Approximate direct and inverse scattering for the AKNS system

We study the direct and inverse scattering problems for the AKNS (Ablowitz-Kaup-Newell-Segur) system. New 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 corresponding unit disks. For the coefficients of the series simple recurrent integration procedures are devised. Solution of the direct scattering problem reduces to computing the coefficients and locating zeros of corresponding analytic functions in the interior of the unit disk. Solution of the inverse scattering problem reduces to the solution of two systems of linear algebraic equations for the power series coefficients, while the potentials are recovered from the first coefficients. 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.