Self-similar Solutions of the Cubic Wave Equation

TL;DR

This paper constructs and analyzes a family of self-similar solutions to the focusing cubic wave equation in three dimensions, revealing their instability and potential non-participation in typical evolutions.

Contribution

It proves the existence of a countable family of smooth self-similar solutions and analyzes their spectral stability properties.

Findings

Existence of countably many self-similar solutions.

All solutions with n>0 are linearly unstable.

Solutions with n>0 have singularities outside the past light cone.

Abstract

We prove that the focusing cubic wave equation in three spatial dimensions has a countable family of self-similar solutions which are smooth inside the past light cone of the singularity. These solutions are labeled by an integer index $n$ which counts the number of oscillations of the solution. The linearized operator around the $n$-th solution is shown to have $n+1$ negative eigenvalues (one of which corresponds to the gauge mode) which implies that all $n>0$ solutions are unstable. It is also shown that all $n>0$ solutions have a singularity outside the past light cone which casts doubt on whether these solutions may participate in the Cauchy evolution, even for non-generic initial data.