Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions

Abstract

We study scattering theory for nonlinear Schrödinger equations with cubic and quadratic nonlinearities in one and two space dimensions, respectively. For example, the nonlinearities are the sum of the gauge-invariant term and non-gauge-invariant terms such as $\lambda_0 \!|u|^2u +\lambda_1 u^3 +\lambda_2 u\bar{u}^2 +\lambda_3 \bar{u}^3$ in the one-dimensional case, where $\lambda_0 \in {\mathbb R}$ and $\lambda_1,\lambda_2,\lambda_3$ $ \in {\mathbb C}$. The scattering theory for these equations belongs to the long-range case. We show the existence and uniqueness of global solutions for those equations which approach a given modified free profile. The same problem for the nonlinear Schrödinger equation with the Stark potentials is also considered.