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 $$4/3\le \alpha <2$$ , 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.