Fredholm Determinant Solution for the Inverse Scattering Transform of the N x N Zakharov-Shabat Equation

Abstract

We give an explicit solution of the Riemann-Hilbert problem with pole singularity for the NxN Zakharov-Shabat equation under the assumption of the existence and the uniqueness of the solution. For a general class of the scattering data the solution of the Marchenko integral equation which was derived in our previous article is exhibited in a quite simple form in terms of the Fredholm deter. minant and its first minors of the integral kernel. It is shown that the potential and the Jost solution derived in this way surely compose the Zakharov-Shabat equation.