Dispersion and collapse of wave maps

Details Dispersion and collapse of wave maps Dispersion and collapse of wave maps

1999-12-13

arxiv.org/abs/math-ph/9912009v2

Nonlinearity 13 (2000) 1411-1423

Physics

Mathematical Physics

14 pages, Latex, 9 eps figures included, typos corrected

Scientific paper

10.1088/0951-7715/13/4/323

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.

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