Focusing dynamics of 2D Bose gases in the instability regime

TL;DR

This paper studies the dynamics of 2D Bose gases with focusing interactions, demonstrating that the many-body quantum evolution aligns with the nonlinear Schrödinger equation before blow-up and validating the Bogoliubov approximation.

Contribution

It provides a rigorous derivation of the focusing 2D Bose gas dynamics and confirms the Bogoliubov approximation's validity in the instability regime.

Findings

Quantum dynamics approximated by NLS before blow-up

Validation of Bogoliubov approximation in focusing regime

Effective description of excitations from condensate

Abstract

We consider the dynamics of a 2D Bose gas with an interaction potential of the form $N^{2\beta-1}w(N^\beta\cdot)$ for $\beta\in (0,3/2)$. The interaction may be chosen to be negative and large, leading to the instability regime where the corresponding focusing cubic nonlinear Schr{\"o}dinger equation (NLS) may blow up in finite time. We show that to leading order, the $N$-body quantum dynamics can be effectively described by the NLS prior to the blow-up time. Moreover, we prove the validity of the Bogoliubov approximation, where the excitations from the condensate are captured in a norm approximation of the many-body dynamics.