Dispersion and collapse of wave maps

TL;DR

This paper investigates the formation of singularities in wave maps from 3+1 Minkowski spacetime into the 3-sphere, proposing universal behavior near singularities and the role of self-similar solutions in threshold phenomena.

Contribution

It formulates two conjectures about the universal approach to singularities and the threshold of formation based on numerical and stability analysis of self-similar solutions.

Findings

Singularities are approached in a universal manner by stable self-similar profiles.

The stable manifold of a self-similar solution determines the threshold for singularity formation.

Numerical evidence supports the conjectures about critical behavior in wave maps.

Abstract

We study numerically the Cauchy problem for equivariant wave maps from 3+1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar solutions we formulate two conjectures. The first conjecture states that singularities which are produced in the evolution of sufficiently large initial data are approached in a universal manner given by the profile of a stable self-similar solution. The second conjecture states that the codimension-one stable manifold of a self-similar solution with exactly one instability determines the threshold of singularity formation for a large class of initial data. Our results can be considered as a toy-model for some aspects of the critical behavior in formation of black holes.