Dynamics of solutions to the Gross-Pitaevskii equation describing dipolar Bose-Einstein condensates

TL;DR

This paper reviews recent findings on the long-term behavior of solutions to the Gross-Pitaevskii equation for dipolar Bose-Einstein condensates, focusing on different initial conditions and their asymptotic dynamics.

Contribution

It provides a comprehensive analysis of the asymptotic behaviors of solutions based on initial data relative to the Mass-Energy threshold, using decay properties of heat kernels and Riesz transforms.

Findings

Solutions exhibit distinct asymptotic behaviors depending on initial data.

Decay properties of heat kernels are crucial for understanding solution dynamics.

Properties of Riesz transforms are revisited to support analysis.

Abstract

We review some recent results on the long time dynamics of solutions to the Gross-Pitaevskii equation governing non-trapped dipolar Quantum Gases. We describe the asymptotic behaviours of solutions for different initial configurations of the initial datum in the energy space, specifically for data below, above, and at the Mass-Energy threshold. We revisit some properties of powers of the Riesz transforms by means of the decay properties of the heat kernel associated to the parabolic biharmonic equation. These decay properties play a fundamental tool to establish the dynamical features of the solutions to the equation.