The 3D nonlinear Schr\"odinger equation with a constant magnetic field revisited

TL;DR

This paper analyzes the 3D nonlinear Schrödinger equation with a magnetic field, establishing conditions for solutions' global existence or blow-up, and exploring the stability and instability of standing waves.

Contribution

It introduces a new approach to prove the existence of normalized solitary waves without using concentration-compactness, extending results to critical and supercritical cases.

Findings

Derived sharp thresholds for global existence versus blow-up.

Proved existence and orbital stability of normalized standing waves.

Established strong instability of ground state standing waves.

Abstract

In this paper, we revisit the Cauchy problem for the three dimensional nonlinear Schr\"odinger equation with a constant magnetic field. We first establish sufficient conditions that ensure the existence of global in time and finite time blow-up solutions. In particular, we derive sharp thresholds for global existence versus blow-up for the equation with mass-critical and mass-supercritical nonlinearities. We next prove the existence and orbital stability of normalized standing waves which extend the previous known results to the mass-critical and mass-supercritical cases. To show the existence of normalized solitary waves, we present a new approach that avoids the celebrated concentration-compactness principle. Finally, we study the existence and strong instability of ground state standing waves which greatly improve the previous literature.