Self-similar blowup for the cubic Schrödinger equation

Abstract

We give a rigorous proof for the existence of a finite-energy, self-similar solution to the focusing cubic Schrödinger equation in three spatial dimensions. The proof is computer-assisted and relies on a fixed point argument that shows the existence of a solution in the vicinity of a numerically constructed approximation. The latter is obtained by a standard pseudo-spectral method. The computer-assisted part of the rigorous proof uses nothing but fraction arithmetic in order to obtain quantitative bounds for the fixed point argument.

83 pages, significant improvements in presentation, Wolfram Mathematica notebooks included