Fast and slow blowup in the S2 σ-model and the (4+1)-dimensional Yang-Mills model

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We study singularity formation in spherically symmetric solutions ofthe charge-one and charge-two sector of the (2+1)-dimensional S2 σ-model and the (4+1)-dimensional Yang-Mills model, near the adiabatic limit. These equationsare nonintegrable, and so studies are performed numerically onrotationally symmetric solutions using an iterative finite differencingscheme that is numerically stable. We evaluate the accuracy of predictions made with thegeodesic approximation. We find that the geodesic approximation is extremelyaccurate for the charge-two σ-model and the Yang-Mills model, bothof which exhibit fast blowup to within experimental error. The charge-one σ-model exhibits slowblowup. There the geodesic approximation must be modified by applyingan infrared cutoff that depends on initial conditions.