Paper detail

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 $\vecβ$. We prove that if the RHP has a solution at some value $\vecβ_0$ then the solution of the RHP is uniquely defined in a some neighborhood of $\vecβ_0$ and is smooth in $\vecβ$. This result is applied to the case of semiclassical focusing NLS.