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 Schroding er 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 ampli tude limit the main contribution to scattering comes from the potentia l V and since moreover, the scattering state has high velocity the cla ssical 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 potenti al V.