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