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

This work studies the direct and inverse fixed energy scattering problem for two-dimensional Schroedinger equation with rather general nonlinear index of refraction. In particular, using the Born approximation we prove that all singularities of the unknown compactly supported potential from $L^2$-space 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 $L^\infty$-space.