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.