The instability of Bourgain-Wang solutions for the L² critical NLS

We consider the two dimensional $L^2$ critical nonlinear Schr\"odinger equation $i\pa_tu+\Delta u+u|u|^2=0$. In the pioneering work \cite{BW}, Bourgain and Wang have constructed smooth solutions which blow up in finite time $T<+\infty$ with the pseudo conformal speed $$\|\nabla u(t)\|_{L^2}\sim \frac{1}{T-t},$$ and which display some decoupling between the regular and the singular part of the solution at blow up time. We prove that this dynamic is unstable. More precisely, we show that any such solution with small super critical $L^2$ mass lies on the boundary of both $H^1$ open sets of global solutions that scatter forward and backwards in time, and solutions that blow up in finite time on the right in the log-log regime exhibited in \cite{MR1}, \cite{MR4}, \cite{R1}. We moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in \cite{Mcpam} in the critical mass case