Inverse scattering  for the linear system associated with the coupled Gerdjikov--Ivanov equations

We consider a certain first-order linear system of ordinary differential equations, and we analyze the direct and inverse scattering problems for that linear system. The linear system involves two potentials in the Schwartz class, and those potentials linearly depend on the spectral parameter. This linear system is related to the integrable system of nonlinear partial differential equations known as the DNLS (derivative nonlinear Schr\"odinger) system III, which is also known as the Gerdjikov--Ivanov system. When analyzing the direct problem, we describe the pertinent properties of the Jost solutions and the scattering coefficients. The bound states poles and the associated normalization constants are represented via a matrix triplet pair, enabling us to deal with any number of bound states and any multiplicities. The inverse scattering problem comprises the determination of the two potentials when the reflection coefficients and the bound-state information are available. To solve the inverse problem, we establish a linear system of integral equation where the kernel and nonhomogeneous term are determined by the Fourier transforms of the reflection coefficients and the matrix triplet pair representing the bound-state information. This system of linear integral equations is the counterpart of the system of Marchenko integral equations available for the AKNS system associated with the integrable NLS (nonlinear Schr\"odinger) system. We recover the potentials from the solution of our established Marchenko integral system. When we use the time-evolved reflection coefficients and the time-evolved matrix triplets, the corresponding time-evolved potential pair yields a solution of the Gerdjikov--Ivanov system.