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

We study singularity formation in spherically symmetric solutions of the 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 equations are nonintegrable, and so studies are performed numerically on rotationally symmetric solutions using an iterative finite differencing scheme that is numerically stable. We evaluate the accuracy of predictions made with the geodesic approximation. We find that the geodesic approximation is extremely accurate for the charge-two σ-model and the Yang-Mills model, both of which exhibit fast blowup to within experimental error. The charge-one σ-model exhibits slow blowup. There the geodesic approximation must be modified by applying an infrared cutoff that depends on initial conditions.