Mass-Energy threshold dynamics for dipolar Quantum Gases

TL;DR

This paper analyzes the long-term behavior of solutions to a dipolar Bose-Einstein condensate model, establishing conditions for scattering and blow-up based on mass-energy thresholds, advancing understanding of quantum gas dynamics.

Contribution

It introduces a scattering criterion for the Gross-Pitaevskii equation in dipolar gases and characterizes solution dynamics at the mass-energy threshold, which was previously less understood.

Findings

Established a scattering criterion for solutions above the threshold

Proved blow-up/grow-up conditions for general data

Analyzed dynamics at the mass-energy threshold

Abstract

We consider a Gross-Pitaevskii equation which appears as a model in the description of dipolar Bose-Einstein condensates, without a confining external trapping potential. We describe the asymptotic dynamics of solutions to the corresponding Cauchy problem in the energy space in different configurations with respect to the mass-energy threshold, namely for initial data above and at the mass-energy threshold. We first establish a scattering criterion for the equation that we prove by means of the concentration/compactness and rigidity scheme. This criterion enables us to show the energy scattering for solutions with data above the mass-energy threshold, for which only blow-up was known. We also prove a blow-up/grow-up criterion for the equation with general data in the energy space. As a byproduct of scattering and blow-up criteria, and the compactness of minimizing sequences for the Gagliardo-Nirenberg's inequality, we study long time dynamics of solutions with data lying exactly at the mass-energy threshold.