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A perturbation of a class of scalar Riemann-Hilbert problems (RHPs) with thejump contour as a finite union of oriented simple arcs in the complex plane andthe jump function with a $z\log z$ type singularity on the jump contour isconsidered. The jump function and the jump contour are assumed to depend on avector of external parameters $\vec\beta$. We prove that if the RHP has asolution at some value $\vec\beta_0$ then the solution of the RHP is uniquelydefined 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.

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