Blowup profile of rotating 2D focusing Bose gases
Abstract
We consider the GrossPitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation $\Omega$. First we study the behavior of the ground state when the coupling constant approaches $a\_*$ , the critical strength of the cubic nonlinearity for the focusing nonlinear Schr{ö}dinger equation. We prove that blowup always happens at the center of the trap, with the blowup profile given by the GagliardoNirenberg solution. In particular, the blowup scenario is independent of $\Omega$, to leading order. This generalizes results obtained by Guo and Seiringer (Lett. Math. Phys., 2014, vol. 104, p. 141156) in the nonrotating case. In a second part we consider the manyparticle Hamiltonian for $N$ bosons, interacting with a potential rescaled in the meanfield manner $a\_N N^{2\beta1} w(N^{\beta} x), with $w$ a positive function such that $\int\_{\mathbb{R}^2} w(x) dx = 1$. Assuming that $\beta < 1/2$ and that $a\_N \to a\_*$ sufficiently slowly, we prove that the manybody system is fully condensed on the GrossPitaevskii ground state in the limit $N \to \infty$.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.01854
 Bibcode:
 2018arXiv180201854L
 Keywords:

 Mathematics  Analysis of PDEs;
 Condensed Matter  Quantum Gases;
 Mathematical Physics