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

2011/08/25 by Sergey Belov, Belov, Sergey, Stephanos Venakides +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.1108.5197

27 pages, 6 figures

arxiv created 2011/08/25 · arxiv updated 2011/08/29

Abstract

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 zlog 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 β. We prove that if the RHP has a solution at some value β₀ then the solution of the RHP is uniquely defined in a some neighborhood of β₀ and is smooth in β. This result is applied to the case of semiclassical focusing NLS.

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