  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.
