ALMOST CUBIC NONLINEAR SCHRODINGER EQUATION: EXISTENCE, UNIQUENESS AND SCATTERING

We give a unified treatment for a class of nonlinear Schrodinger (NLS) equations with non-local nonlinearities in two space dimensions. This class includes the Davey-Stewartson (DS) equations when the second equation is elliptic and the Generalized Davey-Stewartson (GDS) system when the second and the third equations form an elliptic system. We establish local well-posedness of the Cauchy problem in L-2(R-2), H-1(R-2), H-2(R-2) and in Sigma = H-1(R-2) boolean AND L-2(|x|(2) dx). We show that the maximal interval of existence of solutions in all of these spaces coincides. Then we show that the mass is conserved for L-2(R-2)-solutions. Similarly, the energy and the momenta are conserved for the solutions in H-1(R-2). For the solutions in Sigma, we show that the virial identity and the pseudo-conformal conservation hold. We then discuss the global existence and the scattering of solutions when t he underlying Schrodinger equation is of elliptic type. We achieve these results in either of the following three cases: when the initial data is with small enough mass, when an initial data is with subminimal mass and for any initial data in Sigma in the defocusing case. In the focusing case, we show that when the initial energy of the solution in Sigma is negative then this solution blows-up in finite time. We distinguish the focusing and the defocusing cases sharply in terms of a condition on the nonlinearity.