Scattering and inverse scattering for the AKNS system: A rational function approach

We consider the use of rational basis functions to compute the scattering and inverse scattering transforms associated with the AKNS (Ablowitz-Kaup-Newell-Segur) system. The proposed numerical forward scattering transform computes the solution of the AKNS system that is valid on the entire real axis and thereby computes a reflection coefficient at a point by solving a single linear system. The proposed numerical inverse scattering transform makes use of a novel improvement in the rational function approach to the oscillatory Cauchy operator, enabling the efficient solution of certain Riemann-Hilbert problems without contour deformations. The latter development enables access to high-precision computations and this is demonstrated on the inverse scattering transform for the one-dimensional Schrodinger operator with a sech2 potential.

Automated summary

The use of rational basis functions in computing the scattering and inverse scattering transforms of the AKNS system is discussed. The proposed numerical forward scattering transform computes the solution of the AKNS system on the entire real axis, while the numerical inverse scattering transform efficiently solves certain Riemann-Hilbert problems without contour deformations, allowing access to high-precision computations.