A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrodinger equation

This work deals with the inverse scattering problem for a two-dimensional Schrodinger equation -Delta u + qu + alpha vertical bar u vertical bar(2)u/1 + r vertical bar u vertical bar(2) = k(2)u, with a saturation-like nonlinearity, where the real-valued unknown functions q and alpha belong to L(loc)(p)(R(2)) with certain special behaviour at infinity. We prove Saito's formula which implies a uniqueness result and a representation formula for a certain combination of the functions q and alpha in the sense of tempered distributions. What is more, we prove that the leading order singularities of this combination can be obtained exactly by the inverse Born approximation method from general scattering data at arbitrarily large energies.