Scattering of the energy-critical NLS with dipolar interaction

TL;DR

This paper establishes global well-posedness and scattering for a 3D energy-critical nonlinear Schrödinger equation with magnetic dipolar interactions, extending understanding of such equations with nonlocal dipole effects.

Contribution

It provides the first proof of global well-posedness and scattering for the energy-critical NLS with dipolar interactions using advanced induction techniques.

Findings

Proved global well-posedness for the dipolar NLS.

Derived conditions for scattering in the energy-critical setting.

Extended existing methods to nonlocal dipolar interactions.

Abstract

In this paper, we investigate the global well-posedness and $H^{1}$ scattering theory for a 3d energy-critical Schr\"odinger equation under the influence of magnetic dipole interaction $\lambda_{1}|u|^{2}u+\lambda_{2}(K\ast|u|^{2})u$, where $K$ is the dipole-dipole interaction kernel. Our proof of global well-posedness result is based on the argument of Zhang [23]. Moreover, adopting the induction of energy technique of Killip-Oh-Pocovnicu-Visan [20], we obtain a condition for scattering.