Self-similar blow-up profiles for slightly supercritical nonlinear Schrödinger equations

Abstract

We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u+|u{|}^{p-1}u=0$ on ${\mathbf{R}}^d$, close to the mass critical case and for any space dimension $d\ge 1$. These profiles bifurcate from the ground state solitary wave. The argument relies on the classical matched asymptotics method suggested in [Sulem, C.; Sulem, P.-L., The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999] which needs to be applied in a degenerate case due to the presence of exponentially small terms in the bifurcation equation related to the log-log blow-up law observed in the mass critical case.