Slow Blow Up in the (2+1)-dimensional S^2 Sigma Model

Abstract

We study singularity formation in spherically symmetric solitons of the charge one sector of the (2+1) dimensional S^2 sigma model, also known as $\IC P^1$ wave maps, in the adiabatic limit. These equations are non-integrable, and so studies are performed numerically on radially symmetric solutions using an iterative finite differencing scheme. Analytic estimates are made by using an effective Lagrangian cutoff outside a ball of fixed radius. We show the geodesic approximation is valid when the cutoff is applied, with the cutoff approaching infinity linearly as the reciprocal of the initial velocity. Additionally a characterization of the shape of a time slice f(r,T) with T fixed is provided.