Modified Scattering for Nonlocal Nonlinear Schr\"odinger Equations

TL;DR

This paper establishes modified scattering and optimal decay rates for the Hartree and Schr"odinger-Bopp-Podolsky equations in 2D and 3D, improving regularity requirements compared to previous studies.

Contribution

It introduces a new approach using wavepacket testing to prove modified scattering at lower regularity levels for these equations.

Findings

Proves modified scattering for Hartree and Schr"odinger-Bopp-Podolsky equations.

Achieves sharp $L^ abla$ decay results in lower regularity settings.

Extends minimal regularity results to power-type scattering-critical NLS.

Abstract

We prove a modified scattering and sharp $L^\infty$ decay result for both the Hartree and Schr\"odinger-Bopp-Podolsky equations in dimensions $2$ and $3$ using the testing by wavepackets approach due to Ifrim and Tataru. We show that modified scattering and sharp pointwise decay occur for these equations at a regularity much lower than previous results due to Hayashi-Naumkin and Kato-Pusateri, and as a corollary also show that the results on power-type scattering-critical NLS due to Hayashi-Naumkin can be proven under minimal regularity assumptions.