Stabilization against collapse of 2D attractive Bose-Einstein condensates with repulsive, three-body interactions
Dinh-Thi Nguyen, Julien Ricaud
TL;DR
The work provides a rigorous derivation of the 2D cubic–quintic NLSNLS theory as the mean-field limit of a trapped Bose gas with competing attractive two-body and repulsive three-body interactions, and analyzes stability and collapse in both inhomogeneous and homogeneous settings. It establishes existence, blow-up profiles, and energy scalings in the NLSNLS framework, and proves convergence of Hartree and full many-body ground states to NLSNLS minimizers using quantum de Finetti techniques. In the collapse regime, the blow-up profile is shown to coincide with the cubic NLS ground state $Q_0$ under appropriate scaling, and the many-body energy and reduced density matrices converge accordingly. Overall, the paper bridges microscopic many-body dynamics with a cubic–quintic mean-field description, detailing when stabilization by three-body repulsion prevents collapse and when the system exhibits controlled blow-up, with explicit scaling laws and convergence results that are relevant for 2D BEC experiments and mathematical physics.
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 $-aN^{2α-1} U(N^α\cdot)$ and $bN^{4β-2} W(N^β\cdot, N^β\cdot))$, where $a,b,α,β>0$ 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 Schrödinger 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.