Modified scattering for the one-dimensional Schr\"odinger equation with a dissipative nonlinearity for arbitrary large initial data

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrodinger equation with a dissipative nonlinearity $\lambda |u|^\alpha u$, where $0 \frac{\alpha |\text{Re} \lambda |}{2\sqrt{ \alpha +1}}$. For arbitrary large initial data, we present the time decay estimates when $4/3<\alpha \le2$, and the large time asymptotics of the solution when $\frac{7+\sqrt{145}}{12}<\alpha \le2$. The proof is based on the vector fields method and on a semiclassical analysis method.