Global dispersive estimates for defocusing nonlinear Schrodinger equations

TL;DR

This paper establishes global dispersive estimates for the defocusing nonlinear Schrödinger equation under semi-classical scaling, enhancing understanding of scattering and semi-classical behaviors.

Contribution

It introduces a novel approach to obtain uniform bounds and dispersive estimates for the equation, with implications for scattering theory and semi-classical analysis.

Findings

Proved global dispersive estimates in semi-classical scaling.

Established uniform bounds on solution modulus.

Discussed implications for scattering and semi-classical analysis.

Abstract

We consider the defocusing nonlinear Schr{\"o}dinger equation with a gauge invariant power-like nonlinearity. We prove global dispersive estimates in a semi-classical scaling, after rescaling the solution thanks to a suitable distorsion of the natural dispersion rate. We show a uniform bound on the modulus of the solution in some space providing compactness properties. We discuss the consequences of these estimates in the light of both scattering theory and semi-classical analysis.