Schrdinger Equations with nonlinearity of integral type

We consider the Cauchy problem for the nonlinear Schrodinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. Concerning the problem with finite initial time, we prove the global well-posedness in the largest space L2(!R2 ). Concerning the problem with infinite initial time, we prove the existence of modified wave operators on a dense set of small and sufficiently regular asymptotic states