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

TL;DR

This paper investigates the long-term behavior of solutions to a one-dimensional nonlinear Schr"odinger equation with a dissipative nonlinearity, providing decay estimates and asymptotics for large initial data under specific conditions.

Contribution

It introduces new decay estimates and asymptotic descriptions for solutions with subcritical dissipative nonlinearities, using vector fields and semiclassical analysis techniques.

Findings

Established uniform decay estimates for solutions when 4/3<α<2.

Derived large time asymptotics for solutions when (7+√145)/12<α<2.

Analyzed the influence of complex dissipative nonlinearities on solution behavior.

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.