Decay of solutions to one-dimensional inhomogeneous nonlinear Schr\"odinger equations

TL;DR

This paper establishes decay estimates for global solutions of one-dimensional inhomogeneous nonlinear Schr"odinger equations, using localized Virial-Morawetz identities, and explores implications for energy scattering and solutions with external potentials.

Contribution

It introduces new decay estimates in one dimension via localized Virial-Morawetz identities, extending results to equations with external potentials and odd solutions.

Findings

Decay estimates for solutions in $L^r$-norm as time approaches infinity.

Decay results for odd solutions under certain conditions.

Extension of decay results to equations with inverse power and Yukawa-type potentials.

Abstract

We investigate the decay estimates of global solutions for a class of one-dimensional inhomogeneous nonlinear Schr\"odinger equations. While most existing results focus on spatial dimensions $d\geq2$, the decay properties in one dimension remain less explored due to the absence of effective Morawetz inequalities. For equations without external potential, by establishing a localized Virial-Morawetz identity, we derive decay estimates in the context of the $L^r$-norm for global solutions within a compact domain as a time subsequence approaches infinity. This decay result can be applied to obtain a criterion for energy scattering. Additionally, by establishing another type of Virial-Morawetz identity under more strict conditions, we demonstrate the decay result for odd solutions for any time sequence that approaches infinity. Utilizing some results about bound states proved by Barry Simon, we also show that similar decay results hold for the global odd solutions of equations with suitable external potentials that contain inverse power type and Yukawa-type potentials.