Existence and stability of standing waves for nonlinear Schr\"odinger equations with a critical rotational speed

TL;DR

This paper investigates the existence and stability of standing waves in a nonlinear Schrödinger equation with a critical rotational speed, relevant for modeling rotating Bose-Einstein condensates, extending previous results to various nonlinear regimes.

Contribution

It establishes the existence and orbital stability of prescribed mass standing waves for NLS with critical rotational speed across different nonlinearities, broadening prior work on supercritical cases.

Findings

Existence of standing waves for mass-subcritical, critical, and supercritical nonlinearities.

Orbital stability of these standing waves under the equation's dynamics.

Extension of previous results to a broader class of nonlinearities.

Abstract

We study the existence and stability of standing waves associated to the Cauchy problem for the nonlinear Schr\"odinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose-Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with mass-subcritical, mass-critical, and mass-supercritical nonlinearities. Our result extends a recent work of [Bellazzini-Boussa\"id-Jeanjean-Visciglia, Comm. Math. Phys. 353 (2017), no. 1, 229-251], where the existence and stability of standing waves for the supercritical NLS with a partial confinement were established.