Examples of solution of the inverse scattering problem and the equations of the Novikov-Veselov hierarchy from the scattering data of point potentials

The inverse scattering problem is considered for the two- dimensional Schrödinger equation at fixed positive energy. The results include inverse scattering reconstructions from the simplest scattering amplitudes. In particular, a complete analytic solution is given of the phased and phaseless inverse scattering problems for single-point potentials of Bethe- Peierls- Fermi-Zeldovich- Berezin-Faddeev type. Numerical inverse scattering reconstructions from the simplest scattering amplitudes are then studied using the method of the Riemann-Hilbert- Manakov problem in soliton theory. Finally, these numerical inverse scattering results are used to construct corresponding numerical solutions of the non-linear equations of the Novikov-Veselov hierarchy at fixed positive energy. Bibliography: 21 titles.