Remarks on long range scattering for nonlinear Schrödinger equations with Stark effects

Abstract

In this paper, the global existence and asymptotic behavior in time of solutions for the nonlinear Schrödinger equation with the Stark effect in one or two space dimensions are studied. The nonlinearity is cubic and quadratic in one and two dimensional cases, respectively, and it is a summation of a gauge invariant term and non-gauge invariant terms. This nonlinearity is critical between the short range scattering and the long range one. A modified wave operator to this equation is constructed for small final states. Its domain is a certain small ball in $H^{2} \cap \mathcal{F}H^{2}$, where $\mathcal{F}$ is the Fourier transform.