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

Abstract

Abstract 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.