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

TL;DR

This paper proves that solutions to certain scalar Riemann-Hilbert problems depend smoothly on parameters, and applies this to analyze the semiclassical focusing nonlinear Schrödinger equation.

Contribution

It establishes the smooth parametric dependence of Riemann-Hilbert problem solutions with singularities, with applications to semiclassical focusing NLS.

Findings

Solutions are uniquely defined and smooth in parameters near a known solution.

The results apply to RHPs with $z ext{log}z$ singularities on the contour.

Application to semiclassical focusing NLS demonstrates practical relevance.

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 $z\log z$ 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.