Stabilization against collapse of 2D attractive Bose-Einstein condensates with repulsive, three-body interactions

TL;DR

This paper rigorously derives a cubic-quintic nonlinear Schrödinger equation for a 2D Bose gas with attractive two-body and repulsive three-body interactions, analyzing the system's behavior near the collapse threshold and in the homogeneous case.

Contribution

It provides a rigorous derivation of the mean-field limit leading to a cubic-quintic NLS for 2D Bose gases with mixed interactions, including analysis near collapse and in the absence of trapping.

Findings

Derivation of the cubic-quintic nonlinear Schrödinger equation as the mean-field limit.

Analysis of the system's behavior as the attractive interaction approaches the critical value.

Investigation of the homogeneous case without trapping potential.

Abstract

We consider a trapped Bose gas of $N$ identical bosons in two dimensional space with both an attractive, two-body, scaled interaction and a repulsive, three-body, scaled interaction respectively of the form $-aN^{2\alpha-1} U(N^\alpha \cdot)$ and $bN^{4\beta-2} W(N^\beta \cdot, N^\beta \cdot))$, where $a,b,\alpha,\beta>0$ and $\int_{\mathbb R^2}U(x) {\mathop{}\mathrm{d}} x = 1 = \iint_{\mathbb R^{4}} W(x,y) {\mathop{}\mathrm{d}} x {\mathop{}\mathrm{d}} y$. We derive rigorously the cubic--quintic nonlinear Schr\"odinger semiclassical theory as the mean-field limit of the model and we investigate the behavior of the system in the double-limit $a = a_N \to a_*$ and $b = b_N \searrow 0$. Moreover, we also consider the homogeneous problem where the trapping potential is removed.