Riemann-Hilbert factorization of matrices invariant under inversion in a circle

TL;DR

This paper studies the Riemann-Hilbert factorization of matrices invariant under circle inversion, showing that positivity conditions lead to unique solutions and have applications in inverse scattering for nonlinear Schrödinger equations.

Contribution

It establishes conditions under which matrices with inversion invariance have only zero partial indices, ensuring unique Riemann-Hilbert factorizations and solvability of related boundary value problems.

Findings

Matrices with inversion invariance and positivity have only zero partial indices.

Unique solvability of certain Riemann-Hilbert boundary value problems is proven.

Applications include inverse scattering for integrable nonlinear Schrödinger equations.

Abstract

We consider matrix functions with certain invariance under inversion in the unit circle. If such a function satisfies a positivity assumption on the unit circle, then only zero partial indices appear in its Riemann-Hilbert (Wiener-Hopf) factorization. It implies the unique solvability of a certain class of Riemann-Hilbert boundary value problems. It includes the ones associated with the inverse scattering transform of the focusing/defocusing integrable discrete nonlinear Schr\"odinger equations.