Inverse scattering problem on the half-axis for a first order system of ordinary differential equations

TL;DR

This paper addresses the inverse scattering problem for a first order differential system on the half-axis, proposing a method to recover matrix coefficients using Riemann-Hilbert factorization and integral operators.

Contribution

It introduces a novel approach to solve the ISP on the half-axis for systems with triangular matrix coefficients by reducing it to a whole-axis problem.

Findings

Solution via Riemann-Hilbert factorization under specific conditions

Reduction of half-axis ISP to whole-axis ISP with zero extension

Explicit construction of transformation operators for the system

Abstract

In this article, the inverse scattering problem (ISP) of recovering the matrix coefficient of a first order system of ordinary differential equations on the half-axis from its scattering matrix is considered. In the case of a triangular structure of the matrix coefficient, this system has a Volterra-type integral transformation operator at infinity. Such type of transformation operator allows to determine the scattering matrix on the half-axis via the matrix Riemann-Hilbert factorization in the case, where contour is real axis, normalization is canonical and all the partial indices are zero. The ISP on the half-axis is solved by reducing it to ISP on the whole axis for the considered system with the coefficients that are extended to the whole axis as zero.