Dispersion and collapse of wave maps

Abstract

We study numerically the Cauchy problem for equivariantwave maps from 3 + 1 Minkowski spacetime into the 3-sphere. Onthe basis of numerical evidence combined with stability analysis of self-similar solutionswe formulate two conjectures. The first conjecture states thatsingularities which are produced in the evolution of sufficientlylarge initial data are approached in a universal manner given by theprofile of a stable self-similar solution. The second conjecturestates that the codimension-one stable manifold of a self-similarsolution with exactly one instability determines the threshold ofsingularity formation for a large class of initial data. Our resultscan be considered as a toy-model for some aspects of the criticalbehaviour in the formation of black holes.