Focusing dynamics of 2D Bose gases in the instability regime

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 $\beta \in (0,3\mathrm{/}2)$. The interaction may be chosen to be negative and large, leading to the instability regime where the corresponding focusing cubic nonlinear Schr{ö}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.