Modified scattering for the one-dimensional Schr\"odinger equation with a subcritical dissipative nonlinearity

Abstract

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schr\"odinger 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.