Spectral parameter power series for Zakharov-Shabat direct and inverse scattering problems

We study the direct and inverse scattering problems for the Zakharov-Shabat system. Representations for the Jost solutions are obtained in the form of the power series in terms of a transformed spectral parameter. In terms of that parameter, the Jost solutions are convergent power series in the unit disk. The coefficients of the series are computed following a simple recurrent integration procedure. This essentially reduces the solution of the direct scattering problem to the computation of the coefficients and location of zeros of an analytic function inside of the unit disk. The solution of the inverse scattering problem reduces to the solution of a system of linear algebraic equations for the power series coefficients, while the potential is recovered from the first coefficient. The overall approach leads to a simple and efficient method for the numerical solution of both direct and inverse scattering problems, which is illustrated by numerical examples.