Modified scattering for the one-dimensional Schrödinger equation with a subcritical dissipative nonlinearity

Abstract

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrödinger equation with a subcritical dissipative nonlinearity $\lambda |u{|}^{\alpha }u$, where $0<\alpha <2$, and $\lambda$ is a complex constant satisfying $\text{Im}\lambda >\frac{\alpha |\text{Re}\lambda |}{2\sqrt{\alpha +1}}$. For arbitrary large initial data, we present the uniform time decay estimates when $4/3<\alpha <2$, and the large time asymptotics of the solution when $\frac{7+\sqrt{145}}{12}<\alpha <2$. The proof is based on the vector fields method and a semiclassical analysis method.