Perturbation of Riemann-Hilbert jump contours: smooth parametric dependence with application to semiclassical focusing NLS

Details Perturbation of Riemann-Hilbert jump contours: smooth parametric dependence with application to semiclassical focusing NLS Perturbation of Riemann-Hilbert jump contours: smooth parametric dependence with application to semiclassical focusing NLS

2011-08-25

arxiv.org/abs/1108.5197v1

Physics

Mathematical Physics

27 pages, 6 figures

Scientific paper

A perturbation of a class of scalar Riemann-Hilbert problems (RHPs) with the jump contour as a finite union of oriented simple arcs in the complex plane and the jump function with a $z\log z$ type singularity on the jump contour is considered. The jump function and the jump contour are assumed to depend on a vector of external parameters $\vec\beta$. We prove that if the RHP has a solution at some value $\vec\beta_0$ then the solution of the RHP is uniquely defined in a some neighborhood of $\vec\beta_0$ and is smooth in $\vec\beta$. This result is applied to the case of semiclassical focusing NLS.

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