Self-similar blowup for the cubic Schr\"odinger equation

TL;DR

This paper rigorously proves the existence of a finite-energy, self-similar solution to the focusing cubic Schrödinger equation in three dimensions, using a computer-assisted fixed point method based on spectral approximations.

Contribution

It introduces a computer-assisted fixed point approach to establish self-similar solutions for the cubic Schrödinger equation, combining numerical and rigorous analysis.

Findings

Existence of a self-similar solution confirmed

Quantitative bounds obtained via computer-assisted proof

Solution constructed near a spectral approximation

Abstract

We give a rigorous proof for the existence of a finite-energy, self-similar solution to the focusing cubic Schr\"odinger 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.