Stable self similar blow up dynamics for slightly L^2 supercritical NLS equations

Abstract

We consider the focusing nonlinear Schr\"odinger equations $i\partial_t u+\Delta u +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities p_c with $p_c=1+\frac{4}{N}$ and 0<\e\ll 1. We prove the existence and stability in the energy space $H^1$ of a self similar finite time blow up dynamics and provide a qualitative description of the singularity formation near the blow up tim