Renormalization and blow ups for the nonlinear Schr\"odinger equation

Abstract

Existence of finite-time blow ups in the classical one-dimensional nonlinear Schr\"odinger equation (NLS) (1) i \partial_t u + u_{x x} + |u|^{2r} u = 0, u(x,0) = u_0(x) has been one of the central problems in the studies of the singularity formation in the PDEs. We revisit this problem using an approach based on the ideas borrowed from Dynamical Systems. To that end, we reformulate the initial value problem for (1), with r \in \mathbb{N}, r \ge 1, as a fixed point problem for a certain renormalization operator, and use the ideas of apriori bounds to prove existence of a renormalization fixed point. Existence of such fixed points leads to existence of self-similar solutions of the form u(x,t) = (T-t)^{-{1 \over 2 r}} U((T-t)^{-{1 \over 2}} x), whose L^{2 r +2}-norms are bounded up-to a finite time T and whose energy blows up at T.