Inverse fixed energy scattering problem for the two-dimensional nonlinear Schrödinger operator

Abstract

This work studies the direct and inverse fixed energy scattering problem for the two-dimensional Schrödinger equation with a rather limited nonlinear index of refraction. In particular, using the Born approximation, we prove that all singularities of the unknown compactly supported potential belonging in can be obtained uniquely by the scattering data with fixed positive energy. The proof is based on the new estimates for the Faddeev Green’s function in . The main achievement here is the computation of the Born approximation, which is carried out using the total variation regularization method. Numerical examples with noisy data are given to illustrate the effectiveness of the method.