Self-similar solutions of the cubic wave equation

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 labelled by an integer index n which counts the number of oscillations of the solution. The linearized operator around the nth 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.