Inverse scattering for the nonlinear Schrodinger equation

In this paper I present a method to uniquely reconstruct a potential V(x) from the scattering operator associated to the nonlinear Schrodinger equation i partial derivative/partial derivative tu + (H-0 + V)u + f(\u\)u/\u\ = 0, and the corresponding unperturbed equation i partial derivative/partial derivative t + H(0)u = 0, where H-0 = -Delta/2m, m > 0. I uniquely reconstruct the potential V by considering scattering states that have small amplitude and high velocity. In the small amplitude limit the main contribution to scattering comes from the potential V and since moreover, the scattering state has high velocity the classical translation dominates the solution and the quantum spreading is a lower order term. These two effects lead to a simplification of the scattering process that allows me to uniquely reconstruct the potential V.