Modified Scattering for the Hartree Nonlinear Schr\"odinger Equation

TL;DR

This paper establishes sharp decay and modified scattering results for the Hartree nonlinear Schrödinger equation in 2D and 3D, demonstrating scattering at lower regularity than previous studies using wavepacket testing methods.

Contribution

It introduces a novel application of wavepacket testing to prove sharp decay and scattering for the Hartree NLS at lower regularity levels.

Findings

Proves sharp $L^inity$ decay for Hartree NLS in 2D and 3D.

Establishes modified scattering behavior at reduced regularity.

Uses wavepacket testing method of Ifrim and Tataru.

Abstract

We prove sharp $L^\infty$ decay and modified scattering for the Hartree nonlinear Schr\"odinger equation in dimensions $2$ and $3$ using the testing by wavepackets method of Ifrim and Tataru. We show that the scattering behavior happens at a regularity well below that of earlier results of Hayashi-Naumkin and Kato-Pusateri.