Averaging of strong magnetic nonlinear Schr\"odinger equations in the energy space

TL;DR

This paper investigates two models of 3D nonlinear Schrödinger equations under strong magnetic fields, analyzing their solutions in the energy space, and establishes scattering results for the time-averaged model with small energy.

Contribution

It introduces and analyzes two derived models in the energy space, providing global solutions and improved long-time convergence results, including scattering for the nonic case.

Findings

Global solutions in the energy space for both models

Enhanced convergence results over long times

Scattering established for the energy-critical nonic case

Abstract

In this study, we consider two nonlinear Schr\"{o}dinger-type models that are derived by R L. Frank, F. M\'{e}hats, C. Sparber [arXiv:1611.01574] to study 3D nonlinear Schr\"{o}dinger equations under strong magnetic fields. One model is derived by spatial scaling and the other is obtained by averaging the spatial scaled model over time. We study these models in the energy space to obtain global solutions and improve the convergence result over an arbitrarily long time. Regarding the nonic nonlinear power of the time averaged model, we prove a scattering result under a scaling-invariant small-energy condition, which underlines energy-criticality of the nonic case.