105066v1

Nonlinear Sciences

Pattern Formation and Solitons

LaTex file, 1 figure, submitted to J. Phys. A

Scientific paper

10.1088/0305-4470/34/36/316

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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